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From wikipedia:

Suppose we wish to find the expected value of the function $e^{-\int_0^t V(x(\tau)) d\tau}$ in the case where $x(\tau)$ is some realization of a diffusion process starting at $x(0) = 0$. The Feynman–Kac formula says that this expectation is equivalent to the integral of a solution to a diffusion equation. Specifically, under the conditions that $uV(x) \ge 0$,

$$E[e^{-u\int_0^t V(x(\tau)) d\tau}] = \int_{-\infty}^\infty w(x, t) dx$$

where $w(x, 0) = \delta(x)$ and

$$\frac{\partial w}{\partial t} = \frac{1}{2} \frac{\partial^2w}{\partial x^2} - uV(x)w $$

My Questions:

(1) What is $\delta(x)$ in this context? It's not mentioned by the rest of the wiki article.

(2) The "randomness" of the variable whose expectation is being measured is entirely contained within the term $x(\tau)$. But it looks to me like the value of the expectation is independent of the function $x$. It's simply an integral of the function $w$, and the PDE that defines $w$ uses $x$ as a parameter to $w$ but not a function (right?). What am I missing?

Thanks for your help.

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1 Answer

up vote 1 down vote accepted

Check here:

http://www.math.sunysb.edu/~ajt/Teaching/560spring2011/PDF/Covariance%20Operator,%20Support%20of%20Wiener%20Measure.pdf

This should be helpful.

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The link is not valid anymore. Can you upload the document? –  ASDF Feb 25 at 12:26
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