# 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
-∆_u + λu* + |∇_u_|

*q*=

*f*(

*x*) in Ω

*u*= 0 on ∂ Ω,

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