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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:

  1. 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 $\int d\mu_{yx}^{s-\sigma} (\omega(\tau)) = \omega_0(\tau)$, $$\omega_0(\tau) = \left(\sinh(t/2) \right)^{-1}\left(y\sinh(\frac{\tau}{2}) + x\sinh(\frac{t -\tau}{2}) \right).$$

  2. 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 $$e^{t\mathcal{L}}(y,x) = \frac{e^t}{\sqrt{4\pi (1 - e^{-t} )}} exp\left[-\frac{(ye^{-t/2} - x)^2}{4 (1 - e^{-t})}\right].$$

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  • $\begingroup$ I don't really understand this question because I don't know this stuff, but do you have one or two questions? You say "My problem is ..." and then ask "How can we ...". Are these questions related or should they be asked separately? (I honestly don't know.) $\endgroup$
    – user62675
    Jan 3, 2015 at 0:12
  • $\begingroup$ I have 2 questions relating to the calculations of Feynman-Kac formula. They are the derivations of $\nabla_x E$ and $C(\tau, \tau')$. $\endgroup$
    – VTNguyen
    Jan 4, 2015 at 19:26

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