(this is an attempt to refine [a previous question](https://mathoverflow.net/questions/363930/gaussian-bounds-with-exponential-decay-for-discrete-graph-dirichlet-heat-kerne); I was told that it would be better to create a new question than edit the previous one, I hope this is the correct ettiquete.)


Let $\Omega$ be a finite, connected subset of $\mathbb{Z}^n$, $W\_t$ a simple discrete-time 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 might expect a bound (at least for times smaller than the square of the diameter of $\Omega$) of the form
$$
P\_\Omega^D(x,y;t)
\le
C\_\Omega
\frac{\phi\_\Omega(x,t) \phi\_\Omega(y,t) }{t^{n/2}}
e^{- c |x-y|^2/t},
$$
with $\phi\_\Omega:\mathbb{Z}^n \times \mathbb{N} \to [0,1]$ vanishing outside $\Omega$ and under certain circumstances (not too close to reentrant corners?) with $\phi\_\Omega(x,t) \le d(x, \Omega^{\mathsf{c}}) /\sqrt{t}$.


Is this correct/known, in particular for $\Omega$ which is not an interval/rectangle/rectangular solid?



Gaussian bounds for discrete (graph) Dirichlet heat kernel
You can get a reasonable estimate by splitting time in three pieces, say [0,t/3], [t/3,2t/3], [2t/3,t] and estimating from above by the product of


--- the probability starting at x of staying in the domain up to time t/3


--- the probability starting at y of the the reversed walk staying in the domain up to time t/3


--- the conditional probability that S(t/3) = S(2t/3) given this which can be estimated as the supremum over all possibilities for the positions at t/3, 2t/3


If |x-y|^2 is much larger than t you might replace t/3 by the minimum of t/3 and the first time the walk traverses a distance of |x-y|/4 so that the walks in the first two steps are still distance |x-y|/2 away from each other.


