I recently came across the Feynman-Kac formula which states that given an open domain $\Omega\in\mathbb{R}^n$ and $f \in L^2(\Omega)$ where $x \in \Omega$ and $t > 0$, then $e^{t\Delta_D}f(x) = E_x(f(\omega(t))\psi_{\Omega}(\omega, t))$, where t > 0 is arbitrary, $\omega(t)$ denotes an element of the probability space of Brownian motions starting in $x$ and $E_x$ is to be understood with regards to the measure of that probability space and $\psi_{\Omega}(\omega, t) = 1$ if $\omega([0, t]) \subset \Omega$ and $0$ otherwise. I am curious if there are any same kind of formula or generalisations of Feymann–Kac available for other operators for example for fractional Laplacians or higher order Laplacians or even for $p$-Laplacians?
Any insight will be very helpful.
1\ if\ ω([0, t]) \subset \Omega$1\ if\ ω([0, t]) \subset \Omega$, use1\text{ if $\omega([0, t]) \subset \Omega$}$1\text{ if …}$ (although there are differences of opinion about the scope of\text). In fact, there's no need to keep math mode going at all, so$1$ if $\omega([0, t]) \subset \Omega$is best of all. Also, in$p-$ Laplacians, the-goes out of math mode (it's a hyphen, not a minus). I edited accordingly. You might also like to know about the{cases}environment:\psi = \begin{cases} 1 & \text{if …} \\ 0 & \text{if …} \end{cases}. $\endgroup$