A stochastic analogue of Aubry–Mather theory*
نویسندگان
چکیده
In this paper, we discuss a stochastic analogue of Aubry–Mather theory in which a deterministic control problem is replaced by a controlled diffusion. We prove the existence of a minimizing measure (Mather measure) and discuss its main properties using viscosity solutions of Hamilton–Jacobi equations. Then we prove regularity estimates on viscosity solutions of the Hamilton–Jacobi equation using the Mather measure. Finally, we apply these results to prove asymptotic estimates on the trajectories of controlled diffusions and study the convergence of Mather measures as the rate of diffusion vanishes. Mathematics Subject Classification: 35J60, 37J40, 49L25, 60H30
منابع مشابه
un 2 00 5 Symplectic aspects of Aubry - Mather theory 1
On montre que les ensembles d'Aubry et de Mañé introduits par Mather en dynamique Lagrangienne sont des invariants symplectiques. On introduit pour ceci une barriere dans l'espace des phases. Ceci est aussi l'occasion d'´ ebaucher une théorie d'Aubry-Mather pour des Hamiltoniens non convexes. Abstract : We prove that the Aubry and Mañé sets introduced by Mather in Lagrangian dynamics are symple...
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