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

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?

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
• This is not compatible with the bound you wrote because $\lambda_{\Omega}$ equals $1/2$ in this case. Jul 3 '20 at 2:01