Gaussian bounds with exponential decay for discrete (graph) Dirichlet heat kernel

Question

Let $\Omega$ be a finite, connected subset of $\mathbb{Z}^n$, $W_t$ a standard random walk on $\mathbb{Z}^n$ started at $x$, and $T_\Omega$ the first time at which $W_t$ leaves $\Omega$; consider $$ P^D_\Omega(x,y;t) := \mathbb{P}[W_t=y \text{ and } T_\Omega>t], $$ the discrete or graph heat kernel on $\Omega$ with Dirichlet boundary conditions.

By analogy with some known results on infinite graphs and continuous regions with boundaries one would expect a bound of the form $$ P_\Omega^D(x,y;t) \le C_\Omega \frac{\phi_\Omega(x,t) \phi_\Omega(y,t) e^{-\lambda_\Omega t}}{t^{n/2}} e^{- c |x-y|^2/t}, $$ with $\phi:\mathbb{Z}^n \times \mathbb{N} \to [0,1]$ vanishing outside $\Omega$ (also with some bounds near the boundary of $\Omega$).

Is this known?

Answer 1

I assume the question pertains to continuous time random walk; the counterexamples are even simpler in discrete time. There is no reason to expect the power law factor $t^{-n/2}$ in this setting. For the simplest example, consider the case where $\Omega$ consists of two adjacent points $x,y$ in $\mathbb{Z}$. Then $$ P_\Omega^D(x,y;t)=\sum_{k \ge 0} 2^{-2k-1} P({\rm Poisson }(t)=2k+1)= \sum_{k \ge 0}\frac{(t/2)^{2k+1}}{(2k+1)! \, e^t}=\frac{\sinh(t/2)}{e^t}$$ which is asymptotically $\exp(-t/2)\cdot (1/2-o(1))$.

A good discussion of this topic when $\Omega$ is an interval and the walk is discrete can be found in page 243 of [1]. This is easily converted to continuous time, see e.g. Exit time estimate for a simple continuous-time random walk

[1] Spitzer, Frank. Principles of random walk. GTM Vol. 34. Second edition, Springer.