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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.

##### MSC:
 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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