The Feynman-Kac formula says that $$ \exp(-t(-\Delta+V(X)))(x,y) = \mathbb{E}_{\gamma(0)=x,\gamma(t)=y}\left[\exp(-\int_0^t V\circ\gamma)\right] $$ where $\Delta$ is the Laplacian on $L^2(\mathbb{R}^n)$, $X$ is the position operator on $L^2(\mathbb{R}^n)$, $V:\mathbb{R}^n\to\mathbb{R}$ is some potential function, $x,y\in\mathbb{R}^n$, and $\mathbb{P}_{\gamma(0)=x,\gamma(t)=y}$ is the conditional Wiener measure for a Brownian path $\gamma$.

**Question 1**

What is the discrete-time analog of this formula? I mean what are the relevant notions on both the LHS and RHS?

**Question 2**

What is the discrete-space analog of this formula (at continuous time)? I mean, what is the analogous LHS if we change the RHS to make $\gamma$ take values only on $\mathbb{Z}^n$ instead of $\mathbb{R}^n$? I think this might be answered in the body of this question.

**Question 3**

We know that the Gaussian free field (GFF) coincides with the Wiener measure if the domain space happens to be one dimensional. However, if the domain is more than one dimensional, what is the analog of the Feynman-Kac formula? What I mean is the following. Let $\varphi:D\to\mathbb{R}^n$ where $D$ is a compact subset of $\mathbb{R}^m$. Consider the probability measure on random fields $\varphi$ proportional to $$ \exp(-\int_D\varphi\cdot(-\Delta\varphi))\,. $$ Here, the inner product is meant in $\mathbb{R}^n$, and the Laplacian is the Dirichlet Laplacian on $D$. This defines the GFF measure on fields $D\to\mathbb{R}^n$. One way to understand it is via a spectral eigenfunction decomposition (in Dirichlet eigenfunctions of the Laplacian on $D$) of the field $\varphi$. If $D=[0,t]$ this yields (pinned) Brownian motion and we have a Feynman-Kac formula. What would, then, be the analogous LHS of a putative Feynman-Kac formula for the GFF? What I mean is:

$$ \boxed{?} = \frac{\int_{\varphi:D\to\mathbb{R}^n}\exp(-\int_D\varphi\cdot(-\Delta\varphi)-\int_D V\circ\varphi)\mathrm{d}\varphi}{\int_{\varphi:D\to\mathbb{R}^n}\exp(-\int_D\varphi\cdot(-\Delta\varphi))\mathrm{d}\varphi}\,. $$

Here $V:\mathbb{R}^n\to\mathbb{R}$ would be some potential function.

What is the analogous heat equation PDE? Can this work if we replace $\mathbb{R}^n$ with $\mathbb{Z}^n$ (related to Question 2)

I suppose when $D=[0,t]\times\tilde{D}$ then one may write a heat-kernel for a Schroedinger operator on the Hilbert space $L^2(\tilde{D})$? Similar to the Euclidean path-integral prescription of QFT where we have a field that isn't just a point particle. And then the field obeys something like a heat-Klein-Gordon equation? But what if $D$ is general? And is this rigorous as the usual Feynman-Kac formula is?