Homogenization for stochastic Hamilton-Jacobi equations
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.
scientific article; zbMATH DE number 1454839
scientific article