Coupling by reflection for controlled diffusion processes: turnpike property and large time behavior of Hamilton Jacobi Bellman equations

TL;DR

This paper studies the long-term behavior of controlled diffusion processes via Hamilton-Jacobi-Bellman equations, establishing convergence, ergodicity, and regularity results using a probabilistic coupling approach.

Contribution

It introduces a probabilistic coupling method for analyzing the large time behavior of HJB equations and proves new regularity and convergence results.

Findings

Proved existence and uniqueness of solutions for ergodic HJB equations.

Established exponential convergence rates for value functions and controls.

Provided uniform gradient and Hessian estimates for HJB solutions.

Abstract

We investigate the long time behavior of weakly dissipative semilinear Hamilton-Jacobi-Bellman (HJB) equations and the turnpike property for the corresponding stochastic control problems. To this aim, we develop a probabilistic approach based on a variant of coupling by reflection adapted to the study of controlled diffusion processes. We prove existence and uniqueness of solutions for the ergodic Hamilton-Jacobi-Bellman equation and different kind of quantitative exponential convergence results at the level of the value function, of the optimal controls and of the optimal processes. Moreover, we provide uniform in time gradient and Hessian estimates for the solutions of the HJB equation that are of independent interest.