# The ‘‘ergodic limit’’ for a viscous Hamilton−Jacobi equation with Dirichlet conditions

### Alessio Porretta

Università di Roma, Italy

## Abstract

We study the limit, when $λ$ tends to $0$, of the solutions $u_{λ}$ of the Dirichlet problem

when $1<q≤2$ and $f$ is bounded. In case the limit problem does not have any solution, we prove that $u_{λ}$ has a complete blow-up $(u_{λ}→−∞)$ and its behaviour is described in terms of the corresponding ergodic problem with state constraint conditions. In particular, $λu_{λ}$ converges to the ergodic constant $c_{0}$ and $u_{λ}+∥u_{λ}∥_{∞}$ converges to the boundary blow-up solution $ν_{0}$ of the ergodic problem associated to the stochastic optimal control with state constraint.

## Cite this article

Alessio Porretta, The ‘‘ergodic limit’’ for a viscous Hamilton−Jacobi equation with Dirichlet conditions. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 21 (2010), no. 1, pp. 59–78

DOI 10.4171/RLM/561