Ergodic Control of Infinite Dimensional SDEs with Degenerate Noise

TL;DR

This paper investigates the long-term behavior of stochastic control problems in infinite-dimensional spaces with degenerate noise, using probabilistic methods to connect finite, infinite horizon, and ergodic control scenarios.

Contribution

It introduces a probabilistic approach to analyze ergodic control of infinite-dimensional SDEs with degenerate noise, avoiding traditional analytical viscosity solution techniques.

Findings

Established a probabilistic framework for ergodic control in infinite dimensions.

Connected finite and infinite horizon control problems with ergodic limits.

Handled degenerate diffusion coefficients in infinite-dimensional settings.

Abstract

The present paper is devoted to the study of the asymptotic behavior of the value functions of both finite and infinite horizon stochastic control problems and to the investigation of their relation with suitable stochastic ergodic control problems. Our methodology is based only on probabilistic techniques, as for instance the so-called randomization of the control method, thus avoiding completely analytical tools from the theory of viscosity solutions. We are then able to treat with the case where the state process takes values in a general (possibly infinite dimensional) real separable Hilbert space and the diffusion coefficient is allowed to be degenerate.