The ergodic problem for some subelliptic operators with unbounded coefficients

TL;DR

This paper investigates the existence and uniqueness of invariant measures for certain degenerate stochastic processes governed by subelliptic operators with unbounded coefficients, with implications for ergodic problems and long-term behavior analysis.

Contribution

It establishes conditions for invariant measure existence and uniqueness for subelliptic operators with unbounded coefficients, advancing understanding of ergodic properties in degenerate diffusions.

Findings

Existence and uniqueness of invariant measures proved.

Liouville-type theorem established for these operators.

Applications to ergodic problems and long-term solution behavior.

Abstract

We study existence and uniqueness of the invariant measure for a stochastic process with degenerate diffusion, whose infinitesimal generator is a linear subelliptic operator in the whole space R N with coefficients that may be unbounded. Such a measure together with a Liouville-type theorem will play a crucial role in two applications: the ergodic problem studied through stationary problems with vanishing discount and the long time behavior of the solution to a parabolic Cauchy problem. In both cases, the constants will be characterized in terms of the invariant measure.