I have read a paper of J.Bricmont and A. Kupiainen 1994 at [http://iopscience.iop.org/0951-7715/7/2/011], but I didn't understand these calculations concerning to a stochastic process. I hope for kind helps to better understand stochastic calculations. Many thanks!
I have 2 questions:
How can we derive $\nabla_x E(y,x)$ from the integration by parts of Gaussian mearsures, where $$E(y,x) = \int d\mu_{yx}^t(\omega) e^{\int_0^t V(\omega(s),s)ds},$$ and $d\mu_{yx}^t$ is the oscillator measure on the continuous paths $\omega: [0,t] \to \mathbb{R}^N$ with $\omega(0) = x$ and $\omega(t) = y$, i.e., the Gaussian probability measure with covariance kernel \begin{align*} C(\tau, \tau') &= \omega_0(\tau)\omega_0(\tau')\\ &+2 \left(e^{-\frac{1}{2}|\tau - \tau'|} - e^{-\frac{1}{2}|\tau + \tau'|} + e^{-\frac{1}{2}|2t + \tau - \tau'|} - e^{-\frac{1}{2}|2t - \tau - \tau'|} \right), \end{align*} which yields $\int d\mu_{yx}^{s-\sigma} (\omega(\tau)) = \omega_0(\tau)$, $$\omega_0(\tau) = \left(\sinh(t/2) \right)^{-1}\left(y\sinh(\frac{\tau}{2}) + x\sinh(\frac{t -\tau}{2}) \right).$$
How can we obtain the formula of $C(\tau, \tau')$ with the following linear operator $$\mathcal{L} = \Delta - \frac{y}{2}\cdot \nabla + 1,$$ whose kenel is given by Mehler's formula $$e^{t\mathcal{L}}(y,x) = \frac{e^t}{\sqrt{4\pi (1 - e^{-t} )}} exp\left[-\frac{(ye^{-t/2} - x)^2}{4 (1 - e^{-t})}\right].$$