The free-space heat kernel is given by

$K(t,x,y) = \frac{1}{(4\pi t)^{d/2}} e^{-|x-y|^2/4t}$, with $x,y \in \mathbb{R}^d$ and $t>0$.

This expression shows that the heat kernel decreases as the distance from $x$ to $y$ increases.

For a bounded domain $\Omega$ in $\mathbb{R}^d$, there is no general explicit expression for the heat kernel, but it can be expressed as a series of eigenfunctions:

$K(t,x,y) = \sum_{i=1}^{\infty} e^{-\lambda_i t}\phi_i(x)\phi_i(y)$,

where $\phi_i$ and $\lambda_i$ are eigenfunctions and eigenvalues of the problem

$\Delta \phi + \lambda \phi = 0$ in $\Omega$, and $\phi = 0$ on $\partial \Omega$.

Is it possible to say something about how the heat kernel behaves as the distance from $x$ to $y$ increases from the series expansion above?



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