On the Equivalence of Heat Kernels of Second-order parabolic operators

TL;DR

This paper demonstrates that under certain conditions, the heat kernels of a second-order elliptic operator and its small perturbation are equivalent on noncompact Riemannian manifolds, ensuring stability of solutions and boundary structures.

Contribution

It establishes the equivalence of heat kernels and stability of the parabolic Martin boundary under small perturbations for a broad class of elliptic operators on noncompact manifolds.

Findings

Positive minimal heat kernels are equivalent under perturbations.

The parabolic Martin boundary remains stable under small perturbations.

Cones of nonnegative solutions are affine homeomorphic.

Abstract

Let $P$ be a second-order, symmetric, and nonnegative elliptic operator with real coefficients defined on noncompact Riemannian manifold $M$, and let $V$ be a real valued function which belongs to the class of {\em small perturbation potentials} with respect to the heat kernel of $P$ in $M$. We prove that under some further assumptions (satisfying by a large classes of $P$ and $M$) the positive minimal heat kernels of $P-V$ and of $P$ on $M$ are equivalent. Moreover, the parabolic Martin boundary is stable under such perturbations, and the cones of all nonnegative solutions of the corresponding parabolic equations are affine homeomorphic