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.