A generalized conservation property for the heat semigroup on weighted manifolds

TL;DR

This paper investigates a generalized conservation property for the heat semigroup on weighted manifolds, establishing criteria and exploring applications related to Schrödinger operators with nonnegative potentials.

Contribution

It introduces a generalized conservation property for the heat semigroup and provides Khasminskii's criterion for this property on weighted manifolds.

Findings

Established Khasminskii's criterion for the generalized conservation property.

Applied the criterion to various scenarios involving Schrödinger operators.

Extended classical conservation results to weighted manifold settings.

Abstract

In this text we study a generalized conservation property for the heat semigroup generated by a Schr\"odinger operator with nonnegative potential on a weighted manifold. We establish Khasminskii's criterion for the generalized conservation property and discuss several applications.