Homogenization of random Hamilton–Jacobi–Bellman Equations
S. R. Srinivasa Varadhan · 2008
We consider nonlinear parabolic equations of Hamilton–Jacobi– Bellman type. The Lagrangian is assumed to be convex, but with a spatial dependence which is stationary and random. Rescaling in space and time produces a similar equation with a rapidly varying spatial dependence and a small viscosity term. Motivated by corresponding results for the linear elliptic equation with small viscosity, we seek to find the limiting behavior of the solution of the Cauchy (final value) problem in terms of a homogenized problem, described by a convex function of the gradient of the solution. The main idea is to use the principle of dynamic programming to write a variational formula for the solution in terms of solutions of linear problems. We then show that asymptotically it is enough to restrict the optimization to a subclass, one for which the asymptotic behavior can be fully analyzed. The paper outlines these steps and refers to the recently published work of Kosygina, Rezakhanlou and the author for full details. Homogenization is a theory about approximating solutions of a differential equation with rapidly varying coefficients by a solution of a constant coefficient differential equation of a similar nature. The simplest example of its kind is the solution u ϵ of the equation on [0, ∞] ⨯ ℝ. The function a (·) is assumed to be uniformly positive, continuous and periodic of period 1.