Ellipticity and Ergodicity

TL;DR

This paper characterizes when a submarkovian semigroup generated by an elliptic operator preserves functions supported in a subset, linking invariance to flows generated by associated vector fields.

Contribution

It establishes a precise criterion for invariance of $L_2$ spaces under the semigroup, connecting elliptic operator properties with flow invariance.

Findings

Invariance of $L_2( i^d)$ subsets is equivalent to invariance under specific flows.

Provides conditions under which the semigroup preserves functions supported in open subsets.

Links the invariance property to the structure of the elliptic operator and its associated vector fields.

Abstract

Let $S=\{S_t\}_{t\geq0}$ be the submarkovian semigroup on $L_2(\Ri^d)$ generated by a self-adjoint, second-order, divergence-form, elliptic operator $H$ with Lipschitz continuous coefficients $c_{ij}$. Further let $\Omega$ be an open subset of $\Ri^d$. Under the assumption that $C_c^\infty(\Ri^d)$ is a core for $H$ we prove that $S$ leaves $L_2(\Omega)$ invariant if, and only if, it is invariant under the flows generated by the vector fields $Y_i=\sum^d_{j=1}c_{ij}\partial_j$.