Kleinert · 제20장 부록·심화

$$ \frac{\dot{S}(t)}{S(t)}=r_{S}+\eta(t) $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x)=\alpha e^{-2 a} \int_{-\infty}^{\infty} \frac{d y}{2 \pi} e^{i \alpha y x} e^{a\left[(1-i y)^{\lambda}+(1+i y)^{\lambda}\right]} $$

$$ P\left(x t \mid x_{a} t_{a}\right)=e^{-\Delta x / 2} \int_{-\infty}^{+\infty} \frac{d p}{2 \pi} e^{i p \Delta x-\bar{H}(p, \Delta t)} $$

$$ \langle\eta(t)\rangle=0, \quad\left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle=\sigma^{2} \delta\left(t-t^{\prime}\right) . $$

$$ \begin{align*} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x) & =\alpha e^{-2 a} \int_{-\infty}^{\infty} \frac{d y}{2 \pi} e^{i y \alpha x} \sum_{n=0}^{\infty} \frac{a^{n}}{n!}\left[(1-i y)^{\lambda}+(1+i y)^{\lambda}\right]^{n}= \\ & =\alpha e^{-2 a} \int_{-\infty}^{\infty} \frac{d y}{2 \pi} e^{i \alpha x y} \sum_{n=0}^{\infty} \frac{a^{n}}{n!} \sum_{m=0}^{n}\binom{n}{m}(1-i y)^{\lambda(n-m)}(1+i y)^{\lambda m} \end{align*} $$

$$ \bar{H}(p, \Delta t)=-\frac{\gamma^{2} \bar{v} t}{\kappa^{2}}+\frac{2 \gamma \bar{v}}{\kappa^{2}} \ln \left[\cosh \frac{\Omega t}{2}+\frac{\Omega^{2}+\gamma^{2}}{2 \gamma \Omega} \sinh \frac{\Omega t}{2}\right] $$

$$ x(t) \equiv \log S(t) $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x)=\alpha e^{-2 a} \sum_{n=0}^{\infty} \frac{a^{n}}{n!} \sum_{m=0}^{n}\binom{n}{m} \int_{-\infty}^{\infty} \frac{d y}{2 \pi} e^{i \alpha x y}(1-i y)^{\lambda(n-m)}(1+i y)^{\lambda m} $$

$$ \Omega=\sqrt{\gamma^{2}+\kappa^{2}\left(p^{2}+1 / 4\right)} $$

$$ \dot{x}(t)=\frac{\dot{S}}{S}-\frac{1}{2} \sigma^{2}=r_{x}+\eta(t) $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x)=\alpha e^{-2 a} \sum_{n=0}^{\infty} \frac{a^{n}}{n!} \sum_{m=0}^{n}\binom{n}{m} \frac{(\alpha x)^{-1-\lambda n / 2} 2^{\lambda n / 2}}{\Gamma(-\lambda m)} W_{\lambda n / 2-\lambda m,(\lambda n+1) / 2}(2 \alpha x) $$

$$ P\left(x t \mid x_{a} t_{a}\right) \approx e^{-\Delta x / 2}\left[\mu_{0}-\frac{1}{2} \mu_{2}(\Delta x)^{2}\right] \approx \mu_{0} e^{-\Delta x / 2} e^{-\mu_{2}(\Delta x)^{2} / 2 \mu_{0}} $$

$$ r_{x} \equiv r_{S}-\frac{1}{2} \sigma^{2} $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x)=-\frac{\alpha}{\pi} e^{-2 a} \sum_{n=1}^{\infty} a^{n} \sum_{m=0}^{n} \frac{2^{\lambda n / 2} \Gamma(1+\lambda m) \sin (\pi \lambda m)}{(\alpha x)^{1+\lambda n / 2} m!(n-m)!} W_{\lambda n / 2-\lambda m,(\lambda n+1) / 2}(2 \alpha x) $$

$$ \mu_{0}(\Delta t)=\int_{-\infty}^{+\infty} \frac{d p}{2 \pi} e^{-\bar{H}(p, \Delta t)}, \quad \mu_{2}(\Delta t)=\int_{-\infty}^{+\infty} \frac{d p}{2 \pi} p^{2} e^{-\bar{H}(p, \Delta t)} $$

$$ \begin{align*} d x(t) & =\frac{d x}{d S} d S(t)+\frac{1}{2} \frac{d^{2} x}{d S^{2}} d S^{2}(t)+\ldots \\ & =\frac{\dot{S}(t)}{S(t)} d t-\frac{1}{2}\left[\frac{\dot{S}(t)}{S(t)}\right]^{2} d t^{2}+\ldots \end{align*} $$

$$ W_{\lambda, \gamma}(x)=e^{-x / 2} x^{\lambda}\left\{1+\sum_{k=1}^{\infty} \frac{1}{k!x^{k}} \prod_{j=1}^{k}\left[\gamma^{2}-(\lambda-j+1 / 2)^{2}\right]\right\} $$

$$ f(\Delta t)=\int_{-\infty}^{+\infty} d \Delta x \mu_{0} e^{-\Delta x / 2-\mu_{2} \Delta x^{2} / 2 \mu_{0}}=\sqrt{\frac{2 \pi \mu_{0}^{3}}{\mu_{2}}} e^{\mu_{0} / 8 \mu_{2}} $$

$$ \left[\frac{\dot{S}(t)}{S(t)}\right]^{2} d t \rightarrow \dot{x}^{2}(t) d t=\sigma^{2} $$

$$ \begin{align*} \prod_{j=1}^{k}\left[\gamma^{2}-(\lambda-j+1 / 2)^{2}\right] & =\prod_{j=1}^{k}\left\{\left(\frac{\lambda n+1}{2}\right)^{2}-\left[\frac{\lambda(n-2 m)}{2}-j+\frac{1}{2}\right]^{2}\right\} \\ & =\prod_{j=1}^{k}(\lambda m+j)(\lambda n+1-\lambda m-j) \end{align*} $$

$$ \langle S(t)\rangle=S(0) e^{r_{S} t}=S(0)\left\langle e^{r_{x} t+\int_{0}^{t} d t^{\prime} \eta\left(t^{\prime}\right)}\right\rangle=S(0) e^{\left(r_{x}+\sigma^{2} / 2\right) t} $$

$$ \begin{gather*} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x)=-\frac{1}{\pi} e^{-2 a} \frac{e^{-\alpha x}}{x} \sum_{n=1}^{\infty} a^{n} 2^{\lambda n} \sum_{m=1}^{n} \frac{\Gamma(1+\lambda m) \sin (\pi \lambda m)}{m!(n-m)!}(2 \alpha x)^{-\lambda m} \\ \times\left[1+\sum_{k=1}^{\infty} \frac{\prod_{j=1}^{k}(\lambda m+j)(\lambda n+1-\lambda m-j)}{k!(2 \alpha x)^{k}}\right] \end{gather*} $$

$$ \tilde{L}_{\sigma^{2}}^{\lambda}(z) \equiv \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z} L_{\sigma^{2}}^{\lambda}(p) $$

$$ \sum_{k=0}^{\infty} \frac{1}{k!(2 \alpha x)^{k}} \prod_{j=1}^{k}(j+\lambda m)(1-\lambda m-j+\lambda n) $$

$$ L_{\sigma^{2}}^{\lambda}(p) \equiv \exp \left[-\left(\sigma^{2} p^{2}\right)^{\lambda / 2} / 2\right] $$

$$ \begin{align*} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x)= & -\frac{1}{\pi} e^{-2 a} \frac{e^{-\alpha x}}{x} \sum_{m=1}^{\infty} \frac{\Gamma(1+\lambda m) \sin (\pi \lambda m)}{m!(2 \alpha x)^{\lambda m}} \sum_{n=m}^{\infty} \frac{a^{n} 2^{\lambda n}}{(n-m)!} \\ & \times \sum_{k=0}^{\infty} \frac{1}{k!(2 \alpha x)^{k}} \prod_{j=1}^{k}(j+\lambda m)(1-\lambda m-j+\lambda n) \end{align*} $$

$$ \tilde{D}(z)=\int \frac{d p}{2 \pi} e^{i p z} D(p) $$

$$ \begin{gather*} =-\frac{1}{\pi} e^{-2 a} \frac{e^{-\alpha x}}{x} \sum_{k=0}^{\infty} \frac{1}{k!(2 \alpha x)^{k}} \sum_{m=1}^{\infty} \frac{\Gamma(1+\lambda m) \sin (\pi \lambda m)\left(2^{\lambda} a\right)^{m}}{(2 \alpha x)^{\lambda m} m!} \\ \times \prod_{j^{\prime}=1}^{k}\left(j^{\prime}+\lambda m\right) \sum_{n=0}^{\infty} \frac{\left(2^{\lambda} a\right)^{n}}{n!} \prod_{j=1}^{k}(\lambda n+1-j) \end{gather*} $$

$$ D(p) \equiv e^{-H(p)}, $$

$$ f^{(k)}\left(y^{\lambda}\right) \equiv \frac{d^{k}}{d y^{k}} e^{y^{\lambda}} $$

$$ \tilde{D}(z)=e^{-\tilde{H}(z)} $$

$$ f^{(k)}\left(y^{\lambda}\right)=\frac{d^{k}}{d y^{k}} \sum_{n=0}^{\infty} \frac{y^{\lambda n}}{n!}=\sum_{n=0}^{\infty} \frac{1}{n!} \prod_{j=1}^{k}(\lambda n-j+1) y^{\lambda n-k}, $$

$$ e^{-H(p)} \equiv\left\langle e^{-i p z}\right\rangle $$

$$ \begin{align*} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x) & =\frac{1}{\pi} e^{-2 a} \frac{e^{-\alpha x}}{x} \sum_{k=0}^{\infty} \frac{a^{k / \lambda} 2^{k} f^{(k)}\left(2^{\lambda} a\right)}{k!(2 \alpha x)^{k}} \\ & \times \sum_{m=1}^{\infty} \frac{\Gamma(1+\lambda m) \sin (\pi \lambda m)\left(2^{\lambda} a\right)^{m}}{m!}(2 \alpha x)^{-\lambda m} \prod_{j=1}^{k}(\lambda m+j) \end{align*} $$

$$ H(p)=\frac{1}{2}\left(\sigma^{2} p^{2}\right)^{\lambda / 2} $$

$$ \begin{align*} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x)=-\frac{e^{-2 a}}{\pi} e^{-\alpha x} & \sum_{k=0}^{\infty} \frac{(-s)^{k / \lambda} f^{(k)}\left(-s(2 \alpha)^{\lambda}\right)}{k!} \\ & \times \sum_{m=1}^{\infty} \frac{\Gamma(1+\lambda m+k) \sin (\pi \lambda m)(-s)^{m}}{m!x^{1+\lambda m+k}} \end{align*} $$

$$ \tilde{L}_{\sigma^{2}}^{\lambda}(z) \rightarrow A_{\sigma^{2}}^{\lambda} \frac{\lambda}{|z|^{1+\lambda}} $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(x)=-\frac{e^{-2 a}}{\pi} \frac{e^{-\alpha x}}{x} \sum_{k=0}^{\infty} A_{k} \sum_{m=1}^{\infty} B_{k m} \frac{(-s)^{m}}{x^{\lambda m+k}}, $$

$$ \tilde{L}_{\sigma^{2}}^{\lambda}(z) \approx \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z}\left[1-\frac{1}{2}\left(\sigma^{2} p^{2}\right)^{\lambda / 2}\right] \underset{z \rightarrow \infty}{\rightarrow} A_{\sigma^{2}}^{\lambda} \frac{\lambda}{|z|^{1+\lambda}} $$

$$ A_{k}=\frac{(-s)^{k / \lambda} f^{(k)}\left(-s(2 \alpha)^{\lambda}\right)}{k!}, \quad B_{k m}=\frac{\Gamma(1+\lambda m+k) \sin (\pi \lambda m)}{m!} . $$

$$ A_{\sigma^{2}}^{\lambda}=-\frac{\sigma^{\lambda}}{2 \lambda} \int_{0}^{\infty} \frac{d p^{\prime}}{\pi} p^{\prime \lambda} \cos p^{\prime}=\frac{\sigma^{\lambda}}{2 \pi \lambda} \sin (\pi \lambda / 2) \Gamma(1+\lambda) $$

$$ \begin{align*} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)(0)}(x) & =-\frac{e^{-2 a}}{\pi} e^{-\alpha x} f^{(0)}\left(-s(2 \alpha)^{\lambda}\right) \frac{\Gamma(1+\lambda) \sin (\pi \lambda)(-s)}{x^{1+\lambda}} \\ & =s \frac{e^{\alpha^{\lambda} s\left(2-2^{\lambda}\right)}}{\pi} \frac{\Gamma(1+\lambda) \sin (\pi \lambda)}{x^{1+\lambda}} e^{-\alpha x} \end{align*} $$

$$ A_{\sigma^{2}}^{3 / 2}=\frac{1}{4} \frac{\sigma^{3 / 2}}{\sqrt{2 \pi}} $$

$$ \tilde{L}_{\sigma^{2}}^{\lambda}(z)=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} \int_{0}^{\infty} \frac{d p}{\pi} \frac{\sigma^{\lambda n} p^{\lambda n}}{2^{n}} \cos p z=\sum_{n=0}^{\infty} \frac{(-1)^{n+1}}{n!} \frac{\sigma^{\lambda n}}{2^{n} \pi} \Gamma(1+n \lambda) \frac{\sin \frac{\pi \lambda}{2}}{|z|^{1+\lambda}} $$

$$ H_{\lambda, \sigma, \beta}(p)=\frac{1}{2}|\sigma p|^{\lambda}\left[1-i \beta \epsilon(p) F_{\lambda, \sigma}(p)\right] $$

$$ F_{\lambda, \sigma}(p)=\left\{\begin{array}{cc} \tan (\pi \lambda / 2) & \text { for } \lambda \neq 1 \\ -(1 / \pi) \log p^{2} & \text { for } \lambda=1 \end{array}\right. $$

$$ \sigma^{2}=\left\langle z^{2}\right\rangle \equiv \int_{-\infty}^{\infty} d z z^{2} \tilde{L}_{\sigma^{2}}^{\lambda}(z)=-\left.\frac{d^{2}}{d p^{2}} L_{\sigma^{2}}^{\lambda}(p)\right|_{p=0} $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(z) \equiv \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z} L_{\sigma^{2}}^{(\lambda, \alpha)}(p)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z-H(p)} $$

$$ \begin{align*} H(p) & \equiv \frac{\sigma^{2}}{2} \frac{\alpha^{2-\lambda}}{\lambda(1-\lambda)}\left[(\alpha+i p)^{\lambda}+(\alpha-i p)^{\lambda}-2 \alpha^{\lambda}\right] \\ & =\sigma^{2} \frac{\left(\alpha^{2}+p^{2}\right)^{\lambda / 2} \cos [\lambda \arctan (p / \alpha)]-\alpha^{\lambda}}{\alpha^{\lambda-2} \lambda(1-\lambda)} \end{align*} $$

$$ \begin{align*} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(z) & \approx e^{2 s \alpha^{\lambda}} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z}\left\{1-s\left[(\alpha+i p)^{\lambda}+(\alpha-i p)^{\lambda}\right]\right\} \\ & \rightarrow e^{2 s \alpha^{\lambda}} \Gamma(1+\lambda) \frac{\sin (\pi \lambda)}{\pi} s \frac{e^{-\alpha|z|}}{|z|^{1+\lambda}} \end{align*} $$

$$ s \equiv \frac{\sigma^{2}}{2} \frac{\alpha^{2-\lambda}}{\lambda(1-\lambda)} $$

$$ \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z}(\alpha+i p)^{\lambda}=\frac{\Theta(z)}{\Gamma(-\lambda)} \frac{e^{-\alpha z}}{z^{1+\lambda}}, \quad \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z}(\alpha-i p)^{\lambda}=\frac{\Theta(-z)}{\Gamma(-\lambda)} \frac{e^{-\alpha|z|}}{|z|^{1+\lambda}}, $$

$$ \frac{1}{\Gamma(-z)}=-\Gamma(1+z) \sin (\pi z) / \pi $$

$$ \begin{align*} & \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z}(\alpha+i p)^{\lambda}(\alpha-i p)^{\nu} \\ & \quad=(2 \alpha)^{\lambda / 2+\nu / 2} \frac{1}{|z|^{1+\lambda / 2+\nu / 2}}\left\{\begin{array}{l} \frac{1}{\Gamma(-\lambda)} W_{(\nu-\lambda) / 2,(1+\lambda+\nu) / 2}(2 \alpha z) \quad z>0 \\ \frac{1}{\Gamma(-\nu)} W_{(\lambda-\nu) / 2,(1+\lambda+\nu) / 2}(2 \alpha z) \quad \text { for } \quad z<0 \end{array}\right. \end{align*} $$

$$ \begin{align*} W_{\delta, \kappa}(x) & =\frac{\Gamma(-2 \kappa)}{\Gamma(1 / 2-\kappa-\delta)} x^{\kappa+1 / 2} e^{-x / 2}{ }_{1} F_{1}(1 / 2+\kappa-\delta ; 2 \kappa+1 ; x) \\ & +\frac{\Gamma(2 \kappa)}{\Gamma(1 / 2+\kappa-\delta)} x^{-\kappa+1 / 2} e^{-x / 2}{ }_{1} F_{1}(1 / 2-\kappa-\delta ;-2 \kappa+1 ; x) \end{align*} $$

$$ \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z}\left(\alpha^{2}+p^{2}\right)^{\nu}=(2 \alpha)^{\nu / 2} \frac{1}{|z|^{1+\nu}} \frac{1}{\Gamma(-\nu)} W_{0,1 / 2+\nu}(2 \alpha|z|) $$

$$ W_{0,1 / 2+\nu}(x)=\sqrt{\frac{2 z}{\pi}} K_{1 / 2+\nu}(x / 2) $$

$$ \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z}\left(\alpha^{2}+p^{2}\right)^{\nu}=\left(\frac{2 \alpha}{|z|}\right)^{1 / 2+\nu} \frac{1}{\sqrt{\pi} \Gamma(-\nu)} K_{1 / 2+\nu}(\alpha|z|) $$

$$ \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z} \frac{1}{\alpha^{2}+p^{2}}=\frac{1}{2 \alpha} e^{-\alpha|z|} $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(z) \approx e^{2 s \alpha^{\lambda}} \int_{-\infty}^{\infty} \frac{d p}{2 \pi}\left\{1+\sum_{n=1}^{\infty} \frac{(-s)^{n}}{n!}\left[(\alpha+i p)^{\lambda}+(\alpha-i p)^{\lambda}\right]^{n}\right\} e^{i p z} $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(z) \underset{z \rightarrow \infty}{\rightarrow} e^{\left(2-2^{\lambda}\right) s \alpha^{\lambda}} \Gamma(1+\lambda) \frac{\sin (\pi \lambda)}{\pi} s \frac{e^{-\alpha|z|}}{|z|^{1+\lambda}} $$

$$ P_{<}(z)=\int_{-\infty}^{z} d z^{\prime} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}\left(z^{\prime}\right), \quad P_{>}(z)=\int_{z}^{\infty} d z^{\prime} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}\left(z^{\prime}\right)=1-P_{<}(z) $$

$$ H(p)=\frac{1}{2} c_{2} p^{2}-\frac{1}{4!} c_{4} p^{4}+\frac{1}{6!} c_{6} p^{6}-\frac{1}{8!} c_{8} p^{8}+\ldots . $$

$$ \begin{align*} \left\langle z^{2}\right\rangle_{c} & =c_{2}=\sigma^{2} \\ \left\langle z^{4}\right\rangle_{c} & =c_{4}=\sigma^{2}(2-\lambda)(3-\lambda) \alpha^{-2}, \\ \left\langle z^{6}\right\rangle_{c} & =c_{6}=\sigma^{2}(2-\lambda)(3-\lambda)(4-\lambda)(5-\lambda) \alpha^{-4}, \\ & \vdots \\ \left\langle z^{2 n}\right\rangle_{c} & =c_{2 n}=\sigma^{2} \frac{\Gamma(2 n-\lambda)}{\Gamma(2-\lambda)} \alpha^{2-2 n} . \end{align*} $$

$$ \left\langle z^{2}\right\rangle \equiv \int_{-\infty}^{\infty} d z z^{2} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(z)=-\left.\frac{d^{2}}{d p^{2}} e^{-H(p)}\right|_{p=0}=c_{2}=\sigma^{2} $$

$$ \left\langle z^{4}\right\rangle \equiv \int_{-\infty}^{\infty} d z z^{4} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(z)=\left.\frac{d^{4}}{d p^{4}} e^{-H(p)}\right|_{p=0}=c_{4}+3 c_{2}^{2} $$

$$ \kappa \equiv \bar{c}_{4} \equiv \frac{c_{4}}{c_{2}^{2}}=\frac{\left\langle z^{4}\right\rangle_{c}}{\left\langle z^{2}\right\rangle_{c}^{2}}=\frac{\left\langle z^{4}\right\rangle_{c}}{\sigma^{4}} $$

$$ \kappa=\frac{(2-\lambda)(3-\lambda)}{\sigma^{2} \alpha^{2}} . $$

$$ \alpha=\frac{1}{\sigma} \sqrt{\frac{(2-\lambda)(3-\lambda)}{\kappa}} . $$

$$ \begin{align*} \bar{c}_{4} & =\kappa, \quad \bar{c}_{6}=\kappa^{2} \frac{(5-\lambda)(4-\lambda)}{(3-\lambda)(2-\lambda)}, \quad \bar{c}_{8}=\kappa^{2} \frac{(7-\lambda)(6-\lambda)(5-\lambda)(4-\lambda)}{(3-\lambda)^{2}(2-\lambda)^{2}} \\ \vdots & \\ \bar{c}_{n} & =\kappa^{n / 2-1} \frac{\Gamma(n-\lambda) / \Gamma(4-\lambda)}{(3-\lambda)^{n / 2-2}(2-\lambda)^{n / 2-2}} \end{align*} $$

$$ \alpha=\frac{1}{2} \sqrt{\frac{3}{\sigma^{2} \kappa}}, $$

$$ \begin{align*} \bar{c}_{4} & =\kappa, \quad \bar{c}_{6}=\frac{5 \cdot 7}{3} \kappa^{2}, \quad \bar{c}_{8}=5 \cdot 7 \cdot 11 \kappa^{2} \\ \vdots & \\ \bar{c}_{n} & =\frac{\Gamma(n-3 / 2) / \Gamma(5 / 2)}{3^{n / 2-2} / 2^{n-4}} \kappa^{n / 2-1} \end{align*} $$

$$ \begin{align*} & \bar{c}_{6}=\frac{\left\langle z^{6}\right\rangle}{\left\langle z^{2}\right\rangle^{3}}-15 \frac{\left\langle z^{4}\right\rangle}{\left\langle z^{2}\right\rangle^{2}}+30 \\ & \bar{c}_{8}=\frac{\left\langle z^{8}\right\rangle}{\left\langle z^{2}\right\rangle^{4}}-28 \frac{\left\langle z^{6}\right\rangle}{\left\langle z^{2}\right\rangle^{3}}-35 \frac{\left\langle z^{4}\right\rangle^{2}}{\left\langle z^{2}\right\rangle^{4}}+420 \frac{\left\langle z^{4}\right\rangle}{\left\langle z^{2}\right\rangle^{2}}-630, \ldots \end{align*} $$

$$ L_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(p) \equiv e^{-H(p)} $$

$$ \begin{align*} H(p) & \equiv \frac{\sigma^{2}}{2} \frac{\alpha^{2-\lambda}}{\lambda(1-\lambda)}\left[(\alpha+i p)^{\lambda}(1+\beta)+(\alpha-i p)^{\lambda}(1-\beta)-2 \alpha^{\lambda}\right] \\ & =\sigma^{2} \frac{\left(\alpha^{2}+p^{2}\right)^{\lambda / 2}\{\cos [\lambda \arctan (p / \alpha)]+i \beta \sin [\lambda \arctan (p / \alpha)]\}-\alpha^{\lambda}}{\alpha^{\lambda-2} \lambda(1-\lambda)} \end{align*} $$

$$ H(p)=i c_{1} p+\frac{1}{2} c_{2} p^{2}-i \frac{1}{3!} c_{3} p^{3}-\frac{1}{4!} c_{4} p^{4}+i \frac{1}{5!} c_{5} p^{5}+\ldots $$

$$ \begin{align*} & \langle z\rangle \equiv \int_{-\infty}^{\infty} d z z \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(z)=\left.i \frac{d}{d p} e^{-H(p)}\right|_{p=0}=c_{1} \\ & \left\langle z^{2}\right\rangle \equiv \int_{-\infty}^{\infty} d z z^{2} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(z)=-\left.\frac{d^{2}}{d^{2} p} e^{-H(p)}\right|_{p=0}=c_{2}+c_{1}^{2} \\ & \left\langle z^{3}\right\rangle \equiv \int_{-\infty}^{\infty} d z z^{3} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(z)=-\left.i \frac{d^{3}}{d^{3} p} e^{-H(p)}\right|_{p=0}=c_{3}+3 c_{2} c_{1}+c_{1}^{3} \\ & \left\langle z^{4}\right\rangle \equiv \int_{-\infty}^{\infty} d z z^{4} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(z)=\left.\frac{d^{4}}{d^{4} p} e^{-H(p)}\right|_{p=0}=c_{4}+4 c_{3} c_{1}+3 c_{2}^{2}+6 c_{2} c_{1}^{2}+c_{1}^{4} \\ & \quad \vdots \end{align*} $$

$$ \begin{align*} c_{1} & =\langle z\rangle_{c}=\langle z\rangle \\ c_{2} & =\left\langle z^{2}\right\rangle_{c}=\left\langle z^{2}\right\rangle-\langle z\rangle^{2}=\left\langle\left(z-\langle z\rangle_{c}\right)\right\rangle^{2} \\ c_{3} & =\left\langle z^{3}\right\rangle_{c}=\left\langle z^{3}\right\rangle-3\langle z\rangle\left\langle z^{2}\right\rangle+2\langle z\rangle^{3}=\left\langle\left(z-\langle z\rangle_{c}\right)\right\rangle^{3} \\ c_{4} & =\left\langle z^{4}\right\rangle_{c}=\left\langle z^{4}\right\rangle-3\left\langle z^{2}\right\rangle^{2}-4\langle z\rangle\left\langle z^{3}\right\rangle+12\langle z\rangle^{2}\left\langle z^{2}\right\rangle-6\langle z\rangle^{4} \\ & =\left\langle\left(z-\langle z\rangle_{c}\right)\right\rangle^{4}-3\left\langle z^{2}-\langle z\rangle_{c}^{2}\right\rangle^{2}=\left\langle\left(z-\langle z\rangle_{c}\right)\right\rangle^{4}-3 c_{2}^{2} \end{align*} $$

$$ \mu \equiv\langle z\rangle=c_{1} $$

$$ \sigma^{2} \equiv\left\langle z^{2}\right\rangle-\langle z\rangle^{2}=\left\langle(z-\langle z\rangle)^{2}\right\rangle=c_{2} $$

$$ \begin{align*} \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha)}(z) & \approx \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p z}\left\{1-\frac{\sigma^{2}}{2} \frac{\alpha^{2-\lambda}}{\lambda(1-\lambda)}\left[(\alpha+i p)^{\lambda}(1+\beta)+(\alpha-i p)^{\lambda}(1-\beta)-2 \alpha^{\lambda}\right]\right\} \\ & \underset{z \rightarrow \infty}{\rightarrow} \sigma^{2} e^{2 s \alpha^{\lambda}} \Gamma(1+\lambda) \frac{\sin (\pi \lambda)}{\pi} s \frac{e^{-\alpha|z|}}{|z|^{1+\lambda}}[1+\beta \operatorname{sgn}(z)] \end{align*} $$

$$ s \equiv \frac{\left\langle(z-\langle z\rangle)^{3}\right\rangle}{\sigma^{3}}=\bar{c}_{3}=\frac{c_{3}}{c_{2}^{3 / 2}} . $$

$$ s=\frac{(2-\lambda) \beta}{\sigma \alpha} $$

$$ \kappa \equiv \bar{c}_{4} \equiv \frac{c_{4}}{c_{2}^{2}}=\frac{\left\langle z^{4}\right\rangle_{c}}{\left\langle z^{2}\right\rangle_{c}^{2}}=\frac{\left\langle z^{4}\right\rangle_{c}}{\sigma^{4}}=\frac{\left\langle(z-\langle z\rangle)^{4}\right\rangle}{\sigma^{4}}-3 $$

$$ \bar{L}_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(z) \equiv \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(z-\mu) $$

$$ \bar{H}(p) \equiv H(p)-H^{\prime}(0) p $$

$$ \bar{H}(p) \equiv H(p)-H^{\prime}(0) p=\frac{1}{2} c_{2} p^{2}-i \frac{1}{3!} c_{3} p^{3}-\frac{1}{4!} c_{4} p^{4}+i \frac{1}{5!} c_{5} p^{5}+\ldots . $$

$$ H_{+}(p)=\nu \log (1-i p / \mu) $$

$$ \tilde{D}_{\mu, \nu}^{\mathrm{Gamma}}(z)=\frac{1}{\Gamma(\nu)} \mu^{\nu} z^{\nu-1} e^{-\mu z}, \quad \int_{0}^{\infty} d z \tilde{D}_{\mu, \nu}^{\mathrm{Gamma}}(z)=1 $$

$$ c_{n}=(n-1)!\nu / \mu^{n} $$

$$ \bar{z}=\nu / \mu, \quad \sigma^{2}=\nu / \mu^{2}, \quad s=2 / \sqrt{\nu}, \quad \kappa=6 / \nu $$

$$ z_{\max }=(\nu-1) / \mu=\bar{z}-1 / \mu $$

$$ \tilde{D}_{\mu, \nu}^{\mathrm{Chi}}(\sigma) \equiv 2 \sigma \tilde{D}_{\mu, \nu}^{\mathrm{Gamma}}\left(\sigma^{2}\right), \quad \int_{0}^{\infty} d \sigma \tilde{D}_{\mu, \nu}^{\mathrm{Chi}}(\sigma)=1 $$

$$ \tilde{D}_{\mu, \nu}^{\mathrm{Gamma}}(z) \rightarrow \delta\left(z-\frac{\nu}{\mu}\right) $$

$$ H_{-}(p)=\nu \log (1+i p / \mu) $$

$$ \tilde{D}_{\mu, \nu}^{\mathrm{Gamma}}(z)=\frac{1}{\Gamma(\nu)} \mu^{-\nu}|z|^{\nu-1} e^{-\mu|z|}, \quad z \leq 0 $$

$$ \tilde{B}(z)=\frac{1}{2 T} e^{-|z| / T} . $$

$$ B(p)=\int_{-\infty}^{\infty} d z e^{i p z} \frac{1}{2 T} e^{-|z| / T}=\frac{1}{1+(T p)^{2}}=e^{-H(p)} $$

$$ H(p)=\log \left[1+(T p)^{2}\right] $$

$$ B(p)=e^{-H(p)}=\frac{1}{1+(T p)^{2}}=\int_{0}^{\infty} d \tau e^{-\tau\left(1+T^{2} p^{2}\right)} $$

$$ B(p)=e^{-H(p)}=\frac{1}{2 T^{2}} \int_{0}^{\infty} d \sigma^{2} e^{-\sigma^{2} / 2 T^{2}} e^{-\sigma^{2} p^{2} / 2} . $$

$$ \tilde{B}(z)=\frac{1}{2 T^{2}} \int_{0}^{\infty} d \sigma^{2} e^{-\sigma^{2} / 2 T^{2}} \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-z^{2} / 2 \sigma^{2}} $$

$$ \tilde{D}_{\delta}(z)=N_{\delta} \frac{1}{\sqrt{2 \pi \sigma_{\delta}^{2}}} e_{\delta}^{-z^{2} / 2 \sigma_{\delta}^{2}} $$

$$ N_{\delta}=\frac{\sqrt{\delta} \Gamma(1 / \delta)}{\Gamma(1 / \delta-1 / 2)} $$

$$ e_{\delta}^{z} \equiv(1-\delta z)^{-1 / \delta} $$

$$ Z_{N}(W)=\left[\prod_{n=1}^{N} \int_{0}^{W} d w_{n}\right] \delta\left(w_{1}+\ldots+w_{N}-W\right) $$

$$ Z_{N}(W)=\int_{-\infty}^{\infty} \frac{d \lambda}{2 \pi}\left[\prod_{n=1}^{N} \int_{0}^{W} d w_{n}\right] e^{-i \lambda\left(w_{1}+\ldots+w_{N}\right)} e^{i \lambda W}=\int_{-\infty}^{\infty} \frac{d \lambda}{2 \pi} \frac{\left(e^{-i \lambda W}-1\right)^{N} e^{i \lambda W}}{(-i \lambda+\epsilon)^{N}} $$

$$ \int_{-\infty}^{\infty} \frac{d \lambda}{2 \pi} \frac{1}{(-i \lambda+\epsilon)^{\nu}} e^{-i p x}=\frac{p^{\nu-1}}{\Gamma(\nu)} e^{-\epsilon p} \Theta(p) $$

$$ Z_{N}(W)=\frac{W^{N-1}}{\Gamma(N)} \sum_{k=0}^{N-1}(-1)^{k}\binom{N}{N-k}(N-k-1)^{N-1} $$

$$ Z_{N}(W)=\frac{W^{N-1}}{\Gamma(N)} $$

$$ P_{N}\left(w_{n}\right)=Z_{N}^{-1} \frac{\left(W-w_{n}\right)^{N-2}}{\Gamma(N-1)}=\frac{N-1}{W}\left(1-\frac{w_{n}}{E}\right)^{N-2} . $$

$$ P_{N}\left(w_{n}\right)=\frac{N-1}{W}\left[1-\frac{w_{n}}{(N-2) w}\right]^{N-2}=\frac{N-1}{W} e_{-1 /(N-2)}^{-w_{n} / w} $$

$$ e_{\delta}^{-z^{2} / 2 \sigma_{\delta}^{2}}=\frac{1}{\Gamma(1 / \delta)} \int_{0}^{\infty} \frac{d s}{s} s^{1 / \delta} e^{-s} e^{-s \delta z^{2} / 2 \sigma_{\delta}^{2}} $$

$$ e_{\delta}^{-z^{2} / 2 \sigma_{\delta}^{2}}=\frac{\mu^{1 / \delta}}{\Gamma(1 / \delta)} \int_{0}^{\infty} \frac{\mathrm{d} v}{v} v^{1 / \delta} e^{-\mu v} e^{-v z^{2} / 2}, \quad \mu=\sigma_{\delta}^{2} / \delta $$

$$ \tilde{D}_{\mu, 1 / \delta}^{\mathrm{Gamma}}(v)=\frac{1}{\Gamma(1 / \delta)} \mu^{1 / \delta} v^{\nu} e^{-\mu v} $$

$$ e^{-H_{\delta, \beta}(p)} \equiv e_{\delta}^{-\beta p^{2} / 2}=\left(1+\beta \delta p^{2} / 2\right)^{-1 / \delta}, $$

$$ e_{\delta}^{-\beta p^{2} / 2}=\frac{1}{\Gamma(1 / \delta)} \int_{0}^{\infty} \frac{d s}{s} s^{1 / \delta} e^{-s} e^{-s \beta \delta p^{2} / 2}, \quad \beta \equiv \nu / \mu=1 / \mu \delta, $$

$$ e_{\delta}^{-\beta p^{2} / 2}=\frac{\mu^{\nu}}{\Gamma(\nu)} \int_{0}^{\infty} \frac{\mathrm{d} v}{v} v^{\nu} e^{-\mu v} e^{-v p^{2} / 2}, \quad \nu=1 / \delta, \mu=\nu / \beta=1 / \beta \delta . $$

$$ e_{\delta}^{-\beta p^{2} / 2}=\int_{0}^{\infty} d v D_{\mu, \nu}^{\mathrm{Gamma}}(v) e^{-v p^{2} / 2}, \quad \nu=1 / \delta, \mu=\nu / \beta=1 / \beta \delta $$

$$ c_{2}=\frac{\nu}{\mu}, c_{4}=3 \frac{\nu}{\mu^{2}}, c_{6}=\frac{5!!}{\mu^{3}}\left(2+3 \nu-2 \nu^{2}-\nu^{3}\right), c_{8}=\frac{7!!}{\mu^{3}}\left(2+\nu-3 \nu^{2}+\nu^{3}+n u^{4}\right) . $$

$$ \tilde{D}_{\delta, \beta}(z)=\frac{\mu^{1 / \delta}}{\Gamma(1 / \delta)} \int_{0}^{\infty} \frac{\mathrm{d} v}{v} v^{1 / \delta} e^{-\mu v} \frac{1}{\sqrt{2 \pi v}} e^{-z^{2} / 2 v}, \quad \mu=1 / \beta \delta $$

$$ \tilde{D}_{\delta, \beta}(z)=\frac{\sqrt{\mu}}{\Gamma(1 / \delta)} \frac{1}{\sqrt{2 \pi}}\left(\frac{\mu z^{2}}{2}\right)^{1 / 2 \delta-1 / 4} 2 K_{1 / \delta-1 / 2}(\sqrt{2 \mu} z), \quad \mu=1 / \beta \delta $$

$$ \tilde{D}_{\delta, \beta}(0)=\frac{\sqrt{\mu} \Gamma(1 / \delta-1 / 2)}{\Gamma(1 / \delta)} $$

$$ e^{-\beta \sqrt{p^{2}+M^{2}}}=\int_{0}^{\infty} d v \omega_{\beta}(v) e^{-\beta v\left(p^{2}+M^{2}\right) / 2} $$

$$ \omega_{\beta}(v) \equiv \sqrt{\frac{\beta}{2 \pi v^{3}}} e^{-\beta / 2 v} $$

$$ \tilde{D}_{\mathrm{W}}(x)=\frac{b}{\Gamma(a / b)} x^{a-1} e^{-x^{b}} $$

$$ \left\langle x^{n}\right\rangle=\Gamma((a+n) / b) / \Gamma(a / b) $$

$$ \begin{align*} \tilde{M}(z) & =\frac{[2 \cos (b / 2)]^{2 d}}{2 a \pi \Gamma(2 d)}|\Gamma(d+i z / a)|^{2} \exp [b z / a] \\ M(p) & =\left\{\frac{\cos (b / 2)}{\cosh [(a p-i b) / 2]}\right\}^{2 d} . \end{align*} $$

$$ \tilde{M}(z) \rightarrow C_{ \pm}|z|^{\rho} e^{-\sigma_{ \pm}|z|} \quad \text { for } \quad z \rightarrow \pm \infty $$

$$ C_{ \pm}=\frac{[2 \cos (b / 2)]^{2 d}}{2 a \pi \Gamma(2 d)} \frac{2 \pi}{a^{2 d-1}} e^{ \pm 2 \pi d \tan (b / 2)}, \rho=2 d-1, \sigma_{ \pm} \equiv(\pi \pm b) / a $$

$$ \mu=a d \tan (b / 2), \quad \sigma^{2}=a^{2} d / 2 \cos ^{2}(b / 2), \quad s=\sqrt{2} \sin (b / 2) / \sqrt{d}, \quad \kappa=[2-\cos b] / d $$

$$ a^{2}=\sigma^{2}\left(2 \kappa-3 s^{2}\right), \quad d=\frac{1}{\kappa-s^{2}}, \quad b=2 \arcsin (s \sqrt{d / 2}) $$

$$ a=0.029828, b=0.12716, d=0.57295,\langle z\rangle=-0.0011243 . $$

$$ \begin{align*} \tilde{M}(0) & =\frac{2^{2 d-1} \Gamma^{2}(d)}{\pi a \Gamma(2 d)} \\ \tilde{M}^{\prime}(0) & =b \frac{2^{2 d-1} \Gamma^{2}(d)[1-d \psi(d)]}{\pi a^{2} \Gamma(2 d)} \\ \tilde{M}^{\prime \prime}(0) & =-\frac{2^{2 d} \Gamma^{2}(d) \psi(d)}{\pi a^{3} \Gamma(2 d)} \\ \tilde{M}^{(3)}(0) & =-\frac{b}{2} \frac{2^{2 d} \Gamma^{2}(d)\left[6 \psi(d)-6 d \psi^{2}(d)-d \psi^{(3)}(d)\right]}{\pi a^{4} \Gamma(2 d)} \\ \tilde{M}^{(4)}(0) & =\frac{2^{2 d} \Gamma^{2}(d)\left[6 \psi^{2}(d)+\psi^{(3)}(d)\right]}{\pi a^{5} \Gamma(2 d)} \end{align*} $$

$$ \tilde{H}_{G}(z)=\frac{\left(\gamma^{2}-\beta^{2}\right)^{\lambda / 2} e^{\beta z}}{\gamma^{\lambda-1 / 2} \delta^{\lambda} \sqrt{2 \pi}}\left[\delta^{2}+z^{2}\right]^{\lambda / 2-1 / 4} \frac{K_{\lambda-1 / 2}\left(\gamma \sqrt{\delta^{2}+z^{2}}\right)}{K_{\lambda}\left(\delta \sqrt{\gamma^{2}-\beta^{2}}\right)} $$

$$ G(p)=\frac{\left(\delta \sqrt{\gamma^{2}-\beta^{2}}\right)^{\lambda}}{K_{\lambda}\left(\delta \sqrt{\gamma^{2}-\beta^{2}}\right)} \frac{K_{\lambda}\left(\delta \sqrt{\gamma^{2}-(\beta+i p)^{2}}\right)}{\left[\delta \sqrt{\gamma^{2}-(\beta+i p)^{2}}\right]^{\lambda}}, $$

$$ H_{G}(p) \equiv-\log G(p) . $$

$$ \tilde{H}_{G}(z) \rightarrow \sqrt{\frac{\pi}{2 \gamma}} \frac{\left(\gamma^{2}-\beta^{2}\right)^{\lambda / 2} e^{\beta z}}{\gamma^{\lambda-1 / 2} \delta^{\lambda} \sqrt{2 \pi}} \frac{1}{K_{\lambda}\left(\delta \sqrt{\gamma^{2}-\beta^{2}}\right)} z^{\lambda-1} e^{-\gamma z} $$

$$ \begin{align*} c_{1} & =\beta \frac{\delta^{2}}{\zeta} \frac{K_{1+\lambda}(\zeta)}{K_{\lambda}(\zeta)} \\ c_{2} & =\frac{\delta^{2}}{\zeta} \frac{K_{1+\lambda}(\zeta)}{K_{\lambda}(\zeta)}+\frac{\beta^{2} \delta^{4}}{\zeta^{2}}\left\{\frac{K_{2+\lambda}(\zeta)}{K_{\lambda}(\zeta)}-\left[\frac{K_{1+\lambda}(\zeta)}{K_{\lambda}(\zeta)}\right]^{2}\right\} \end{align*} $$

$$ K_{\nu+1}(z)-K_{\nu-1}(z)=\frac{2 \nu}{z} K_{\nu}(z) $$

$$ \rho=\rho(\zeta)=\frac{K_{1+\lambda}(\zeta)}{K_{\lambda}(\zeta)} $$

$$ c_{2}=\frac{\delta^{2}}{\zeta} \rho+\frac{\beta^{2} \delta^{4}}{\zeta^{3}}\left[\zeta+2(1+\lambda) \rho-\zeta \rho^{2}\right] $$

$$ \sigma_{s}^{2} \equiv \delta^{2} \rho / \zeta $$

$$ c_{1}=\beta \sigma_{s}, \quad c_{2}=\sigma^{2}=\sigma_{s}^{2}+\beta^{2}\left[\frac{\delta^{4}}{\zeta^{2}}+2(1+\lambda) \frac{\delta^{2}}{\zeta^{2}} \sigma_{s}-\sigma_{s}^{2}\right] $$

$$ \begin{align*} c_{3} & =\beta\left[\frac{3 \delta^{4}}{\zeta^{2}}+6(1+\lambda) \frac{\delta^{2}}{\zeta^{2}} \sigma_{s}^{2}-3 \sigma_{s}^{4}\right] \\ & +\beta^{3}\left\{2(2+\lambda) \frac{\delta^{6}}{\zeta^{4}}+\left[4(1+\lambda)(2+\lambda)-2 \zeta^{2}\right] \frac{\delta^{4}}{\zeta^{4}} \sigma_{s}^{2}-6(1+\lambda) \frac{\delta^{2}}{\zeta^{2}} \sigma_{s}^{4}+2 \sigma_{s}^{6}\right\} \end{align*} $$

$$ \begin{align*} c_{4}= & \kappa \sigma^{4}=\frac{3 \delta^{4}}{\zeta^{2}}+\frac{6 \delta^{2}}{\zeta^{2}}(1+\lambda) \sigma_{s}^{2}-3 \sigma_{s}^{4} \\ + & 6 \beta^{2}\left\{2(2+\lambda) \frac{\delta^{6}}{\zeta^{4}}+\left[4(1+\lambda)(2+\lambda)-2 \zeta^{2}\right] \frac{\delta^{4}}{\zeta^{4}} \sigma_{s}^{2}-6(1+\lambda) \frac{\delta^{2}}{\zeta^{2}} \sigma_{s}^{4}+2 \sigma_{s}^{6}\right\} \\ + & \beta^{4}\left\{\left[4(2+\lambda)(3+\lambda)-\zeta^{2}\right] \frac{\delta^{8}}{\zeta^{6}}+\left[4(1+\lambda)(2+\lambda)(3+\lambda)-2(5+4 \lambda) \zeta^{2}\right] \frac{\delta^{6}}{\zeta^{6}} \sigma_{s}^{2}\right. \\ & \left.-2\left[(1+\lambda)(11+7 \lambda)-2 \zeta^{2}\right] \frac{\delta^{4}}{\zeta^{4}} \sigma_{s}^{4}+12(1+\lambda) \frac{\delta^{2}}{\zeta^{2}} \sigma_{s}^{6}-3 \sigma_{s}^{8}\right\} \end{align*} $$

$$ \kappa_{s} \equiv \frac{3 \delta^{4}}{\zeta^{2} \sigma_{s}^{4}}+\frac{6 \delta^{2}}{\zeta^{2} \sigma_{s}^{2}}(1+\lambda)-3 $$

$$ \kappa_{s} \equiv \frac{3}{r^{2}(\zeta)}+(1+\lambda) \frac{6}{\zeta r(\zeta)}-3 $$

$$ \delta^{2}=\frac{\sigma_{s}^{2} \zeta}{\rho(\zeta)} $$

$$ \beta \approx \frac{s}{\kappa_{s} \sigma_{s}} $$

$$ \begin{align*} G(0) & =\left(\frac{\delta}{\gamma}\right)^{\lambda-1 / 2} \zeta^{-\lambda} k_{-} \\ G^{\prime}(0) & =\beta G(0) \\ G^{\prime \prime}(0) & =-\left(\frac{\delta}{\gamma}\right)^{\lambda-3 / 2} \zeta^{-\lambda} k_{+}+\left(\frac{\delta}{\gamma}\right)^{\lambda-1 / 2} \delta^{-2} \zeta^{-\lambda}\left(1-2 \lambda-\beta^{2} \delta^{2}\right) k_{-} \\ G^{(3)}(0) & =-\frac{\beta}{2}\left[3\left(\frac{\delta}{\gamma}\right)^{\lambda-3 / 2} \zeta^{-\lambda} k_{+}+\left(\frac{\delta}{\gamma}\right)^{\lambda-1 / 2} \delta^{-2} \zeta^{-\lambda}\left(3-6 \lambda-\beta^{2} \delta^{2}\right)\right] k_{-}, \end{align*} $$

$$ k_{ \pm} \equiv \frac{1}{\sqrt{2 \pi}} \frac{K_{\lambda \pm 1 / 2}(\delta \gamma)}{K_{\lambda}(\delta \gamma)} $$

$$ \left\langle e^{P z}\right\rangle \equiv \int d z \frac{1}{\sqrt{2 \pi \sigma^{2}}} e^{-z^{2} / 2 \sigma^{2}} e^{P z}=\int d z \int \frac{d p}{2 \pi} e^{-\sigma^{2} p^{2} / 2} e^{i p z+P z}=e^{\sigma^{2} P^{2} / 2} $$

$$ \left\langle e^{P z}\right\rangle \equiv \int d z \int \frac{d p}{2 \pi} e^{-H(p)} e^{i p z+P z}=e^{-H(i P)} $$

$$ \dot{x}(t)=r_{x}+\eta(t) $$

$$ \begin{align*} H(p) \rightarrow H_{r_{x}}(p) & \equiv H(p)-H^{\prime}(0) p+i r_{x} p \equiv \bar{H}(p)+i r_{x} p \\ & \equiv i r_{x} p+\frac{1}{2} c_{2} p^{2}-i \frac{1}{3!} c_{3} p^{3}-\frac{1}{4!} c_{4} p^{4}+i \frac{1}{5!} c_{5} p^{5}+\ldots \end{align*} $$

$$ c_{1} \rightarrow r_{x} $$

$$ \dot{x}(t)=\eta(t) $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} \eta \int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right] \delta[\dot{x}-\eta] $$

$$ \tilde{H}(\eta)=-\log \tilde{D}(\eta) $$

$$ \mathcal{D} \mu \equiv \mathcal{D} \eta P[\eta]=\mathcal{D} \eta e^{-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))} $$

$$ \int_{x\left(t_{a}\right)=x_{a}}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \delta[\dot{x}-\eta] $$

$$ \mathcal{D} \mu^{\prime}=\mathcal{D} \eta P[\eta-r]=\mathcal{D} \eta e^{-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t)-r)} $$

$$ \mathcal{D} \mu^{\prime} / \mathcal{D} \mu=e^{-\int_{t_{a}}^{t_{b}}[\tilde{H}(\eta-r)-\tilde{H}(\eta)]} $$

$$ \mathcal{D} \mu^{\prime} / \mathcal{D} \mu=e^{\int_{t_{a}}^{t_{b}}\left[r \eta(t)-r^{2} t / 2\right]} $$

$$ \left[\int_{x(t)=x}^{x\left(t_{b}\right)=x_{b}} \mathcal{D} x \delta[\dot{x}-\eta]\right] \times\left[\int_{x\left(t_{a}\right)=x_{a}}^{x(t)=x} \mathcal{D} x \delta[\dot{x}-\eta]\right], $$

$$ \langle f(x(t))\rangle=\int d x P\left(x_{b} t_{b} \mid x t\right) f(x) P\left(x t \mid x_{a} t_{a}\right) $$

$$ P[\eta]=\int \frac{\mathcal{D} p}{2 \pi} \exp \left\{\int_{t_{a}}^{t_{b}} d t[i p(t) \eta(t)-H(p(t))]\right\} $$

$$ \begin{align*} \left\langle\eta\left(t_{1}\right) \cdots \eta\left(t_{n}\right)\right\rangle & =i^{n} \int \mathcal{D} \eta \int \frac{\mathcal{D} p}{2 \pi} e^{i \int_{t_{a}}^{t_{b}} d t p(t) \eta(t)} \frac{\delta}{\delta p\left(t_{1}\right)} \cdots \frac{\delta}{\delta p\left(t_{n}\right)} e^{-\int_{t_{a}}^{t_{b}} d t H(p(t))} \\ & =i^{n}\left[\frac{\delta}{\delta p\left(t_{1}\right)} \cdots \frac{\delta}{\delta p\left(t_{n}\right)} e^{-\int_{t_{a}}^{t_{b}} d t H(p(t))}\right]_{p(t) \equiv 0} \end{align*} $$

$$ \begin{align*} \left\langle\eta\left(t_{1}\right)\right\rangle & \equiv Z^{-1} \int \mathcal{D} \eta \eta\left(t_{1}\right) \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right]=0 \\ \left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right)\right\rangle & \equiv Z^{-1} \int \mathcal{D} \eta \eta\left(t_{1}\right) \eta\left(t_{2}\right) \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right] \\ & =c_{2} \delta\left(t_{1}-t_{2}\right)+c_{1}^{2} \\ \left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right)\right\rangle & \equiv Z^{-1} \int \mathcal{D} \eta \eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right] \\ & =c_{3} \delta\left(t_{1}-t_{2}\right) \delta\left(t_{1}-t_{3}\right) \\ & +c_{2} c_{1}\left[\delta\left(t_{1}-t_{2}\right)+\delta\left(t_{2}-t_{3}\right)+\delta\left(t_{1}-t_{3}\right)\right]+c_{1}^{3}, \\ \left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \eta\left(t_{4}\right)\right\rangle & \equiv Z^{-1} \int \mathcal{D} \eta \eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \eta\left(t_{4}\right) \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right] \\ & =c_{4} \delta\left(t_{1}-t_{2}\right) \delta\left(t_{1}-t_{3}\right) \delta\left(t_{1}-t_{4}\right) \\ & +c_{3} c_{1}\left[\delta\left(t_{1}-t_{2}\right) \delta\left(t_{1}-t_{3}\right)+3 \text { cyclic perms }\right] \\ & +c_{2}^{2}\left[\delta\left(t_{1}-t_{2}\right) \delta\left(t_{3}-t_{4}\right)+\delta\left(t_{1}-t_{3}\right) \delta\left(t_{2}-t_{4}\right)+\delta\left(t_{1}-t_{4}\right) \delta\left(t_{2}-t_{3}\right)\right] \\ & +c_{2} c_{1}^{2}\left[\delta\left(t_{1}-t_{2}\right)+5 \text { pair terms }\right]+c_{1}^{4}, \end{align*} $$

$$ Z \equiv \int \mathcal{D} \eta \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right] $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} \eta \int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi} \exp \left\{\int_{t_{a}}^{t_{b}} d t[i p(t) \dot{x}(t)-i p(t) \eta(t)-\tilde{H}(\eta(t))]\right\} . $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi} \exp \left\{\int_{t_{a}}^{t_{b}} d t[i p(t) \dot{x}(t)-H(p(t))]\right\} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} \exp \left[i p\left(x_{b}-x_{a}\right)-\left(t_{b}-t_{a}\right) H(p)\right] $$

$$ \tilde{D}(z)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i k z-H(p)} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\tilde{L}_{\sigma^{2}\left(t_{b}-t_{a}\right)}^{(\lambda, \alpha)}\left(x_{b}-x_{a}\right) $$

$$ P(x, t)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} \exp [i p x-t H(p)] $$

$$ \tilde{D}(z, t)=\int \frac{d p}{2 \pi} e^{i p z} e^{-t H(p)} $$

$$ \begin{align*} \tilde{D}(z, t) & \xrightarrow[t \text { large }]{ } e^{i p_{z} z-t H\left(p_{z}\right)} \int \frac{d p}{2 \pi} e^{i\left(p-p_{z}\right) z-t \sigma^{2}\left(p-p_{z}\right)^{2} / 2}=\frac{e^{i p_{z} z-t H\left(p_{z}\right)}}{\sqrt{2 \pi \sigma^{2}}} e^{-z^{2} / 2 t \sigma^{2}} \\ & =\frac{e^{\sigma^{2} p_{z}^{2} / 2-t H\left(p_{z}\right)}}{\sqrt{2 \pi \sigma^{2}}} e^{-\left(z-t \sigma^{2} p_{z}\right)^{2} / 2 t \sigma^{2}} . \end{align*} $$

$$ H(p)=-i r p+H_{\lambda, \sigma, \beta}(p) . $$

$$ \dot{x}(t)=r_{x}+\eta_{1}(t)+\eta_{2}(t) $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} p \int \mathcal{D} x \exp \left\{\int_{t_{a}}^{t_{b}} d t\left[i p \dot{x}-i r_{x} p-H_{1}(p)-H_{2}(p)\right]\right\} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} d x d x_{c} P_{1}\left(x_{b} t_{b} \mid x_{c} t_{a}\right) P_{2}\left(x_{c} t_{b} \mid x_{a} t_{a}\right) $$

$$ H(p)=\int d z e^{i p z} F(z) $$

$$ H_{r}(p)=i r p+\int d z\left(e^{i p z}-1-i p z\right) F(z) $$

$$ H_{r}(p)=i r p+\frac{\sigma^{2}}{2} p^{2}+\int d z\left(e^{i p z}-1\right) \bar{F}(z) $$

$$ P(x, t) \underset{t \text { large }}{\rightarrow} \frac{e^{-(x-t r)^{2} / 2 t \sigma^{2}}}{\sqrt{2 \pi \sigma^{2}}} $$

$$ F(z)=\frac{e^{\beta z}}{|z|}\left\{\frac{1}{\pi^{2}} \int_{0}^{\infty} \frac{d y}{y} \frac{e^{-\sqrt{y+\gamma^{2}}}|z|}{J_{\lambda}^{2}(\delta \sqrt{y})+Y_{\lambda}^{2}(\delta \sqrt{y})}+\lambda e^{-\gamma|z|}\right\} $$

$$ \dot{x}=r_{x} t+\eta_{\mathrm{G}}+\eta_{\leq 1}+\eta_{>1} $$

$$ \eta_{\leq 1}=\int_{|x| \leq 1} d z\left(e^{i p z}-1\right) \bar{F}(z) $$

$$ \eta_{>1}=\int_{|x| \leq 1} d z\left(e^{i p z}-1\right) \bar{F}(z) $$

$$ \tilde{D}_{Z}(z)=\int \frac{d p}{2 \pi} e^{i p z} e^{\tau_{Z}\left(e^{-i p Z}-1\right)} $$

$$ \tilde{D}_{Z}(z)=\int \frac{d p}{2 \pi} e^{i p z} \sum_{n=0}^{\infty} e^{-\tau_{Z}} \frac{\tau_{Z}^{n}}{n!} \tau_{Z}^{n} e^{-i p n Z}=\sum_{n=0}^{\infty} e^{-\tau_{Z}} \frac{\tau_{Z}^{n}}{n!} \delta(z-n Z) . $$

$$ P\left(n, \tau_{Z}\right)=e^{-\tau_{Z}} \frac{\tau_{Z}^{n}}{n!}, $$

$$ \left\langle n^{k}\right\rangle=\sum_{n=0}^{\infty} n^{k} P\left(n, \tau_{Z}\right)=e^{-\tau_{Z}}\left(\tau_{Z} \partial_{\tau_{z}}\right)^{k} e^{\tau_{Z}} \sum_{n=0}^{\infty} P\left(n, \tau_{Z}\right)=e^{-\tau_{Z}}\left(\tau_{Z} \partial_{\tau_{z}}\right)^{k} e^{\tau_{Z}}=\frac{\Gamma\left(\lambda_{Z}+k\right)}{\Gamma\left(\lambda_{Z}\right)} . $$

$$ \tilde{D}(z)=\int_{-1}^{1} d Z F(Z) \sum_{n=0}^{\infty} e^{-\tau_{Z}} \frac{\tau_{Z}^{n}}{n!} \delta(z-n Z) $$

$$ P\left(x_{c} t_{c} \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} d x_{b} P\left(x_{c} t_{c} \mid x_{b} t_{b}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ P\left(x_{c} t_{a} \mid x_{a} t_{a}\right)=\delta\left(x_{b}-x_{a}\right) . $$

$$ \begin{align*} \left\langle f\left(x\left(t_{b}\right), t_{b}\right)\right\rangle & =\int d x_{b} f\left(x_{b}, t_{b}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \\ & =\int d x\left[\int d x_{b} f\left(x_{b}, t_{b}\right) P\left(x_{b} t_{b} \mid x t\right)\right] P\left(x t \mid x_{a} t_{a}\right) \end{align*} $$

$$ \mathbb{E}\left[f\left(x\left(t_{b}\right), t_{b}\right) \mid x_{a} t_{a}\right] \equiv \int d x_{b} f\left(x_{b}, t_{b}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \int d x \mathbb{E}\left[f\left(x_{b}, t_{b}\right) \mid x t\right] P\left(x t \mid x_{a} t_{a}\right)=\mathbb{E}\left[\mathbb{E}\left[f\left(x_{b}, t_{b}\right) \mid x t\right] \mid x_{a} t_{a}\right] $$

$$ \mathbb{E}\left[f\left(x\left(t_{b}, t_{b}\right)\right) \mid x_{a} t_{a}\right]=\mathbb{E}\left[\mathbb{E}\left[f\left(x_{b}, t_{b}\right) \mid x t\right] \mid x_{a} t_{a}\right] . $$

$$ \mathbb{E}\left[f\left(x\left(t_{a}\right), t_{a}\right) \mid x_{a} t_{a}\right]=f\left(x_{a}, t_{a}\right) $$

$$ \left\langle x^{n}\right\rangle(t) \equiv \int_{-\infty}^{\infty} d x x^{n} P(x, t) $$

$$ \left\langle x^{n}\right\rangle(t)=\int_{\infty}^{\infty} \frac{d p}{2 \pi} e^{-t H(p)} \int_{-\infty}^{\infty} d x x^{n} e^{i p x}=\int_{\infty}^{\infty} d p e^{-t H(p)}\left(-i \partial_{p}\right)^{n} \delta(p) $$

$$ \left\langle x^{n}\right\rangle(t)=\left.\left(i \partial_{p}\right)^{n} e^{-t H(p)}\right|_{p=0} $$

$$ \left\langle x^{n}\right\rangle_{c}(t)=-t H^{(n)}(0)=t\left\langle x^{n}\right\rangle_{c}(1)=t c_{n} $$

$$ \left\langle x^{2}\right\rangle_{c}(t)=t 2 T^{2}, \quad \kappa(t)=\frac{c_{4}}{t c_{2}^{2}}=\frac{3}{t} . $$

$$ T=\sqrt{\left\langle x^{2}\right\rangle_{c}\left(t_{0}\right) / 2 t_{0}} . $$

$$ \begin{align*} \langle x\rangle(t) & =t\langle x\rangle_{c} \\ \left\langle x^{2}\right\rangle(t) & =t\left\langle x^{2}\right\rangle_{c}+t^{2}\langle x\rangle_{c}^{2} \\ \left\langle x^{3}\right\rangle(t) & =t\left\langle x^{3}\right\rangle_{c}+3 t^{2}\langle x\rangle_{c}\left\langle x^{2}\right\rangle_{c}+t^{3}\langle x\rangle_{c}^{3} \\ \left\langle x^{4}\right\rangle(t) & =t\left\langle x^{4}\right\rangle_{c}+3 t^{2}\left\langle x^{2}\right\rangle_{c}^{2}-4 t^{2}\langle x\rangle_{c}\left\langle x^{3}\right\rangle_{c}+6 t^{3}\langle x\rangle_{c}^{2}\left\langle x^{2}\right\rangle_{c}+t^{4}\langle x\rangle_{c}^{4} \\ & \vdots \end{align*} $$

$$ e^{-t H(p)}=\frac{1}{\left[1+(T p)^{2}\right]^{t}} $$

$$ P(x, t)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p x-t H(p)}=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} \frac{1}{\left(1+T^{2} p^{2}\right)^{t}} e^{i p x} $$

$$ e^{-t H(p)}=\frac{1}{\left[1+(T p)^{2}\right]^{t}}=\frac{1}{\Gamma(t)} \int_{0}^{\infty} \frac{d \tau}{\tau} \tau^{t} e^{-\tau\left(1+T^{2} p^{2}\right)} $$

$$ e^{-t H(p)}=\left(\frac{t}{2 T^{2}}\right)^{t} \frac{1}{\Gamma(t)} \int_{0}^{\infty} \frac{d v}{v} v^{t} e^{-t v / 2 T^{2}} e^{-t v p^{2} / 2} $$

$$ P(x, t)=\left(\frac{t}{2 T^{2}}\right)^{t} \frac{1}{\Gamma(t)} \int_{0}^{\infty} \frac{d v}{v} v^{t} e^{-t v / 2 T^{2}} \frac{1}{\sqrt{2 \pi t v}} e^{-x^{2} / 2 t v} $$

$$ P(x, t)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p x-t H(p)}=\frac{1}{T \sqrt{\pi} \Gamma(t)}\left(\frac{|x|}{2 T}\right)^{t-1 / 2} K_{t-1 / 2}(|x| / T) $$

$$ v^{t-1} e^{-t v / 2 T^{2}}=\left[2 T^{2}(1-1 / t)\right]^{t-1} e^{-(t-1)} e^{-t \delta v^{2} / 2\left[2 T^{2}(1-1 / t)\right]^{2}}\left[1+\mathcal{O}\left(\delta v^{3}\right)\right] $$

$$ e^{-t z^{2} / 2}\left[1+\frac{a}{3} z^{3}+\ldots\right]=\sqrt{2 \pi / t}\left[\delta(z)+\frac{1}{2 t} \delta^{\prime \prime}(z)+\frac{a}{t^{2}} \delta^{\prime}(z) \ldots\right] $$

$$ e^{-t \delta v^{2} / 2\left[2 T^{2}(1-1 / t)\right]^{2}} \rightarrow \sqrt{\frac{2 \pi}{t}} 2 T^{2}(1-1 / t) \delta(\delta v) $$

$$ v^{t-1} e^{-t v / 2 T^{2}} \rightarrow \sqrt{\frac{2 \pi}{t}}\left(2 T^{2}\right)^{t} e^{-t}\left\{\delta(\delta v)+\frac{\left[2 T^{2}(1-1 / t)\right]^{2}}{2 t} \delta^{\prime \prime}(\delta v)+\ldots\right\} $$

$$ \left(\frac{z}{2}\right)^{\nu} K_{\nu}(z)=\frac{\pi}{2 \sin \pi \nu \Gamma(1-\nu)}\left[1+\frac{\Gamma(1-\nu)}{1!\Gamma(2-\nu)} \frac{z^{2}}{4}+\mathcal{O}\left(z^{4}, z^{2 \nu}\right)\right] $$

$$ \left(\frac{z}{2}\right)^{\nu} K_{\nu}(z) \approx \frac{\Gamma(\nu)}{2} e^{-z^{2} / 4(t-3 / 2)} $$

$$ P(x, t) \underset{\text { small }|x|}{\approx} \frac{1}{\sqrt{2 \pi 2 T^{2} t}} e^{-x^{2} / 22 T^{2}(t-3 / 2)} $$

$$ e^{-t H_{\delta, \beta}(p)}=\left[1-\beta \delta p^{2} / 2\right]^{-t / \delta}=\frac{1}{\Gamma(t / \delta)} \int_{0}^{\infty} \frac{d s}{s} s^{t / \delta} e^{-s} e^{-s \beta \delta p^{2} / 2}, \beta \equiv \frac{\nu}{\mu}=\frac{1}{\mu \delta}, $$

$$ e^{-t H_{\delta, \beta}(p)}=\frac{\mu^{t / \delta}}{\Gamma(t / \delta)} \int_{0}^{\infty} \frac{d v}{v} v^{t / \delta} e^{-\mu v} e^{-v p^{2} / 2}, \quad \mu=1 / \beta \delta $$

$$ P_{\delta, \beta}(x, t)=\frac{\mu^{t / \delta}}{\Gamma(t / \delta)} \int_{0}^{\infty} \frac{d v}{v} v^{t / \delta} e^{-\mu v} \frac{1}{\sqrt{2 \pi v}} e^{-x^{2} / 2 v}, \quad \mu=1 / \beta \delta $$

$$ P_{\delta, \beta}(x, t)=\frac{\mu^{t / \delta}}{\Gamma(t / \delta)} \frac{1}{\sqrt{2 \pi}}\left(\frac{x^{2}}{2 \mu}\right)^{t / 2 \delta-1 / 4} 2 K_{t / \delta-1 / 2}(\sqrt{2 \mu} x), \quad \mu=\frac{1}{\beta \delta} $$

$$ P_{\delta, \beta}(0, t)=\sqrt{\mu} \Gamma(t / \delta-1 / 2) / \Gamma(t / \delta) $$

$$ e^{-H(p)}=\int_{0}^{\infty} d v w(v) e^{-v p^{2} / 2} $$

$$ e^{-t H(p)}=\int_{0}^{\infty} d v^{\prime} w_{t}\left(v^{\prime}\right) e^{-v^{\prime} p^{2} / 2}=\int_{0}^{\infty} d v \omega_{t}(v) e^{-t v p^{2} / 2} $$

$$ \omega_{t}(v) \equiv t w_{t}(v t) $$

$$ \int d v_{12} w_{t_{1}+t_{2}}\left(v_{12}\right) e^{-v_{12} p^{2} / 2}=\int d v_{2} w_{t_{2}}\left(v_{2}\right) e^{-v_{2} p^{2} / 2} \int d v_{1} w_{t_{1}}\left(v_{1}\right) e^{-v_{1} p^{2} / 2} $$

$$ \tilde{w}_{t}\left(p_{v}\right) \equiv \int d v e^{-p_{v} v} w_{t}(v) $$

$$ \tilde{w}_{t_{1}+t_{2}}\left(p_{v}\right)=\tilde{w}_{t_{2}}\left(p_{v}\right) \tilde{w}_{t_{1}}\left(p_{v}\right) $$

$$ \tilde{w}_{t}\left(p_{v}\right)=e^{-t H_{v}\left(p_{v}\right)}, $$

$$ w_{t}(v)=\int_{\gamma-i \infty}^{\gamma+i \infty} \frac{d p_{v}}{2 \pi i} e^{p_{v} v-t H_{v}\left(p_{v}\right)} $$

$$ e^{-t H(p)}=\int_{\gamma-i \infty}^{\gamma+i \infty} \frac{d p_{v}}{2 \pi i} \frac{1}{p^{2} / 2-p_{v}} e^{-t H_{v}\left(p_{v}\right)} $$

$$ H_{v}\left(p^{2} / 2\right)=H(p) $$

$$ \tilde{\omega}_{\beta}\left(p_{v}\right)=\int d v e^{-v p_{v}} \omega_{\beta}(v)=\int d v e^{-v p_{v}} \sqrt{\frac{\beta}{2 \pi v^{3}}} e^{-\beta / 2 v}=e^{-\sqrt{2 \beta p_{v}}} $$

$$ \int d v^{\prime} e^{-v^{\prime} p_{v}} w_{\beta}\left(v^{\prime}\right)=\int d v e^{-\beta v p_{v}} \omega_{\beta}(v)=\tilde{\omega}_{\beta}\left(\beta p_{v}\right) $$

$$ \omega_{t}(v)=\left.\lim _{k \rightarrow \infty} \frac{(-1)^{k}}{k!} x^{k+1} \frac{\partial^{k} \tilde{\omega}_{t}(x)}{\partial x^{k}}\right|_{x=k / v} $$

$$ \partial_{t} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=-H\left(-i \partial_{x}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \psi(x, t)=\int \mathcal{D} \eta \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right] \psi\left(x-\int_{t_{a}}^{t} d t^{\prime} \eta\left(t^{\prime}\right)\right) $$

$$ \begin{align*} \psi(x, t+\epsilon) & =\int \mathcal{D} \eta \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right] \psi\left(x-\int_{t_{a}}^{t} d t^{\prime} \eta\left(t^{\prime}\right)-\int_{t}^{t+\epsilon} d t^{\prime} \eta\left(t^{\prime}\right)\right) \\ & =\int \mathcal{D} \eta \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right]\left\{\psi\left(x-\int_{t_{a}}^{t} d t^{\prime} \eta\left(t^{\prime}\right)\right)\right. \\ & -\psi^{\prime}\left(x-\int_{t_{a}}^{t} d t^{\prime} \eta\left(t^{\prime}\right)\right) \int_{t}^{t+\epsilon} d t^{\prime} \eta\left(t^{\prime}\right) \\ & +\frac{1}{2} \psi^{\prime \prime}\left(x-\int_{t_{a}}^{t} d t^{\prime} \eta\left(t^{\prime}\right)\right) \int_{t}^{t+\epsilon} d t_{1} d t_{2} \eta\left(t_{1}\right) \eta\left(t_{2}\right) \\ & -\frac{1}{3!} \psi^{\prime \prime \prime}\left(x-\int_{t_{a}}^{t} d t^{\prime} \eta\left(t^{\prime}\right)\right) \int_{t}^{t+\epsilon} d t_{1} d t_{2} d t_{3} \eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \\ & \left.+\frac{1}{4!} \psi^{(4)}\left(x-\int_{t_{a}}^{t} d t^{\prime} \eta\left(t^{\prime}\right)\right) \int_{t}^{t+\epsilon} d t_{1} d t_{2} d t_{3} d t_{4} \eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \eta\left(t_{4}\right)+\ldots\right\} \end{align*} $$

$$ \begin{align*} \psi(x, t+\epsilon) & =\int \mathcal{D} \eta \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}(\eta(t))\right] \\ & \times\left[-\epsilon c_{1} \partial_{x}+\left(\epsilon c_{2}+\epsilon^{2} c_{1}\right) \frac{1}{2} \partial_{x}^{2}-\left(\epsilon c_{3}+3 \epsilon^{2} c_{2} c_{1}\right) \frac{1}{3!} \partial_{x}^{3}\right. \\ & \left.+\left(\epsilon c_{4}+\epsilon^{2} 4 c_{3} c_{1}+\epsilon^{2} 3 c_{2}^{2}+\epsilon^{3} c_{2} c_{1}^{2}+\epsilon^{4} c_{1}^{2}\right) \frac{1}{4!} \partial_{x}^{4}+\ldots\right] \psi\left(x-\int_{t_{a}}^{t} d t^{\prime} \eta\left(t^{\prime}\right)\right) \end{align*} $$

$$ \partial_{t} \psi(x, t)=\left[-c_{1} \partial_{x}+\frac{c_{2}}{2!} \partial_{x}^{2}-\frac{c_{3}}{3!} \partial_{x}^{3}+\frac{c_{4}}{4!} \partial_{x}^{4}+\ldots\right] \psi(x, t) $$

$$ \partial_{t} \psi(x, t)=-H_{r_{x}}\left(-i \partial_{x}\right) \psi(x, t) $$

$$ -H\left(-i \partial_{x}\right) \psi(x, t)=\int d x^{\prime} e^{-x^{\prime} \partial_{x}} F\left(x^{\prime}\right) \psi(x, t)=\int d x^{\prime} F\left(x^{\prime}\right) \psi\left(x-x^{\prime}, t\right) $$

$$ \partial_{t} \psi(x, t)=\left[-c_{1} \partial_{x}-\frac{c_{2}}{2} \partial_{x}^{2}\right] \psi(x, t)+\int d x^{\prime} \bar{F}\left(x^{\prime}\right) \psi\left(x-x^{\prime}, t\right) $$

$$ P\left(x_{c} t_{b}+\epsilon \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} d x_{b} P\left(x_{c} t_{b}+\epsilon \mid x_{b} t_{b}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ P\left(x_{c} t_{b}+\epsilon \mid x_{b} t_{b}\right)=\int d x \delta\left(x-x_{b}+x_{b}-x_{c}\right) P\left(x t_{b}+\epsilon \mid x_{b} t_{b}\right) $$

$$ P\left(x_{c} t_{b}+\epsilon \mid x_{b} t_{b}\right)=\int d x \sum_{n=0}^{\infty} \frac{\left(x-x_{b}\right)^{n}}{n!} P\left(x t_{b}+\epsilon \mid x_{b} t_{b}\right) \partial_{x_{b}}^{n} \delta\left(x_{b}-x_{c}\right) $$

$$ C_{n}\left(x_{b} t_{b}\right) \equiv \int d x\left(x-x_{b}\right)^{n} \partial_{t_{b}} P\left(x t_{b} \mid x_{b} t_{b}\right) $$

$$ P\left(x_{c} t_{b} \mid x_{b} t_{b}\right)+\epsilon \sum_{n=1}^{\infty} \int d x_{b} \frac{C_{n}\left(x_{b} t_{b}\right)}{n!}\left[\partial_{x_{b}}^{n} \delta\left(x_{b}-x_{c}\right)\right] \partial_{t_{b}} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)+\mathcal{O}\left(\epsilon^{2}\right) $$

$$ \partial_{t_{b}} P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=-H\left(-i \partial_{x_{b}}, x_{b}, t_{b}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ H\left(-i \partial_{x_{b}}, x_{b}, t_{b}\right) \equiv-\sum_{n=1}^{\infty}\left(-\partial_{x_{b}}\right)^{n} \frac{C_{n}\left(x_{b} t_{b}\right)}{n!} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\hat{T} e^{-\int_{t_{a}}^{t_{b}} d t H\left(-i \partial_{x_{b}}, x_{b}, t\right)} \delta\left(x_{b}-x_{a}\right) $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\left\langle x_{b}\right| \hat{T} e^{-\int_{t_{a}}^{t_{b}} d t H\left(-i \partial_{x_{b}}, x_{b}, t\right)}\left|x_{a}\right\rangle $$

$$ C_{n}(x t) \underset{\epsilon \rightarrow 0}{=} \frac{1}{\epsilon} \int d x\left(x^{\prime}-x\right)^{n} P\left(x^{\prime} t+\epsilon \mid x t\right) \underset{\epsilon \rightarrow 0}{=} \frac{1}{\epsilon}\left\langle[x(t+\epsilon)-x(t)]^{n}\right\rangle, n \geq 1 $$

$$ C_{n}(x t) \underset{\epsilon \rightarrow 0}{=} \frac{1}{\epsilon} \int_{t}^{t+\epsilon} d t_{1} \cdots \int_{t}^{t+\epsilon} d t_{n}\left\langle\dot{x}\left(t_{1}\right) \cdots \dot{x}\left(t_{n}\right)\right\rangle, \quad n \geq 1 $$

$$ C_{n}(x t)=c_{n} $$

$$ \dot{x}(t)=r(x, t)+\sigma(x, t) \eta(t) $$

$$ C_{1}(x t)=a(x, t)+b^{\prime}(x, t) b(x, t)=a(x, t)+\frac{\partial_{x} C_{2}(x t)}{2}, C_{2}(x t)=b^{2}(x, t) $$

$$ \begin{align*} f(x(t+\epsilon))=f(x(t)) & +f^{\prime}(x(t)) \Delta x(t) \\ & +\frac{1}{2} f^{\prime \prime}(x(t))[\Delta x(t)]^{2}+\frac{1}{3!} f^{(3)}[\Delta x(t)]^{3}+\ldots \end{align*} $$

$$ \begin{align*} \langle f(x(t+\epsilon))\rangle= & \langle f(x(t))\rangle+\left\langle f^{\prime}(x(t))\right\rangle \epsilon c_{1}+\frac{1}{2}\left\langle f^{\prime \prime}(x(t))\right\rangle\left(\epsilon c_{2}+\epsilon^{2} c_{1}^{2}\right) \\ & +\frac{1}{3!}\left\langle f^{(3)}(x(t))\right\rangle\left(\epsilon c_{3}+\epsilon^{2} c_{2} c_{1}+\epsilon^{3} c_{1}^{3}\right)+\ldots \\ = & \langle f(x(t))\rangle+\epsilon\left[c_{1} \partial_{x}+c_{2} \frac{1}{2} \partial_{x}^{2}+c_{3} \frac{1}{3!} \partial_{x}^{3}+\ldots\right]\langle f(x(t))\rangle+\mathcal{O}\left(\epsilon^{2}\right) \end{align*} $$

$$ \langle\dot{f}(x(t))\rangle=-H_{r_{x}}\left(i \partial_{x}\right)\langle f(x(t))\rangle $$

$$ \langle\dot{f}(x(t))\rangle=\left\langle f^{\prime}(x(t))\right\rangle\langle\dot{x}(t)\rangle-\bar{H}_{r_{x}}\left(i \partial_{x}\right)\langle f(x(t))\rangle $$

$$ \dot{f}(x(t))=f^{\prime}(x(t)) \dot{x}(t)-\bar{H}_{r_{x}}\left(i \partial_{x}\right) f(x(t)), $$

$$ \dot{x}(t)=\langle\dot{x}(t)\rangle+\eta(t)=c_{1}+\eta(t) $$

$$ \begin{align*} \left\langle\eta\left(t_{1}\right)\right\rangle & =c_{1} \\ \left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right)\right\rangle & =c_{2} \delta\left(t_{1}-t_{2}\right) \\ \left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right)\right\rangle & =c_{3} \delta\left(t_{1}-t_{2}\right) \delta\left(t_{1}-t_{3}\right) \\ \left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \eta\left(t_{4}\right)\right\rangle & =c_{4} \delta\left(t_{1}-t_{2}\right) \delta\left(t_{1}-t_{3}\right) \delta\left(t_{1}-t_{4}\right) \\ & +c_{2}^{2}\left[\delta\left(t_{1}-t_{2}\right) \delta\left(t_{3}-t_{4}\right)+\delta\left(t_{1}-t_{3}\right) \delta\left(t_{2}-t_{4}\right)+\delta\left(t_{1}-t_{4}\right) \delta\left(t_{2}-t_{3}\right)\right] \\ & \vdots \end{align*} $$

$$ \left\langle\left[z_{2,2}(t)\right]^{2}\right\rangle=\int_{t}^{t+\epsilon} d t_{1} \int_{t}^{t+\epsilon} d t_{2} \int_{t}^{t+\epsilon} d t_{3} \int_{t}^{t+\epsilon} d t_{4}\left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \eta\left(t_{4}\right)\right\rangle=\epsilon c_{4}+\epsilon^{2} c_{2}^{2} $$

$$ [\Delta x(t)]^{3}=\epsilon^{3} c_{1}^{3}+3 \epsilon^{2} c_{1}^{2} z_{1}(t)+\epsilon c_{1}\left[z_{1}(t)\right]^{2}+\left[z_{1}(t)\right]^{3} $$

$$ \int_{t}^{t+\epsilon} d t_{1} \int_{t}^{t+\epsilon} d t_{2} \int_{t}^{t+\epsilon} d t_{3} \int_{t}^{t+\epsilon} d t_{4} \int_{t}^{t+\epsilon} d t_{5} \int_{t}^{t+\epsilon} d t_{6}\left\langle\eta\left(t_{1}\right) \eta\left(t_{2}\right) \eta\left(t_{3}\right) \eta\left(t_{4}\right) \eta\left(t_{5}\right) \eta\left(t_{6}\right)\right\rangle $$

$$ \left\langle\left\{\left[z_{1}(t)\right]^{3}\right\}^{2}\right\rangle=\epsilon c_{6}+\mathcal{O}\left(\epsilon^{2}\right) $$

$$ \frac{d}{d t}\left\langle e^{P x(t)}\right\rangle=\left[P\langle\dot{x}(t)\rangle-\bar{H}_{r_{x}}(i P)\right]\left\langle e^{P x(t)}\right\rangle $$

$$ r_{S}=r_{x}-\bar{H}(i)=r_{x}-\left[H(i)-i H^{\prime}(0)\right]=-H_{r_{x}}(i) $$

$$ \left\langle\frac{\dot{S}}{S}\right\rangle=\langle\dot{x}(t)\rangle-\bar{H}(i)=\langle\dot{x}(t)\rangle-\left[H(i)-i H^{\prime}(0)\right]=\langle\dot{x}(t)\rangle-r_{x}-H_{r_{x}}(i) $$

$$ \langle S(t)\rangle=S(0) e^{r_{S} t}=S(0)\left\langle e^{r_{x} t+\int_{0}^{t} d t^{\prime} \eta\left(t^{\prime}\right)}\right\rangle=S(0) e^{-H_{r_{x}}(i) t}=S(0) e^{\left\{r_{x} t-\left[H(i)-i H^{\prime}(0)\right] t\right\}} $$

$$ \langle S(t)\rangle=S(0) e^{r_{S} t}=S(0)\left\langle e^{\int_{0}^{t} d t^{\prime} \eta\left(t^{\prime}\right)}\right\rangle=S(0) e^{-H_{r_{x}}(i) t} $$

$$ \left\langle e^{P \int_{0}^{t} d t^{\prime} \eta\left(t^{\prime}\right)}\right\rangle=e^{-H_{r_{x}}(i P) t} $$

$$ \begin{align*} \langle f(x(t+d t))\rangle & =\langle f(x(t))\rangle+\left\langle f^{\prime}(x(t))\right\rangle\langle\dot{x}(t)\rangle d t+\frac{1}{2}\left\langle f^{\prime \prime}(x(t))\right\rangle\left\langle\dot{x}^{2}(t)\right\rangle d t^{2} \\ & +\frac{1}{3!}\left\langle f^{(3)}(x(t))\right\rangle\left\langle\dot{x}^{3}(t)\right\rangle d t^{3}+\ldots \end{align*} $$

$$ \langle\dot{x}(t)\rangle d t \rightarrow c_{1} d t, \quad\left\langle\dot{x}^{2}(t)\right\rangle d t^{2} \rightarrow c_{2} d t, \quad\left\langle\dot{x}^{3}(t)\right\rangle d t^{3} \rightarrow c_{3} d t, \ldots . $$

$$ \Delta f\left(x\left(t_{n}\right)\right)=f^{\prime}\left(x\left(t_{n}\right)\right) \Delta x\left(t_{n}\right)-\bar{H}_{r_{x}}\left(i \partial_{x}\right) f\left(x\left(t_{n}\right)\right)+\mathcal{O}(\sigma \sqrt{\Delta t}) $$

$$ e^{-r_{S} t} S(t)=e^{-r_{S} t} e^{x(t)}=e^{-r_{S} t} e^{r_{x} t+\int^{t} d t^{\prime} \eta\left(t^{\prime}\right)} $$

$$ H(p)=i r_{x} p+\frac{\sigma^{2}}{2} p^{2} $$

$$ P^{r_{x}}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} \exp \left[i p\left(x_{b}-x_{a}\right)-\Delta t\left(i r_{x} p+\sigma^{2} p^{2} / 2\right)\right] $$

$$ e^{-r_{S} t_{b}} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} \delta(p-i) e^{-i p x_{a}} e^{\Delta t \sigma^{2} p^{2} / 2}=e^{-r_{S} t} e^{x_{a}} e^{\left(r_{x}-\sigma^{2} / 2\right) \Delta t}=e^{-r_{S} t_{a}} e^{x_{a}} $$

$$ \left\langle e^{-r_{S} t_{b}} S\left(t_{b}\right)\right\rangle^{r_{x}}=\left\langle e^{-r_{S} t_{b}} e^{x\left(t_{b}\right)}\right\rangle^{r_{x}}=e^{-r_{S} t_{a}} S\left(t_{a}\right) $$

$$ \mathbb{E}\left[e^{-r_{S}\left(t_{b}-t_{a}\right)} S\left(t_{b}\right) \mid x_{a} t_{a}\right]=S\left(t_{a}\right) $$

$$ \mathbb{E}\left[\mathbb{E}\left[f\left(x_{b}, t_{b}\right) \mid x t\right] \mid x_{a} t_{a}\right]=\mathbb{E}\left[\mathbb{E}[f(x, t) \mid x t] \mid x_{a} t_{a}\right]=\mathbb{E}\left[f(x, t) \mid x_{a} t_{a}\right] $$

$$ \mathbb{E}\left[\mathbb{E}\left[f\left(x_{b}, t_{b}\right) \mid x t\right] \mid x_{a} t_{a}\right]=f\left(x_{a}, t_{a}\right) $$

$$ P^{r_{x}}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \equiv \frac{1}{\sqrt{2 \pi \sigma^{2}\left(t_{b}-t_{a}\right)}} \exp \left\{-\frac{\left[x_{b}-x_{a}-r_{x}\left(t_{b}-t_{a}\right)\right]^{2}}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\right\} $$

$$ P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r_{S}\left(t_{b}-t_{a}\right)} P^{r_{x}}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \left\langle f\left(x_{b}\right)\right\rangle^{\left(M, r_{x}\right)} \equiv \int d x_{b} f\left(x_{b}\right) P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \left\langle e^{x_{b}}\right\rangle^{\left(M, r_{x}\right)}=e^{x_{a}} $$

$$ P^{(M, r)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r \Delta t} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} \exp \left[i p\left(x_{b}-x_{a}\right)-\Delta t H_{r}(p)\right] $$

$$ P^{(M, r)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \equiv \frac{e^{-r\left(t_{b}-t_{a}\right)}}{\sqrt{2 \pi \sigma^{2}\left(t_{b}-t_{a}\right)}} \exp \left\{-\frac{\left[x_{b}-x_{a}-r\left(t_{b}-t_{a}\right)\right]^{2}}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\right\} $$

$$ e^{-r_{S} t} S(t)=e^{-r_{S} t} e^{r_{x} t+\int_{0}^{t} d t^{\prime} \eta\left(t^{\prime}\right)} $$

$$ P^{r_{x}}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int \mathcal{D} \eta \int \mathcal{D} x \exp \left\{-\int_{t_{a}}^{t_{b}} d t \tilde{H}_{r_{x}}(\eta(t))\right\} \delta\left[\dot{x}-r_{x}\left(t_{b}-t_{a}\right) \eta\right] $$

$$ P^{r_{x}}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} \exp \left[i p\left(x_{b}-x_{a}\right)-\Delta t H_{r_{x}}(p)\right] $$

$$ P^{(M, r)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r \Delta t} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} \exp \left[i p\left(x_{b}-x_{a}\right)-\Delta t H_{r_{x}}(p)\right] $$

$$ \tilde{D}(z)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{-H(p)} e^{i p z} $$

$$ D^{\theta}(z) \equiv e^{H(i \theta)} e^{\theta z} \tilde{D}(z) $$

$$ D^{\theta}(z)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{-H^{\theta}(p)} e^{i p z} $$

$$ H^{\theta}(p) \equiv H(p+i \theta)-H(i \theta) $$

$$ \langle F(z)\rangle^{\theta} \equiv \int d z D^{\theta}(z) F(z) $$

$$ \langle F(z)\rangle^{\theta} \equiv e^{H(i \theta)}\left\langle e^{\theta z} F(z)\right\rangle $$

$$ \left\langle e^{z}\right\rangle^{\theta}=e^{-H^{\theta}(i)} \equiv e^{H^{\theta}(i \theta)}\left\langle e^{(\theta+1) z}\right\rangle=e^{H^{\theta}(i \theta)-H^{\theta}(i \theta+i)} $$

$$ P^{\theta}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r_{S} \Delta t} e^{H_{r_{x}}(i \theta) \Delta t} \int \mathcal{D} \eta \int \mathcal{D} x \exp \left\{\int_{t_{a}}^{t_{b}} d t\left[\theta \eta(t)-\tilde{H}_{r_{x}}(\eta(t))\right]\right\} \delta[\dot{x}-\eta] $$

$$ P^{\theta}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r_{S} \Delta t} e^{H_{r_{x}}(i \theta) \Delta t} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} \exp \left[i p\left(x_{b}-x_{a}\right)-\Delta t H_{r_{x}}(p+i \theta)\right] $$

$$ \langle S(t)\rangle^{\theta}=e^{-H_{r_{x}}^{\theta}(i) t} $$

$$ r^{\theta} \equiv-H_{r_{x}}^{\theta}(i)=-H_{r_{x}}(i+i \theta)+H_{r_{x}}(i \theta) $$

$$ P^{M \theta}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \equiv e^{-r^{\theta} t} P^{\theta}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ \dot{x}(t)=-\frac{v(t)}{2}+\sqrt{v(t)} \eta(t) $$

$$ \langle\eta(t)\rangle=0, \quad\left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle=\delta\left(t-t^{\prime}\right) $$

$$ \dot{v}(t)=-\gamma[v(t)-\bar{v}]+\varepsilon \sqrt{v(t)} \eta_{v}(t) $$

$$ \eta_{v}(t) \equiv \rho \eta(t)+\sqrt{1-\rho^{2}} \eta^{\prime}(t) . $$

$$ \left\langle\eta(t) \eta\left(t^{\prime}\right)\right\rangle=\delta\left(t-t^{\prime}\right), \quad\left\langle\eta(t) \eta_{v}\left(t^{\prime}\right)\right\rangle=\rho \delta\left(t-t^{\prime}\right), \quad\left\langle\eta_{v}(t) \eta_{v}\left(t^{\prime}\right)\right\rangle=\delta\left(t-t^{\prime}\right) . $$

$$ P_{\boldsymbol{\eta}}\left(x v t \mid x_{a} v_{a} t_{a}\right)=\delta\left(x_{\eta}(t)-x\right) \delta\left(v_{\eta_{v}}(t)-v\right) . $$

$$ \partial_{t} P_{\boldsymbol{\eta}}\left(x v t \mid x_{a} v_{a} t_{a}\right)=-\left[\partial_{x} \dot{x}_{\eta}(t)+\partial_{v} \dot{v}_{\eta_{v}}(t)\right] P_{\boldsymbol{\eta}}\left(x v t \mid x_{a} v_{a} t_{a}\right) $$

$$ \begin{align*} \partial_{t} P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right) & =\left\{\partial_{x}\left[\frac{v(t)}{2}-\sqrt{v(t)} \eta(t)\right]\right. \\ & \left.+\partial_{v}\left[\gamma(v(t)-\bar{v})-\varepsilon \sqrt{v(t)} \eta_{v}(t)\right]\right\} P_{\eta}\left(x v t \mid x_{a} v_{a} t_{a}\right) \end{align*} $$

$$ P[\boldsymbol{\eta}]=\exp \left\{-\frac{1}{2} \int d t\left[\eta^{2}(t)+\eta^{\prime 2}(t)\right]\right\}=\exp \left\{-\frac{1}{1-\rho^{2}} \int d t\left[\eta^{2}(t)+\eta_{v}^{2}(t)-2 \rho \eta \eta_{v}\right]\right\} $$

$$ \begin{align*} \eta(t) & \rightarrow-\delta / \delta \eta(t)-\rho \delta / \delta \eta_{v}(t) \\ \eta_{v}(t) & \rightarrow-\rho \delta / \delta \eta(t)-\delta / \delta \eta_{v}(t) \end{align*} $$

$$ \frac{\delta}{\delta \eta\left(t^{\prime}\right)} \delta\left(x_{\eta}(t)-x\right) \delta\left(v_{\eta_{v}}(t)-v\right)=-\left[\frac{\delta x_{\eta}(t)}{\delta \eta\left(t^{\prime}\right)} \partial_{x}+\frac{\delta v_{\eta_{v}}(t)}{\delta \eta\left(t^{\prime}\right)} \partial_{v}\right] \delta\left(x_{\eta}(t)-x\right) \delta\left(v_{\eta_{v}}(t)-v\right) $$

$$ \partial_{t} P\left(x v t \mid x_{a} v_{a} t_{a}\right)=-\hat{H} P\left(x v t \mid x_{a} v_{a} t_{a}\right) $$

$$ \hat{H}=-\frac{1}{2} \partial_{x}^{2} v-\frac{1}{2} \partial_{x} v-\frac{\varepsilon^{2}}{2} \partial_{v}^{2} v-\gamma \partial_{v}(v-\bar{v})-\rho \varepsilon \partial_{x} \partial_{v} v $$

$$ \begin{align*} P\left(x_{b} v_{b} t_{b} \mid x_{a} v_{a} t_{a}\right) & =\int \mathcal{D} x \int \frac{\mathcal{D} p}{2 \pi} \int \mathcal{D} v \int \frac{\mathcal{D} p_{v}}{2 \pi} \\ & \times \exp \left\{\int_{t_{a}}^{t_{b}} d t\left[i\left(p \dot{x}+p_{v} \dot{v}\right)-H\left(p, p_{v}, v\right)\right]\right\} \end{align*} $$

$$ H\left(p, p_{v}, v\right)=\frac{p^{2}}{2} v-i \frac{1}{2} p v+\varepsilon^{2} \frac{p_{v}^{2}}{2} v-i \gamma p_{v}(v-\bar{v})+\rho \varepsilon p p_{v} v-\frac{3 \varepsilon^{2}}{4} i p_{v}-\frac{\gamma}{2}-\frac{i}{2} \rho \epsilon p . $$

$$ \frac{\varepsilon^{2}}{2 \gamma \bar{v}} \leq 1 $$

$$ \left(\partial_{t}+\alpha \partial_{v}\right) P\left(x v t \mid x_{a} v_{a} t_{a}\right)=\left(\gamma+\rho \varepsilon \partial_{x}\right) P\left(x v t \mid x_{a} v_{a} t_{a}\right) $$

$$ P\left(x v t \mid x_{a} v_{a} t_{a}\right)=f(v-\alpha t, x), $$

$$ P\left(v t \mid v_{a} t_{a}\right) \equiv \int d x P\left(x v t \mid x_{a} v_{a} t_{a}\right) $$

$$ \frac{\partial}{\partial t} P\left(v t \mid v_{a} t_{a}\right)=\left[\frac{\varepsilon^{2}}{2} \partial_{v}^{2} v+\gamma \partial_{v}(v-\bar{v})\right] P\left(v t \mid v_{a} t_{a}\right) $$

$$ P^{*}(v)=\frac{\mu^{\nu}}{\Gamma(\nu)} v^{\nu-1} e^{-\mu v}, \quad \text { where } \quad \mu \equiv \frac{2 \gamma}{\varepsilon^{2}}, \quad \nu \equiv \mu \bar{v} $$

$$ \frac{v_{\max }}{w}=\sqrt{\frac{2 \gamma \bar{v}}{\varepsilon^{2}}-1} $$

$$ P\left(x v t \mid x_{a} v_{a} t_{a}\right)=\int \frac{d p}{2 \pi} e^{i p\left(x-x_{a}\right)} \bar{P}_{p}\left(v t \mid v_{a} t_{a}\right) . $$

$$ \partial_{t} \bar{P}_{p}\left(v t \mid v_{a} t_{a}\right)=\left[\gamma \frac{\partial}{\partial v}(v-\bar{v})-\frac{p^{2}-i p}{2} v-i \rho \varepsilon p \frac{\partial}{\partial v} v-\frac{\varepsilon^{2}}{2} \frac{\partial^{2}}{\partial v^{2}} v\right] \bar{P}_{p}\left(v t \mid v_{a} t_{a}\right) . $$

$$ \tilde{P}_{p}\left(v t \mid v_{a} t_{a}\right)=\int \frac{d p_{v}}{2 \pi} e^{i p_{v} v} \tilde{P}_{p}\left(p_{v} t \mid v_{a} t_{a}\right) $$

$$ \left[\frac{\partial}{\partial t}+\left(\Gamma p_{v}+\frac{i \varepsilon^{2}}{2} p_{v}^{2}+\frac{i p^{2}+p}{2}\right) \frac{\partial}{\partial p_{v}}\right] \tilde{P}_{p}\left(p_{v} t \mid v_{a} t_{a}\right)=-i \gamma \bar{v} p_{v} \tilde{P}_{p}\left(p_{v} t \mid v_{a} t_{a}\right), $$

$$ \Gamma(p) \equiv \gamma+i \rho \varepsilon p $$

$$ \tilde{P}_{p}\left(p_{v} t_{a} \mid v_{a} t_{a}\right)=e^{-i p_{v} v_{a}} $$

$$ \tilde{P}_{p}\left(p_{v} t \mid v_{a} t_{a}\right)=\exp \left[-i \tilde{p}_{v}\left(t_{a}\right) v_{a}-i \gamma \bar{v} \int_{t_{a}}^{t} d t^{\prime} \tilde{p}_{v}\left(t^{\prime}\right)\right] $$

$$ \frac{d \tilde{p}_{v}(t)}{d t}=\Gamma(p) \tilde{p}_{v}(t)+\frac{i \varepsilon^{2}}{2} \tilde{p}_{v}^{2}(t)+\frac{i}{2}\left(p^{2}-i p\right) $$

$$ \tilde{p}_{v}(t)=-i \frac{2 \Omega(p)}{\varepsilon^{2}} \frac{1}{\zeta\left(p, p_{v}\right) e^{\Omega(p)\left(t_{b}-t\right)}-1}+i \frac{\Gamma(p)-\Omega(p)}{\varepsilon^{2}} $$

$$ \Omega(p)=\sqrt{\Gamma^{2}(p)+\varepsilon^{2}\left(p^{2}-i p\right)} $$

$$ \zeta\left(p, p_{v}\right)=1-i \frac{2 \Omega(p)}{\varepsilon^{2} p_{v}-i[\Gamma(p)-\Omega(p)]} $$

$$ \begin{align*} & P\left(x v t \mid x_{a} v_{a} t_{a}\right)=\iint_{-\infty}^{+\infty} \frac{d p}{2 \pi} \frac{d p_{v}}{2 \pi} e^{i p x+i p_{v} v} \\ & \quad \times \exp \left\{-i \tilde{p}_{v}\left(t_{a}\right) v_{a}+\frac{\gamma \bar{v}[\Gamma(p)-\Omega(p)]}{\varepsilon^{2}} \Delta t-\frac{2 \gamma \bar{v}}{\varepsilon^{2}} \ln \frac{\zeta\left(p, p_{v}\right)-e^{-\Omega(p) \Delta t}}{\zeta\left(p, p_{v}\right)-1}\right\},( \end{align*} $$

$$ P\left(x t \mid x_{a} v_{a} t_{a}\right) \equiv \int_{-\infty}^{+\infty} d v P\left(x v t \mid x_{a} v_{a} t_{a}\right) $$

$$ P\left(x t \mid x_{a} v_{a} t_{a}\right)=\int_{-\infty}^{+\infty} \frac{d p}{2 \pi} e^{i p x-\frac{p^{2}-i p}{\Gamma+\Omega \operatorname{coth}(\Omega \Delta t / 2)} v_{a}+\frac{\gamma \Gamma \bar{v}}{\varepsilon^{2}} \Delta t-\frac{2 \gamma \bar{v}}{\varepsilon^{2}} \ln \left(\cosh \frac{\Omega \Delta t}{2}+\frac{\Gamma}{\Omega} \sinh \frac{\Omega \Delta t}{2}\right)}, $$

$$ v(t)=\bar{v}+\left(v_{a}-\bar{v}\right) e^{-\gamma \Delta t} $$

$$ P\left(x t \mid x_{a} t_{a} v_{a}\right)=\frac{1}{\sqrt{2 \pi\left(t-t_{a}\right) \bar{v}(t)}} \exp \left\{-\frac{\left[x+\bar{v}(t)\left(t-t_{a}\right) / 2\right]^{2}}{2 \bar{v}(t)}\right\}, $$

$$ \bar{v}(t) \equiv \frac{1}{\Delta t} \int_{t_{a}}^{t} d t^{\prime} v\left(t^{\prime}\right) $$

$$ P\left(x t \mid x_{a} t_{a}\right)=\int_{0}^{\infty} d v_{a} P^{*}\left(v_{a}\right) P\left(x t \mid x_{a} t_{a} v_{a}\right) $$

$$ P\left(x t \mid x_{a} t_{a}\right)=\int_{-\infty}^{+\infty} \frac{d p}{2 \pi} e^{i p \Delta x-\Delta t H(p, \Delta t)} $$

$$ H(p, \Delta t)=-\frac{\gamma \Gamma(p) \bar{v}}{\varepsilon^{2}}+\frac{2 \gamma \bar{v}}{\varepsilon^{2} \Delta t} \ln \left[\cosh \frac{\Omega(p) \Delta t}{2}+\frac{\Omega^{2}(p)-\Gamma^{2}(p)+2 \gamma \Gamma(p)}{2 \gamma \Omega(p)} \sinh \frac{\Omega(p) \Delta t}{2}\right] $$

$$ c_{1}(\Delta t)=-\frac{\bar{v}}{2}-\frac{\rho}{\epsilon \Delta t}\left(1-e^{-\gamma \Delta t}\right) $$

$$ \bar{H}(p, \Delta t) \equiv H(p, \Delta t)-i c_{1}(\Delta t) p=H(p, \Delta t)+i\left[\frac{\bar{v}}{2}+\frac{\rho}{\epsilon \Delta t}\left(1-e^{-\gamma \Delta t}\right)\right] p $$

$$ H(p, \Delta t) \approx \frac{\gamma \bar{v}}{\varepsilon^{2}}[\Omega(p)-\Gamma(p)] $$

$$ p=C \tilde{p}+i p_{0} $$

$$ C \equiv \frac{\omega_{0}}{\varepsilon \sqrt{1-\rho^{2}}}, \quad \omega_{0} \equiv \sqrt{\gamma^{2}+\varepsilon^{2}\left(1-\rho^{2}\right) p_{0}^{2}}, \quad p_{0} \equiv \frac{\varepsilon-2 \rho \gamma}{2 \varepsilon\left(1-\rho^{2}\right)} $$

$$ P\left(x t \mid x_{a} t_{a}\right)=\frac{C}{\pi} e^{-p_{0} \Delta x+\Lambda \Delta t} \int_{0}^{\infty} d \tilde{p} \cos (A \tilde{p}) e^{-B \sqrt{1+\tilde{p}^{2}}} $$

$$ A=C\left(\Delta x+\rho \frac{\gamma \bar{v}}{\varepsilon} \Delta t\right), \quad B=\frac{\gamma \bar{v} \omega_{0}}{\varepsilon^{2}} \Delta t $$

$$ \Lambda=\frac{\gamma \bar{v}}{2 \varepsilon^{2}} \frac{2 \gamma-\rho \varepsilon}{1-\rho^{2}} $$

$$ P\left(x t \mid x_{a} t_{a}\right)=N(\Delta t) e^{-p_{0} \Delta x} F^{*}(y), \quad F^{*}(y)=K_{1}(y) / y, $$

$$ y \equiv \sqrt{A^{2}+B^{2}}=\frac{\omega_{0}}{\varepsilon} \sqrt{\frac{(\Delta x+\rho \gamma \bar{v} \Delta t / \varepsilon)^{2}}{1-\rho^{2}}+\left(\frac{\gamma \bar{v} \Delta t}{\varepsilon}\right)^{2}} \text {, } $$

$$ N(\Delta t)=\frac{\omega_{0}^{2} \gamma \bar{v}}{\pi \varepsilon^{3} \sqrt{1-\rho^{2}}} \Delta t e^{\Lambda \Delta t} $$

$$ \ln \frac{P\left(x t \mid x_{a} t_{a}\right)}{N(\Delta t)} \approx-p_{0} \Delta x-y . $$

$$ \ln \frac{P\left(x t \mid x_{a} t_{a}\right)}{N(\Delta t)} \approx-p_{0} \Delta x-c|\Delta x| . $$

$$ \ln \frac{P\left(x t \mid x_{a} t_{a}\right)}{N^{\prime}(\Delta t)} \approx-p_{0} \Delta x-\frac{\omega_{0}(\Delta x+\rho \gamma \bar{v} \Delta t / \varepsilon)^{2}}{2\left(1-\rho^{2}\right) \gamma \bar{v} \Delta t}, $$

$$ \sigma^{2}=\frac{\left(1-\rho^{2}\right) \gamma \bar{v}}{\omega_{0}} \Delta t $$

$$ \Delta x_{m}(t)=\Delta r_{S} \Delta t, \quad \text { with } \quad \Delta r_{S} \equiv-\frac{\gamma \bar{v}}{2 \omega_{0}}\left[1+2 \frac{\rho\left(\omega_{0}-\gamma\right)}{\varepsilon}\right] $$

$$ \bar{H}(p, \Delta t) \approx \frac{2 \gamma \bar{v}}{\varepsilon^{2} \Delta t} \log \left(p-p_{1}^{ \pm}\right) $$

$$ q_{*}^{ \pm}= \pm p_{0}+\frac{\omega_{0}}{\varepsilon \sqrt{1-\rho^{2}}} \quad \text { for } \quad \gamma \Delta t \gg 1 $$

$$ q^{ \pm}(\Delta t) \approx \pm p_{0}+\sqrt{p_{0}^{2}+\frac{4 \gamma}{\varepsilon^{2}\left(1-\rho^{2}\right) \Delta t}} \text { for } \gamma \Delta t \ll 1 $$

$$ \bar{P}_{p}\left(v_{b}, t_{b} \mid v_{a} t_{a}\right)=\int \mathcal{D} v \frac{\mathcal{D} p_{v}}{2 \pi} e^{\mathcal{A}_{p}\left[p_{v}, v\right]} $$

$$ \mathcal{A}_{p}\left[p_{v}, v\right]=\int_{t_{a}}^{t_{b}} d t\left[i p_{v} \dot{v}-H\left(p, p_{v}, v\right)\right] $$

$$ \begin{align*} \mathcal{A}_{p}\left[p_{v}, v\right]= & i\left[p_{v}\left(t_{b}\right) v_{b}-p_{v}\left(t_{a}\right) v_{a}\right]-i \gamma \bar{v} \int_{t_{a}}^{t_{b}} d t p_{v}(t)\left(t_{b}-t_{a}\right) \\ & -\int_{t_{a}}^{t_{b}} d t\left[i \dot{p}_{v}(t)+\frac{\partial H}{\partial v(t)}\right] v(t) \end{align*} $$

$$ \bar{P}_{p}\left(v_{b}, t_{b} \mid v_{a} t_{a}\right)=\int_{-\infty}^{+\infty} \frac{d p_{v}}{2 \pi} J e^{i\left[p_{v} v-\tilde{p}_{v}\left(t_{a}\right) v_{a}\right]-i \gamma \theta \int_{0}^{t} d t \tilde{p}_{v}(t)+(\gamma-i \rho \epsilon p)\left(t_{b}-t_{a}\right) / 2} $$

$$ J=\operatorname{Det}^{-1}\left(i \partial_{t}+\frac{\partial^{2} H\left(p, p_{v}, v\right)}{\partial p_{v} \partial v}\right) $$

$$ \frac{\partial^{2} H\left(p, p_{v}, v\right)}{\partial p_{v} \partial v}=-i \gamma+\rho \epsilon p $$

$$ J=e^{-(\gamma-i \rho \epsilon p)\left(t_{b}-t_{a}\right) / 2} $$

$$ H^{\mathrm{tot}}(p, \Delta t)=H(p, \Delta t)+i r_{S} p $$

$$ H(i, \Delta t) \approx \frac{\bar{v}}{\epsilon \Delta t}\left(1-e^{-\gamma \Delta t}\right) \rho+\frac{\bar{v}}{4 \gamma \Delta t}\left[3-2 e^{-\gamma \Delta t}(1+2 \gamma \Delta t)-e^{-2 \gamma \Delta t}\right] \rho^{2} $$

$$ \bar{H}(i, \Delta t)=H(i, \Delta t)+c_{1}(\Delta t) . $$

$$ P^{\left(M, r_{S}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r(\Delta t) \Delta t} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} \exp \left[i p\left(x_{b}-x_{a}\right)-\Delta t H_{r_{x}(\Delta t)}^{\mathrm{tot}}(p, \Delta t)\right] . $$

$$ r_{x}(\Delta t)=r_{S}+c_{1}(\Delta t)+\bar{H}(i, \Delta t) $$

$$ r(\Delta t)=r_{S}+c_{1}(\Delta t) $$

$$ x\left(t_{n}\right)=\sqrt{v\left(t_{n-1}\right)} \eta\left(t_{n}\right), \quad v\left(t_{n}\right)=v_{0}+\sum_{k=1}^{q} \alpha_{k}\left(t_{n-k}\right) x^{2}\left(t_{n-k}\right) $$

$$ v\left(t_{n}\right)=v_{0}+\sum_{k=1}^{q} \alpha_{k}\left(t_{n-k}\right) x^{2}\left(t_{n-k}\right)+\sum_{k=1}^{p} \beta_{k}\left(t_{n-k}\right) v\left(t_{n-k}\right) . $$

$$ \sigma^{2}=\left\langle v\left(t_{n}\right)\right\rangle=\frac{v_{0}}{1-\alpha_{1}}, \quad \kappa=\frac{\left\langle x^{4}\left(t_{n}\right)\right\rangle_{c}}{\left\langle x^{2}\left(t_{n}\right)\right\rangle_{c}^{2}}=\frac{6 \alpha_{1}^{2}}{1-3 \alpha_{1}^{2}} $$

$$ \sigma^{2}=\left\langle v\left(t_{n}\right)\right\rangle=\frac{v_{0}}{1-\alpha_{1}-\beta_{1}}, \quad \kappa=\frac{\left\langle x^{4}\left(t_{n}\right)\right\rangle_{c}}{\left\langle x^{2}\left(t_{n}\right)\right\rangle_{c}^{2}}=\frac{6 \alpha_{1}^{2}}{1-3 \alpha_{1}^{2}-2 \alpha_{1} \beta_{1}-\beta_{1}^{2}} $$

$$ \dot{x}(t)=r_{x}+\sum_{\lambda} \eta_{\lambda}(t), $$

$$ P_{\lambda}\left[\eta_{\lambda}\right]=\exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}_{\lambda}\left(\eta_{\lambda}(t)\right)\right]=\int \frac{\mathcal{D} p}{2 \pi} \exp \left\{\int_{t_{a}}^{t_{b}} d t\left[i p(t) \eta_{\lambda}(t)-H_{\lambda}(p(t))\right]\right\}, $$

$$ H_{\lambda}(p) \equiv \frac{\sigma_{\lambda}^{2}|p|^{\lambda}}{2} . $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\prod_{\lambda}\left\{\int \mathcal{D} \eta_{\lambda} \int \mathcal{D} x \exp \left[-\int_{t_{a}}^{t_{b}} d t \tilde{H}_{\lambda}\left(\eta_{\lambda}(t)\right)\right]\right\} \delta\left[\dot{x}-\sum_{\lambda} \eta_{\lambda}\right] . $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} \frac{\mathcal{D} p}{2 \pi} \prod_{\lambda}\left[\int \mathcal{D} \eta_{\lambda}\right] \int \mathcal{D} x e^{\int_{t_{a}}^{t_{b}} d t\left[i p(t) \dot{x}(t)-\tilde{H}_{\lambda}\left(\eta_{\lambda}(t)\right)\right]} e^{-i \sum_{\lambda} \int_{-\infty}^{\infty} d t p(t) \eta_{\lambda}(t)} $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\int_{-\infty}^{\infty} \frac{\mathcal{D} p}{2 \pi} \int \mathcal{D} x e^{\int_{t_{a}}^{t_{b}} d t\left[i p(t) \dot{x}(t)-H_{\lambda}(p)\right]} $$

$$ H(p) \equiv \sum_{\lambda} H_{\lambda}(p) $$

$$ H(p)=H(|p|) \equiv \int_{0}^{\infty} d \lambda \sigma_{\lambda}^{2}|p|^{\lambda} $$

$$ \sigma_{\lambda}^{2}=\int_{-i \infty}^{i \infty} \frac{d \log |p|}{2 \pi i} p^{-\lambda} H(p) $$

$$ W(t)=N_{S}(t) S(t)+N_{O}(t) O(S, t)+N_{B}(t) B(t) $$

$$ \dot{W}(t) \approx r_{W} W(t) . $$

$$ \dot{B}(t) \approx r_{B} B(t) . $$

$$ r_{W} \approx r_{B} $$

$$ \begin{align*} N_{S}(t) \dot{S}(t)+ & N_{O}(t) \dot{O}(S, t)+N_{B}(t) \dot{B}(t)+\dot{N}_{S}(t) S(t)+\dot{N}_{O}(t) O(S, t)+\dot{N}_{B}(t) B(t) \\ & =r_{W}\left[N_{S}(t) S(t)+N_{O}(t) O(S, t)+N_{B}(t) B(t)\right] \end{align*} $$

$$ \dot{N}_{S}(t) S(t)+\dot{N}_{O}(t) O(S, t)+\dot{N}_{B}(t) B(t)=0 $$

$$ \dot{W}(t)=N_{S} \dot{S}+N_{O} \dot{O}+N_{B} \dot{B}=r_{W}\left(N_{S} S+N_{O} O+N_{B} B\right) $$

$$ N_{S} \dot{S}+N_{O} \dot{O}=r_{W}\left(N_{S} S+N_{O} O\right) $$

$$ \frac{N_{S}(t)}{N_{O}(t)}=-\Delta(S(t), t)=-\frac{\partial O(S(t), t)}{\partial S(t)} . $$

$$ N_{S} \dot{S}+N_{O} \dot{O}=N_{O} r_{W}\left(-\frac{\partial O}{\partial x}+O\right) $$

$$ N_{S} \dot{S}=-N_{O} \frac{\partial O(S, t)}{\partial S} \dot{S}=-N_{O} \frac{\partial O(S, t)}{\partial x} \frac{\dot{S}}{S} $$

$$ \begin{align*} \frac{d O}{d t} & =\frac{1}{d t}[O(x(t)+\dot{x}(t) d t, t+d t)-O(x(t), t)] \\ & =\frac{\partial O}{\partial t}+\frac{\partial O}{\partial x} \dot{x}+\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} \dot{x}^{2} d t+\frac{1}{3!} \frac{\partial^{3} O}{\partial x^{3}} \dot{x}^{3} d t^{2}+\ldots \end{align*} $$

$$ \begin{align*} N_{S} \dot{S}+N_{O} \dot{O}= & -N_{O} \frac{\partial O}{\partial x} \frac{\dot{S}}{S} \\ & +N_{O}\left(\frac{\partial O}{\partial t}+\frac{\partial O}{\partial x} \dot{x}+\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} \dot{x}^{2} d t+\frac{1}{3!} \frac{\partial O}{\partial x} \dot{x}^{3} d t+\ldots\right) \\ = & N_{O}\left[\frac{\partial O}{\partial t}+\left(\dot{x}-\frac{\dot{S}}{S}\right) \frac{\partial O}{\partial x}+\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} \dot{x}^{2} d t+\frac{1}{3!} \frac{\partial^{3} O}{\partial x^{3}} \dot{x}^{3} d t+\ldots\right] . \end{align*} $$

$$ \frac{\partial O}{\partial t}=-r_{W} O-\left(\dot{x}-\frac{\dot{S}}{S}+r_{W}\right) \frac{\partial O}{\partial x}-\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} \dot{x}^{2} d t-\frac{1}{3!} \frac{\partial^{3} O}{\partial x^{3}} \dot{x}^{3} d t+\ldots=0 $$

$$ r_{W}-\frac{\sigma^{2}}{2} \equiv r_{x_{W}} $$

$$ -\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} \dot{x}^{2} d t-\frac{1}{6} \frac{\partial^{3} O}{\partial x^{3}} \dot{x}^{3} d t^{2}+\ldots $$

$$ \frac{\partial O}{\partial t}=r_{W} O-r_{x_{W}} \frac{\partial O}{\partial x}-\frac{\sigma^{2}}{2} \frac{\partial^{2} O}{\partial x^{2}} $$

$$ \Theta=r_{W} O-r_{x_{W}} S \Delta-\frac{\sigma^{2}}{2}\left(S \frac{\partial O}{\partial S}+S^{2} \frac{\partial^{2} O}{\partial S^{2}}\right)=r_{W} O-r_{W} S \Delta-\frac{\sigma^{2}}{2} S^{2} \Gamma $$

$$ \delta \Delta=\frac{d}{d t} \frac{\partial^{2} O(S(t), t)}{\partial S(t)} \delta t=\left[\frac{\partial}{\partial t} \frac{\partial O(S(t), t)}{\partial S(t)}+\frac{\partial^{2} O(S(t), t)}{\partial S(t)^{2}}\right] \delta t=\left[\frac{\partial \Theta}{\partial S}+\Gamma\right] \delta t . $$

$$ \delta \frac{N_{S}}{N_{O}}=-\delta \Delta=-\left(\frac{\partial \Theta}{\partial S}+\Gamma\right) \delta t $$

$$ \left\langle\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} \dot{x}^{2} d t+\frac{1}{6} \frac{\partial^{3} O}{\partial x^{3}} \dot{x}^{3} d t^{2}+\ldots\right\rangle=-\bar{H}\left(i \partial_{x}\right) O $$

$$ \frac{\partial}{\partial t}\langle O\rangle=\left[r_{W}-r_{x_{W}} \frac{\partial}{\partial x}+\bar{H}\left(i \partial_{x}\right)\right]\langle O\rangle $$

$$ r_{x_{W}} \equiv r_{W}+\bar{H}(i) $$

$$ P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p\left(x_{b}-x_{a}\right)} e^{-\left(\sigma^{2} p^{2} / 2+i r_{x_{W}} p\right)\left(t_{b}-t_{a}\right)} . $$

$$ P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p\left(x_{b}-x_{a}\right)} e^{-\left[\bar{H}(p)+i r_{x_{W}} p\right]\left(t_{b}-t_{a}\right)}, $$

$$ P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=\frac{e^{-r_{W}\left(t_{b}-t_{a}\right)}}{\sqrt{2 \pi \sigma^{2}\left(t_{b}-t_{a}\right)}} \exp \left\{-\frac{\left[x_{b}-x_{a}-r_{x_{W}}\left(t_{b}-t_{a}\right)\right]^{2}}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\right\} . $$

$$ P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} \int \mathcal{D} x \exp \left\{-\frac{1}{2 \sigma^{2}} \int_{t_{a}}^{t_{b}}\left[\dot{x}-r_{x_{W}}\right]^{2}\right\} $$

$$ O\left(x_{b}, t_{b}\right)=\Theta\left(x_{b}-x_{E}\right)\left(e^{x_{b}}-e^{x_{E}}\right) $$

$$ x_{E} \equiv \log E $$

$$ O\left(x_{a}, t_{a}\right)=\int_{-\infty}^{\infty} d x_{b} O\left(x_{b}, t_{b}\right) P^{\left(M, r_{W}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ O\left(x_{a}, t_{a}\right)=O_{S}\left(x_{a}, t_{a}\right)-O_{E}\left(x_{a}, t_{a}\right) $$

$$ O_{S}\left(x_{a}, t_{a}\right)=\frac{e^{-r_{W}\left(t_{b}-t_{a}\right)}}{\sqrt{2 \pi \sigma^{2}\left(t_{b}-t_{a}\right)}} \int_{x_{E}}^{\infty} d x_{b} e^{x_{b}} \exp \left\{-\frac{\left[x_{b}-x_{a}-r_{x_{W}}\left(t_{b}-t_{a}\right)\right]^{2}}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\right\} $$

$$ O_{E}\left(x_{a}, t_{a}\right)=E e^{-r_{W}\left(t_{b}-t_{a}\right)} \frac{1}{\sqrt{2 \pi \sigma^{2}\left(t_{b}-t_{a}\right)}} \int_{x_{E}}^{\infty} d x_{b} \exp \left\{-\frac{\left[x_{b}-x_{a}-r_{x_{W}}\left(t_{b}-t_{a}\right)\right]^{2}}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\right\} $$

$$ x_{-} \equiv x_{a}+r_{x_{W}}\left(t_{b}-t_{a}\right)=x_{a}+\left(r_{W}-\frac{1}{2} \sigma^{2}\right)\left(t_{b}-t_{a}\right) $$

$$ O_{E}\left(x_{a}, t_{a}\right)=E \frac{e^{-r_{W}\left(t_{b}-t_{a}\right)}}{\sqrt{2 \pi \sigma^{2}\left(t_{b}-t_{a}\right)}} \int_{x_{E}-x_{-}}^{\infty} d x_{b} \exp \left\{-\frac{x_{b}^{2}}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\right\} $$

$$ O_{E}\left(x_{a}, t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} E N\left(y_{-}\right) $$

$$ N(y) \equiv \int_{-\infty}^{y} \frac{d \xi}{\sqrt{2 \pi}} e^{-\xi^{2} / 2}=\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{y}{\sqrt{2}}\right)\right] $$

$$ \begin{align*} y_{-} & \equiv \frac{x_{-}-x_{E}}{\sqrt{\sigma^{2}\left(t_{b}-t_{a}\right)}}=\frac{\log \left[S\left(t_{a}\right) / E\right]+r_{x_{W}}\left(t_{b}-t_{a}\right)}{\sqrt{\sigma^{2}\left(t_{b}-t_{a}\right)}} \\ & =\frac{\log \left[S\left(t_{a}\right) / E\right]+\left(r_{W}-\frac{1}{2} \sigma^{2}\right)\left(t_{b}-t_{a}\right)}{\sqrt{\sigma^{2}\left(t_{a}-t_{b}\right)}} \end{align*} $$

$$ \begin{align*} x_{b} & -\frac{\left[x_{b}-x_{a}-r_{x_{W}}\left(t_{b}-t_{a}\right)\right]^{2}}{2 \sigma^{2}\left(t_{b}-t_{a}\right)} \\ & =-\frac{\left[x_{b}-x_{a}-\left(r_{x_{W}}+\sigma^{2}\right)\left(t_{b}-t_{a}\right)\right]^{2}-2 r_{W} \sigma^{2}\left(t_{b}-t_{a}\right)-2 x_{a} \sigma^{2}\left(t_{b}-t_{a}\right)}{2 \sigma^{2}\left(t_{b}-t_{a}\right)} \end{align*} $$

$$ x_{+} \equiv x_{a}+\left(r_{x_{W}}+\sigma^{2}\right)\left(t_{b}-t_{a}\right)=x_{a}+\left(r_{W}+\frac{1}{2} \sigma^{2}\right)\left(t_{b}-t_{a}\right), $$

$$ O_{S}\left(x_{a}, t_{a}\right)=S\left(t_{a}\right) N\left(y_{+}\right), $$

$$ \begin{align*} y_{+} & \equiv \frac{x_{+}-x_{E}}{\sqrt{\sigma^{2}\left(t_{a}-t_{b}\right)}}=\frac{\log \left[S\left(t_{a}\right) / E\right]+\left(r_{x_{W}}+\sigma^{2}\right)\left(t_{b}-t_{a}\right)}{\sqrt{\sigma^{2}\left(t_{a}-t_{b}\right)}} \\ & =\frac{\log \left[S\left(t_{a}\right) / E\right]+\left(r_{W}+\frac{1}{2} \sigma^{2}\right)\left(t_{b}-t_{a}\right)}{\sqrt{\sigma^{2}\left(t_{a}-t_{b}\right)}} \end{align*} $$

$$ O\left(x_{a}, t_{a}\right)=S\left(t_{a}\right) N\left(y_{+}\right)-e^{-r_{W}\left(t_{b}-t_{a}\right)} E N\left(y_{-}\right) $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)^{x_{0}}=\frac{\sqrt{3}}{\pi \sigma^{2}\left(t_{b}-t_{a}\right)} \exp \left\{-\frac{1}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\left[\left(x_{b}-x_{a}\right)^{2}+12\left(x_{0}-\frac{x_{b}+x_{a}}{2}\right)^{2}\right]\right\} . $$

$$ \begin{align*} & P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)^{x_{0}}=e^{-r_{W}\left(t_{b}-t_{a}\right)} \frac{\sqrt{3}}{\pi \sigma^{2}\left(t_{b}-t_{a}\right)} \\ & \quad \times \exp \left\{-\frac{1}{2 \sigma^{2}\left(t_{b}-t_{a}\right)}\left[\left(x_{b}-x_{a}-r_{x_{W}}\left(t_{b}-t_{a}\right)\right)^{2}+12\left(x_{0}-\frac{x_{b}+x_{a}}{2}\right)^{2}\right]\right\} \end{align*} $$

$$ \int_{-\infty}^{\infty} d x_{0} P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)^{x_{0}}=P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right) $$

$$ O\left(x_{b}, t_{b}\right)=\Theta\left(x_{b}-x_{0}\right)\left(e^{x_{b}}-e^{x_{0}}\right) $$

$$ O_{S}\left(x_{a}, t_{a}\right)=\int_{-\infty}^{\infty} d x_{0} \int_{x_{0}}^{\infty} d x_{b} e^{x_{b}} P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)^{x_{0}} $$

$$ O_{\bar{x}}\left(x_{a}, t_{a}\right)=\int_{-\infty}^{\infty} d x_{0} e^{x_{0}} \int_{x_{0}}^{\infty} d x_{b} P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)^{x_{0}} $$

$$ \begin{align*} O\left(x_{a}, t_{a}\right) & =\int_{-\infty}^{\infty} \frac{d t}{2 \pi i} \frac{e^{i\left(x_{b}-x_{0}\right) t}}{t-i \eta} \\ & \times \int_{-\infty}^{\infty} d x_{0} \int_{-\infty}^{\infty} d x_{b}\left(e^{x_{b}}-e^{x_{0}}\right) P^{\left(M, r_{x}\right)}\left(x_{b} t_{b} \mid x_{a} t_{a}\right)^{x_{0}} \end{align*} $$

$$ e^{-A(t)}-e^{-A_{0}(t)} $$

$$ \begin{align*} A(t) & =-r_{W}\left(t_{b}-t_{a}\right)-\frac{i}{2}\left(r_{W}+\frac{\sigma^{2}}{2}\right) t\left(t_{b}-t_{a}\right)+\frac{\sigma^{2} t^{2}\left(t_{b}-t_{a}\right)}{6} \\ A_{0}(t) & =\left(-r_{W}+\frac{\sigma^{2}}{6}\right) \frac{t_{b}-t_{a}}{2}-\frac{i}{2}\left(r_{W}-\frac{\sigma^{2}}{6}\right) t\left(t_{b}-t_{a}\right)+\frac{\sigma^{2} t^{2}\left(t_{b}-t_{a}\right)}{6} \end{align*} $$

$$ \int_{-\infty}^{\infty} \frac{d t}{2 \pi i} \frac{e^{-a^{2} t^{2} / 2+i b t}}{t-i \eta}=N[b / a] $$

$$ O\left(x_{a}, t_{a}\right)=S\left(t_{a}\right)\left[N\left(z_{+}\right)-e^{-\left(r_{W}+\sigma^{2} / 6\right)\left(t_{b}-t_{a}\right) / 2} N\left(z_{-}\right)\right] $$

$$ z_{+}=\sqrt{\frac{3\left(t_{b}-t_{a}\right)}{4 \sigma^{2}}}\left(r_{W}+\frac{\sigma^{2}}{2}\right), \quad z_{-}=\sqrt{\frac{3\left(t_{b}-t_{a}\right)}{4 \sigma^{2}}}\left(r_{W}-\frac{\sigma^{2}}{6}\right) . $$

$$ O^{B}\left(x_{a}, t_{a}\right)=\left(\frac{t}{\sigma^{2}}\right)^{t} \frac{1}{\Gamma(t)} \int_{0}^{\infty} \frac{d v}{v} v^{t} e^{-t v / \sigma^{2}} O^{v}\left(x_{a}, t_{a}\right) $$

$$ O^{B}\left(x_{a}, t_{a}\right)=O^{v}\left(x_{a}, t_{a}\right)+\frac{\left[2 T^{2}(1-1 / t)\right]^{2}}{2 t} \partial_{v}^{2} O^{v}\left(x_{a}, t_{a}\right)+\ldots $$

$$ H_{r_{x_{W}}}(p) \equiv \bar{H}(p)+i r_{x_{W}} p $$

$$ \begin{align*} O\left(x_{a}, t_{a}\right) & =\int_{x_{E}}^{\infty} d x_{b}\left(e^{x_{b}}-e^{x_{E}}\right) P\left(x_{b} t_{b} \mid x_{a} t_{a}\right) \\ & =e^{-r_{W}\left(t_{b}-t_{a}\right)} \int_{x_{E}}^{\infty} d x_{b}\left(e^{x_{b}}-e^{x_{E}}\right) \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p\left(x_{b}-x_{a}\right)-H_{r_{X}}(p)\left(t_{b}-t_{a}\right)} \end{align*} $$

$$ O\left(x_{a}, t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} \int_{x_{E}}^{\infty} d x_{b} \int_{-\infty}^{\infty} \frac{d p}{2 \pi}\left[e^{x_{a}} e^{i(p-i)\left(x_{b}-x_{a}\right)}-e^{x_{E}} e^{i p\left(x_{b}-x_{a}\right)}\right] e^{-H_{r_{x}}(p)\left(t_{b}-t_{a}\right)} $$

$$ O\left(x_{a}, t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} \int_{x_{E}}^{\infty} d x_{b} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p\left(x_{b}-x_{a}\right)} f(p) $$

$$ f(p) \equiv e^{x_{a}} e^{-H_{r_{x_{W}}}(p+i)\left(t_{b}-t_{a}\right)}-e^{x_{E}} e^{-H_{r_{x_{W}}}(p)\left(t_{b}-t_{a}\right)} $$

$$ \tilde{f}\left(x_{b}-x_{a}\right)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p\left(x_{b}-x_{a}\right)} f(p) $$

$$ \int_{x_{E}}^{\infty} d x_{b} \tilde{f}\left(x_{b}-x_{a}\right)=\int_{-\infty}^{\infty} d x_{b} \int_{-\infty}^{\infty} \frac{d q}{2 \pi} \frac{i}{q+i \eta} e^{-i q\left(x_{b}-x_{E}\right)} \tilde{f}\left(x_{b}-x_{a}\right) $$

$$ O\left(x_{a}, t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p\left(x_{E}-x_{a}\right)} \frac{i}{p+i \eta} f(p) $$

$$ \frac{1}{p+i \eta}=\frac{\mathcal{P}}{p}-i \pi \delta(p) $$

$$ O\left(x_{a}, t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)}\left[\frac{1}{2} f(0)+i \int_{-\infty}^{\infty} \frac{d p}{2 \pi} \frac{e^{i p\left(x_{E}-x_{a}\right)} f(p)-f(0)}{p}\right] $$

$$ \int_{-\infty}^{\infty} \frac{d p}{2 \pi} \frac{e^{i p\left(x_{E}-x_{a}\right)} f(p)-f(0)}{p} \approx \frac{1}{2} \epsilon\left(x_{a}-x_{E}\right) f(0), $$

$$ O\left(x_{a}, t_{a}\right) \approx \frac{e^{-r_{W}\left(t_{b}-t_{a}\right)}}{2}\left[1+\Theta\left(x_{a}-x_{E}\right)\right] f(0) $$

$$ O\left(x_{a}, t_{a}\right) \approx e^{x_{a}}-e^{x_{E}} e^{-r_{W}\left(t_{b}-t_{a}\right)}=S\left(t_{a}\right)-e^{-r_{W}\left(t_{b}-t_{a}\right)} E . $$

$$ \tilde{L}_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(x)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p x-H(p)} $$

$$ \sigma^{2} \rightarrow \sigma^{2}\left(t_{b}-t_{a}\right), \quad r_{x_{W}} \rightarrow r_{x_{W}}\left(t_{b}-t_{a}\right) . $$

$$ \bar{L}_{\sigma^{2}}^{(\lambda, \alpha, \beta)}(x)=\int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p x-\bar{H}(p)} $$

$$ \int_{-\infty}^{\infty} \frac{d p}{2 \pi} e^{i p x-\left[\bar{H}(p)+i r_{x_{W}} p\right]\left(t_{b}-t_{a}\right)}=\tilde{L}_{\sigma^{2}\left(t_{b}-t_{a}\right)}^{(\lambda, \alpha, \beta))}\left(x-r_{x_{W}}\left(t_{b}-t_{a}\right)\right) $$

$$ P\left(x_{b} t_{b} \mid x_{a} t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} \bar{L}_{\sigma^{2}\left(t_{b}-t_{a}\right)}^{(\lambda, \alpha, \beta)}\left(x_{b}-x_{a}-r_{x_{W}}\left(t_{b}-t_{a}\right)\right) . $$

$$ O\left(x_{a}, t_{a}\right)=e^{-r_{W}\left(t_{b}-t_{a}\right)} \int_{x_{E}}^{\infty} d x_{b}\left(e^{x_{b}}-e^{x_{E}}\right) \bar{L}_{\sigma^{2}\left(t_{b}-t_{a}\right)}^{(\lambda, \alpha, \beta)}\left(x_{b}-x_{a}-r_{x_{W}}\left(t_{b}-t_{a}\right)\right) $$

$$ \begin{align*} \frac{d O}{d t} & =\frac{1}{d t}[O(x(t)+\dot{x}(t) d t, v(t)+\dot{v}(t) d t, t+d t)-O(x(t), v(t), t)] \\ & =\frac{\partial O}{\partial t}+\frac{\partial O}{\partial x} \dot{x}+\frac{\partial O}{\partial v} \dot{v}+\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} \dot{x}^{2} d t+\frac{\partial^{2} O}{\partial x \partial v} \dot{x} \dot{v} d t+\frac{1}{2} \frac{\partial^{2} O}{\partial v^{2}} \dot{v}^{2} d t+\ldots \end{align*} $$

$$ \dot{x}^{2} \longrightarrow v(t), \quad \dot{v}^{2} \longrightarrow \epsilon^{2} v(t), \quad \dot{x} \dot{v} \longrightarrow \rho \epsilon v(t) . $$

$$ \begin{align*} \frac{d O}{d t} & =\frac{1}{d t}[O(x(t)+\dot{x}(t) d t, v(t)+\dot{v}(t) d t, t+d t)-O(x(t), v(t), t)] \\ & =\frac{\partial O}{\partial t}+\frac{\partial O}{\partial x} \dot{x}+\frac{\partial O}{\partial v} \dot{v}+\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} v+\frac{\partial^{2} O}{\partial x \partial v} \rho \epsilon v+\frac{1}{2} \frac{\partial^{2} O}{\partial v^{2}} \epsilon^{2} v \end{align*} $$

$$ \begin{align*} & N_{O} r_{W}\left(-\frac{\partial O}{\partial x}+O\right)=-N_{O} \frac{\partial O}{\partial x}\left(\dot{x}+\frac{v^{2}}{2}\right)+N_{O} \dot{O} \\ & \quad=N_{O}\left[-\frac{v^{2}}{2} \frac{\partial O}{\partial x}+\frac{\partial O}{\partial v} \dot{v}+\frac{\partial O}{\partial t}+\frac{1}{2} \frac{\partial^{2} O}{\partial x^{2}} v+\frac{\partial^{2} O}{\partial x \partial v} \rho \epsilon v+\frac{1}{2} \frac{\partial^{2} O}{\partial v^{2}} \epsilon^{2} v+\ldots\right] . \end{align*} $$

$$ W(t)=N_{S}(t) S(t)+N_{O}(t) O(S, t)+N_{V}(t) V(t)+N_{B}(t) B(t) $$

$$ \frac{N_{V}(t)}{N_{O}(t)}=-\frac{\partial O(S(t), v(t), t)}{\partial v(t)} $$

$$ \dot{v}(t) \longrightarrow-\gamma[v(t)-\bar{v}] . $$

$$ \gamma^{*}=\gamma+\lambda, \quad \text { and } \quad \bar{v}^{*}=\gamma \bar{v} / \gamma^{*} $$

$$ \frac{\partial O}{\partial t}=r_{W} O-\left(r_{W}-\frac{v}{2}\right) \frac{\partial O}{\partial x}+\gamma^{*}\left[v(t)-\bar{v}^{*}\right] \frac{\partial O}{\partial v}-\frac{v}{2} \frac{\partial^{2} O}{\partial x^{2}}-\rho \epsilon v \frac{\partial^{2} O}{\partial x \partial v}-\frac{\epsilon^{2} v}{2} \frac{\partial^{2} O}{\partial v^{2}} $$

$$ \frac{\partial O}{\partial t}=r_{W}\left(O-\partial_{x} O\right)+\left(\hat{H}^{*}+\gamma^{*}+\rho \epsilon \partial_{x}+\epsilon^{2} \partial_{v}\right) O $$

$$ P^{v}\left(x_{b}, v_{b}, t_{b} \mid x_{a} v_{a} t_{a}\right)=\delta\left(x_{b}-x_{a}\right) \delta\left(v_{b}-v_{a}\right) $$

$$ P^{v}\left(x_{b}, v_{b}, t_{b} \mid x_{a} v_{a} t_{a}\right)=e^{-\left(r_{W}+\gamma^{*}\right) \Delta t} P_{\mathrm{sh}}\left(x_{b}, v_{b}, t_{b} \mid x_{a} v_{a} t_{a}\right), $$

$$ x_{b}-x_{a} \rightarrow x_{b}-x_{a}-\left(r_{W}-\rho \epsilon\right), \quad v_{b}-v_{a} \rightarrow v_{b}-v_{a}-\epsilon^{2} . $$

$$ \gamma^{*}=2, \quad \bar{v}=0.01, \quad \epsilon=0.1, r_{W}=0 $$

$$ D(p)=\left[1+\frac{a_{4}}{4!} p^{4}-\frac{a_{6}}{6!} p^{6}+\frac{c_{8}}{8!} p^{8}-\ldots-\ldots\right] e^{-\sigma^{2} p^{2} / 2} $$

$$ a_{4}=c_{4}, \quad a_{6}=c_{6}, \quad a_{8}=c_{8}+35 c_{4}^{2}, \quad a_{10}=c_{10}+210 c_{4} c_{6}, \ldots $$

$$ D(p)=\left[1+\frac{a_{4}}{4!} 2^{2}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{2}+\frac{a_{6}}{6!} 2^{3}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{3}+\frac{a_{8}}{8!} 2^{4}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{4}+\ldots\right] e^{-\sigma^{2} p^{2} / 2} $$

$$ \begin{align*} \tilde{D}(x)= & {\left[1+\frac{a_{4}}{4!} 2^{2}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{2}+\frac{a_{6}}{6!} 2^{3}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{3}+\frac{a_{8}}{8!} 2^{4}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{4}+\ldots\right] \frac{e^{-x^{2} / 2 \sigma^{2}}}{\sqrt{2 \pi \sigma^{2}}} } \\ = & {\left[1+\frac{\bar{c}_{4}}{8}-\frac{\bar{c}_{6}}{48}+\frac{35 \bar{c}_{4}^{2}}{384}+\frac{\bar{c}_{8}}{384}+\frac{x^{2}}{\sigma^{2}}\left(-\frac{\bar{c}_{4}}{4}+\frac{\bar{c}_{6}}{16}-\frac{35 \bar{c}_{4}^{2}}{96}-\frac{\bar{c}_{8}}{96}\right)\right.} \\ & +\frac{x^{4}}{\sigma^{4}}\left(\frac{\bar{c}_{4}}{24}-\frac{\bar{c}_{6}}{48}+\frac{35 \bar{c}_{4}^{2}}{192}+\frac{\bar{c}_{8}}{192}\right)+\frac{x^{6}}{\sigma^{6}}\left(\frac{\bar{c}_{6}}{720}-\frac{35 \bar{c}_{4}^{2}}{1440}-\frac{\bar{c}_{8}}{1440}\right) \\ & \left.+\frac{x^{8}}{\sigma^{8}}\left(\frac{35 \bar{c}_{4}^{2}}{40320}+\frac{\bar{c}_{8}}{40320}\right)+\ldots\right] \frac{e^{-x^{2} / 2 \sigma^{2}}}{\sqrt{2 \pi \sigma^{2}}} . \end{align*} $$

$$ \begin{align*} \tilde{D}(x)= & \left\{1-\frac{x^{2}}{2 \sigma^{2}}\left(\frac{\bar{c}_{4}}{2}-\frac{\bar{c}_{6}}{8}+\frac{2 \bar{c}_{4}^{2}}{3}+\frac{\bar{c}_{8}}{48}+\frac{5 \bar{c}_{4} \bar{c}_{6}}{192}-\frac{33 \bar{c}_{4}^{3}}{256}\right)\right. \\ & +\frac{x^{4}}{\sigma^{4}}\left(\frac{\bar{c}_{4}}{24}-\frac{\bar{c}_{6}}{48}+\frac{17 \bar{c}_{4}^{2}}{96}+\frac{\bar{c}_{8}}{192}+\frac{\bar{c}_{4} \bar{c}_{6}}{288}-\frac{239 \bar{c}_{4}^{3}}{9216}\right) \\ & +\frac{x^{6}}{\sigma^{6}}\left(\frac{\bar{c}_{6}}{720}-\frac{7 \bar{c}_{4}^{2}}{288}-\frac{\bar{c}_{8}}{1440}-\frac{\bar{c}_{4} \bar{c}_{6}}{5760}+\frac{7 \bar{c}_{4}^{3}}{2304}\right) \\ & \left.+\frac{x^{8}}{\sigma^{8}}\left(\frac{\bar{c}_{4}^{2}}{1152}+\frac{\bar{c}_{8}}{40320}-\frac{\bar{c}_{4}^{3}}{9216}\right)+\ldots\right\} \frac{e^{-x^{2} / 2 \sigma^{2}}}{\sqrt{2 \pi \sigma_{1}^{2}}}, \end{align*} $$

$$ \tilde{\sigma}_{1}^{2} \equiv \sigma^{2}\left(1-\frac{\bar{c}_{4}}{4}+\frac{\bar{c}_{6}}{24}-\frac{13 \bar{c}_{4}^{2}}{96}-\frac{\bar{c}_{8}}{192}-\frac{\bar{c}_{4} \bar{c}_{6}}{64}-\frac{31 \bar{c}_{4}^{3}}{512}+\ldots\right) $$

$$ \begin{align*} \tilde{D}(x)= & \exp \left\{-\frac{x^{2}}{2 \sigma^{2}}\left(1+\frac{\bar{c}_{4}}{2}+\frac{2 \bar{c}_{4}^{2}}{3}-\frac{33 \bar{c}_{4}^{3}}{256}-\frac{\bar{c}_{6}}{8}+\frac{5 \bar{c}_{4} \bar{c}_{6}}{192}+\frac{\bar{c}_{8}}{48}\right)\right. \\ & +\frac{x^{4}}{\sigma^{4}}\left(\frac{\bar{c}_{4}}{24}+\frac{7 \bar{c}_{4}^{2}}{48}-\frac{1007 \bar{c}_{4}^{3}}{9216}-\frac{\bar{c}_{6}}{48}+\frac{11 \bar{c}_{4} \bar{c}_{6}}{576}+\frac{\bar{c}_{8}}{192}\right) \\ & +\frac{x^{6}}{\sigma^{6}}\left(-\frac{\bar{c}_{4}^{2}}{72}+\frac{43 \bar{c}_{4}^{3}}{768}+\frac{\bar{c}_{6}}{720}-\frac{23 \bar{c}_{4} \bar{c}_{6}}{2880}-\frac{\bar{c}_{8}}{1440}\right) \\ & \left.+\frac{x^{8}}{\sigma^{8}}\left(-\frac{101 \bar{c}_{4}^{3}}{9216}+\frac{7 \bar{c}_{4} \bar{c}_{6}}{5760}+\frac{\bar{c}_{8}}{40320}\right)+\ldots\right\} \frac{1}{\sqrt{2 \pi \sigma_{1}^{2}}} . \end{align*} $$

$$ \tilde{\sigma}_{2}^{2} \equiv \sigma^{2}\left(1-\frac{\bar{c}_{4}}{2}+\frac{\bar{c}_{6}}{8}-\frac{5 \bar{c}_{4}^{2}}{12}-\frac{\bar{c}_{8}}{48}-\frac{29 \bar{c}_{4} \bar{c}_{6}}{192}+\frac{515 \bar{c}_{4}^{3}}{768}+\ldots\right), $$

$$ \begin{align*} \tilde{D}(x)= & \exp \left\{-\frac{x^{2}}{\tilde{\sigma}_{2}^{2}}+\frac{x^{4}}{\tilde{\sigma}_{2}^{4}}\left(\frac{\bar{c}_{4}}{24}-\frac{\bar{c}_{6}}{48}+\frac{5 \bar{c}_{4}^{2}}{48}+\frac{\bar{c}_{8}}{192}+\frac{29 \bar{c}_{4} \bar{c}_{6}}{576}-\frac{2576 \bar{c}_{4}^{3}}{9216}\right)\right. \\ & +\frac{x^{6}}{\tilde{\sigma}_{2}^{6}}\left(\frac{\bar{c}_{6}}{720}-\frac{\bar{c}_{4}^{2}}{72}-\frac{8 \bar{c}_{8}}{1440}-\frac{29 \bar{c}_{4} \bar{c}_{6}}{2880}+\frac{59 \bar{c}_{4}^{3}}{768}\right) \\ & \left.+\frac{x^{8}}{\tilde{\sigma}_{2}^{8}}\left(\frac{\bar{c}_{8}}{40320}+\frac{7 \bar{c}_{4} \bar{c}_{6}}{5760}-\frac{101 \bar{c}_{4}^{3}}{9216}\right)+\ldots\right\} \frac{1}{\sqrt{2 \pi \tilde{\sigma}_{1}^{2}}} . \end{align*} $$

$$ \tilde{D}(x)=\frac{1}{\sqrt{2 \pi \tilde{\sigma}_{1}^{2}}} \exp \left[-\frac{x^{2}}{2 \Sigma^{2}(x)}\right] $$

$$ \begin{align*} & \tilde{\Sigma}(x) \equiv \sigma_{2}^{2}\left[1+\frac{x^{2}}{\sigma_{2}^{2}}\left(\frac{\bar{c}_{4}}{12}+\frac{5 \bar{c}_{4}^{2}}{24}-\frac{2575 \bar{c}_{4}^{3}}{4608}-\frac{\bar{c}_{6}}{24}+\frac{29 \bar{c}_{4} \bar{c}_{6}}{288}+\frac{\bar{c}_{8}}{96}\right)\right. \\ & +\frac{x^{4}}{\sigma_{2}^{4}}\left(\frac{-\bar{c}_{4}^{2}}{48}+\frac{217 \bar{c}_{4}^{3}}{1152}-\frac{1375 \bar{c}_{4}^{4}}{27648}+\frac{\bar{c}_{6}}{360}-\frac{13 \bar{c}_{4} \bar{c}_{6}}{480}-\frac{\bar{c}_{4}^{2} \bar{c}_{6}}{1728}+\frac{\bar{c}_{6}^{2}}{576}-\frac{\bar{c}_{8}}{720}+\frac{\bar{c}_{4} \bar{c}_{8}}{576}\right) \\ & \left.+\frac{x^{6}}{\sigma_{2}^{6}}\left(-\frac{359 \bar{c}_{4}^{3}}{13824}+\frac{127 \bar{c}_{4}^{4}}{6912}+\frac{5 \bar{c}_{4} \bar{c}_{6}}{1728}-\frac{13 \bar{c}_{4}^{2} \bar{c}_{6}}{17280}-\frac{\bar{c}_{6}^{2}}{4320}+\frac{\bar{c}_{8}}{20160}-\frac{\bar{c}_{4} \bar{c}_{8}}{4320}\right)+\ldots\right] . \end{align*} $$

$$ \mathcal{O} \equiv\left[1+\frac{a_{4}}{4!} 2^{2}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{2}+\frac{a_{6}}{6!} 2^{3}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{3}+\frac{a_{8}}{8!} 2^{4}\left(\frac{\partial}{\partial \sigma^{2}}\right)^{4}+\ldots\right] $$

$$ \begin{align*} & a_{4}=\frac{\varepsilon}{M^{2}}, a_{6}=\frac{5 \cdot 7 \varepsilon^{2}}{3 M^{3}}, a_{8}=\frac{5 \cdot 7 \cdot 11 \varepsilon^{2}}{M^{4}}+\frac{5 \cdot 7 \varepsilon^{2}}{M^{4}} \\ & a_{10}=\frac{5^{2} \cdot 7 \cdot 11 \cdot 13 \varepsilon^{4}}{3 M^{5}}+\frac{2^{2} \cdot 5 \cdot 7 \cdot 17 \varepsilon^{4}}{3 M^{5}}, \ldots \end{align*} $$

$$ O^{L}\left(x_{a}, t_{a}\right)=\mathcal{O} O\left(x_{a}, t_{a}\right)=\mathcal{O}\left[S\left(t_{a}\right) N\left(y_{+}\right)-e^{-\left(r_{x_{W}}+\sigma^{2} / 2\right)\left(t_{b}-t_{a}\right)} E N\left(y_{-}\right)\right] $$

$$ \begin{align*} O_{1}\left(x_{a}, t_{a}\right)=-\frac{\varepsilon}{12 M^{2}}\left\{\left(S e^{-y_{+}^{2} / 2}-\right.\right. & \left.e^{-r_{W}\left(t_{b}-t_{a}\right)} E e^{-y_{-}^{2} / 2}\right) \frac{y_{m}}{\sqrt{2 \pi} \sigma^{2}} \\ & \left.-e^{-r_{W}\left(t_{b}-t_{a}\right)}\left(t_{b}-t_{a}\right) E N\left(y_{-}\right)\right\} . \end{align*} $$

$$ \begin{align*} O_{2}\left(x_{a}, t_{a}\right)=\frac{35 \varepsilon^{2}}{3 M^{3}} \frac{1}{2^{3} \cdot 3^{2} \cdot 5} & \left\{\left[S e^{-y_{+}^{2} / 2}\left(y_{+} y_{-}^{2}-3 y_{+}+4 \sqrt{\sigma^{2}\left(t_{b}-t_{a}\right)}\right)\right.\right. \\ & \left.-e^{-r_{W}\left(t_{b}-t_{a}\right)} E e^{-y_{-}^{2} / 2}\left(y_{-}^{3}-3 y_{-}-2 \sigma^{2}\left(t_{b}-t_{a}\right) y_{-}\right)\right] \frac{1}{\sqrt{2 \pi} \sigma^{4}} \\ & \left.+e^{-r_{W}\left(t_{b}-t_{a}\right)}\left(t_{b}-t_{a}\right)^{2} E N\left(y_{-}\right)\right\} \end{align*} $$

$$ \begin{align*} & O_{3}\left(x_{a}, t_{a}\right)=\frac{385 \varepsilon^{3}+35 \varepsilon^{2}}{M^{4}} \frac{1}{2^{6} \cdot 3^{2} \cdot 5 \cdot 7} \\ & \quad \times\left\{\left[S e^{-y_{+}^{2} / 2}\left(-y_{-}^{3} y_{+}^{2}+9 y_{-} y_{+}^{2}+y_{-}^{3}-15 y_{+}+18+\sqrt{\sigma^{2}\left(t_{b}-t_{a}\right)}\left(12 y_{-} y_{+}+18\right)\right)\right.\right. \\ & \quad-e^{-r_{W}\left(t_{b}-t_{a}\right)} E e^{-y_{-}^{2} / 2}\left(y_{-}^{5}-10 y_{-}^{3}+15 y_{-}+\sigma^{2}\left(t_{b}-t_{a}\right)\left(3 y_{-}^{3}-9 y_{-}\right)\right. \\ & \left.\left.\left.\quad+\sigma^{4}\left(t_{b}-t_{a}\right)^{2} 3 y_{-}\right)\right] \frac{1}{\sqrt{2 \pi} \sigma^{6}}+e^{-r_{W}\left(t_{b}-t_{a}\right)}\left(t_{b}-t_{a}\right)^{3} E N\left(y_{-}\right)\right\} \end{align*} $$