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Homogenization for stochastic Hamilton-Jacobi equations. (English) Zbl 0954.35022
Homogenization results for the Hamilton-Jacobi equation $\partial_{t}u^\varepsilon + H({x\over\varepsilon},Du^\varepsilon, \omega) = 0 \;\text{in $$\mathbb{R}^{d}\times \left]0,\infty\right[$$}, \quad u^\varepsilon(0,\cdot) = g \;\text{on $$\mathbb{R}^{d}$$}, \tag{1}$ with a random Hamiltonian $$H$$ are studied. Let $$(\tau_{x}, x\in \mathbb{R}^{d})$$ be a group of ergodic measure preserving transformations of a probability space $$(\Omega, \mathcal F,\mathbf P)$$. Let $$H(x,q,\omega) = \widetilde H(q,\tau_{x}\omega)$$, the function $$\widetilde H(\cdot,\omega)$$ being convex, coercive and continuously differentiable $$\mathbf P$$-almost surely. Let $$g:\mathbb{R}^{d}\to\mathbb{R}$$ be a Lipschitz function, denote by $$u^\varepsilon$$ the viscosity solution to (1). For a $$\delta>0$$ set $$A(\delta) = \{(t,x); t\geq\delta, |x|\leq\delta^{-1}\}$$. Under suitable hypotheses on $$H$$ it is proven that $$\lim_{\varepsilon \to 0}\mathbf E\sup_{(t,x)\in A(\delta)}|u^\varepsilon(t,x, \omega)- \bar u(t,x)|= 0$$ for every $$\delta>0$$, where $$\bar u(t,x) = \inf_{y\in\mathbb{R}^{d}}\{g(y) + t\bar L(t^{-1}(x-y))\}$$ for a convex coercive function $$\bar L$$. For Hamiltonians of a particular form, a central limit theorem for the convergence of $$u^\varepsilon$$ to $$\bar u$$ is established.
Moreover, homogenization results for nonconvex Hamiltonians are obtained in the deterministic case.

##### MSC:
 35B27 Homogenization in context of PDEs; PDEs in media with periodic structure 35R60 PDEs with randomness, stochastic partial differential equations 60H15 Stochastic partial differential equations (aspects of stochastic analysis)
##### Keywords:
central limit theorem; nonconvex Hamiltonians
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