Suppose that $X$ is the continuous-time simple symmetric random walk on the lattice $\mathbb Z^d$ (i.e., a simple symmetric random walk with i.i.d. exponential jump times), and let
$$u(t,x):=\mathbf E\left[\exp\left(\int\_0^tV(X\_s)~ds\right)f(X\_t)\bigg|X\_0=x\right]
\tag{1}$$
for $(t,x)\in[0,\infty)\times\mathbb Z^d$, where $V,f:\mathbb Z^d\to\mathbb R$ are functions.


According to the Feynman-Kac formula, we know that $u(t,x)$ solves the lattice/matrix heat equation
$$\partial\_tu=\tfrac12\Delta u+Vu,\qquad u(0,x)=f(x),\tag{2}$$
where
$$\Delta:=\left[
\begin{array}{ccccc}
&\ddots&\ddots&\\
&\ddots&-2&1&\\
&&1&-2&1&\\
&&&1&-2&\ddots\\
&&&&\ddots&\ddots&
\end{array}\right]$$
is the discrete Laplacian, and we think of $V$ as the diagonal matrix
$$V=\left[
\begin{array}{ccccc}
&\ddots&&\\
&&V(-1)&&\\
&&&V(0)&&\\
&&&&V(1)&&\\
&&&&&\ddots&&
\end{array}\right].$$




---


As an alternative to $(2)$, a common model for a lattice heat equation is to consider
$$\partial\_tu=\tfrac12\Delta u+\tilde Vu,\qquad u(0,x)=f(x),\tag{3}$$
where the potential $\tilde V$ is instead of the form
$$\tilde V=\left[
\begin{array}{ccccc}
&\ddots&\ddots&\\
&\ddots&0&V(-1)&\\
&&V(-1)&0&V(0)&\\
&&&V(0)&0&V(1)&\\
&&&&V(1)&0&\ddots\\
&&&&&\ddots&\ddots&
\end{array}\right],$$
or a more general tridiagonal matrix
$$\tilde V=\left[
\begin{array}{ccccc}
&\ddots&\ddots&\\
&\ddots&U(-1)&V(-1)&\\
&&V(-1)&U(0)&V(0)&\\
&&&V(0)&U(1)&V(1)&\\
&&&&V(1)&U(2)&\ddots\\
&&&&&\ddots&\ddots&
\end{array}\right].$$




---



> 
> **Question.** Does there exist a Feynman-Kac formula *similar to* $(1)$ for lattice operators with non-diagonal potential such as $(3)$?
> 
> 
> 


To clarify a bit what I mean by *similar to* $(1)$: It's easy enough to come up with *some* probabilistic representation of the solution of $(3)$ (for example by using the Trotter-Kato theorem: $e^{\Delta/2+V}\approx(e^{\Delta/2n}e^{V/n})^n$ for large $n$), but I can't get anything *nice* like $(1)$, and It's not clear to me if we should/shouldn't expect such a nice representation in those cases.



Feynman-Kac formula for lattice heat equation with non-diagonal potential
A Feynman-Kac formula for (3) is given by (1) with $V$ replaced with $$\left[
\begin{array}{ccccc}
&\ddots&&\\
&&V(-2)+U(-1)+V(-1)&&\\
&&&V(-1)+U(0)+V(0)&&\\
&&&&V(0)+U(1) + V(1)&&\\
&&&&&\ddots&&
\end{array}\right].$$ and the stochastic process $X\_t$ being the one generated by the following infinitesimal generator $$
L f(i) = \frac{1}{2} (f(i+1) - 2 f(i) + f(i-1)) + V(i) (f(i+1)-f(i)) + V(i-1) (f(i-1)-f(i)) \;.
$$


