Quasistationary Distributions and Ergodic Control Problems

TL;DR

This paper explores two dual ergodic control problems related to quasistationary distributions of diffusions, providing theoretical insights and characterizations of QSD properties through Hamilton-Jacobi-Bellman equations.

Contribution

It introduces and analyzes two dual ergodic control formulations linked to QSDs, including well-posedness and characterization results.

Findings

Proved well-posedness of the Hamilton-Jacobi-Bellman equations.

Characterized the QSD using the control problem's cost potential.

Established duality between generator-based and adjoint-based control problems.

Abstract

We introduce and study the basic properties of two ergodic stochastic control problems associated with the quasistationary distribution (QSD) of a diffusion process $X$ relative to a bounded domain. The two problems are in some sense dual, with one defined in terms of the generator associated with $X$ and the other in terms of its adjoint. Besides proving wellposedness of the associated Hamilton-Jacobi-Bellman equations, we describe how they can be used to characterize important properties of the QSD. Of particular note is that the QSD itself can be identified, up to normalization, in terms of the cost potential of the control problem associated with the adjoint.