Homogenization of Hamilton-Jacobi-Bellman equations with respect to time-space shifts in a stationary ergodic medium. (English) Zbl 1144.35008

Summary: We consider a family \(\{u_\varepsilon(t, x,\omega)\}\), \(\varepsilon> 0\), of solutions to the equation \[ \partial u_\varepsilon/\partial t+\varepsilon\Delta u_\varepsilon/2+ H(t/\varepsilon, x/\varepsilon,\nabla u_\varepsilon, \omega)= 0 \] with the terminal data \(u_\varepsilon(T,x,\omega)= U(x)\). Assuming that the dependence of the Hamiltonian \(H(t,x,p,\omega)\) on time and space is realized through shifts in a stationary ergodic random medium, and that \(H\) is convex in \(p\) and satisfies certain growth and regularity conditions, we show the almost sure locally uniform convergence, in time and space, of \(u_\varepsilon(t,x,\omega)\) as \(\varepsilon\to 0\) to the solution \(u(t, x)\) of a deterministic averaged equation \(\partial u/\partial t+ \overline H(\nabla u)= 0\), \(u(T,x)= U(x)\). The “effective” Hamiltonian \(H\) is given by a variational formula.


35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
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