Rigidity results in Diffusion Markov Triples

TL;DR

This paper proves that under certain curvature conditions in a general diffusion Markov triple setting, stable solutions with integrable carré du champs must be constant, extending previous results and providing new insights.

Contribution

It establishes rigidity results for stable solutions in diffusion Markov triples under curvature conditions, generalizing and unifying prior findings.

Findings

Stable solutions with integrable carré du champs are constant under curvature conditions.

Theorems encompass previous results as special cases and introduce new structural insights.

A geometric Poincaré formula underpins the proofs.

Abstract

We consider stable solutions of semilinear equations in a very general setting. The equation is set on a Polish topological space endowed with a measure and the linear operator is induced by a carr\'e du champs (equivalently, the equation is set in a diffusion Markov triple). Under suitable curvature dimension conditions, we establish that stable solutions with integrable carr\'e du champs are necessarily constant (weaker conditions characterize the structure of the carr\'e du champs and carr\'e du champ it\'er\'e). The proofs are based on a geometric Poincar\'e formula in this setting. From the general theorems established, several previous results are obtained as particular cases and new ones are provided as well.