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](http://www.ems-ph.org/JOURNALS/show_pdf.php?issn=0213-2230&vol=15&iss=1&rank=7) and [continuous regions with boundaries](https://www.sciencedirect.com/science/article/pii/S0022123614000299) 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?



Gaussian bounds with exponential decay for discrete (graph) Dirichlet heat kernel
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](https://mathoverflow.net/questions/363463/exit-time-estimate-for-a-simple-continuous-time-random-walk)


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


