Stochastic homogenization of a class of nonconvex viscous HJ equations in one space dimension

Elena Kosygina
Bernard M. Baruch College, CUNY

We shall present a homogenization result for a class of nonconvex viscous Hamilton-Jacobi equations in stationary ergodic random environment in one space dimension. We consider Hamiltonians of the form G(p)+V(x,w), where the nonlinearity G is a minimum of two or more convex functions with the same absolute minimum, and the potential V is a stationary process satisfying an additional “valley and hill” condition introduced earlier by A.Yilmaz and O.Zeitouni. Our approach is based on PDE methods and does not rely on representation formulas for solutions. Using only comparison with suitably constructed super- and sub-solutions, we obtain tight upper and lower bounds for solutions with linear initial data which is sufficient to get a full homogenization result. Another important ingredient is a general result of P. Cardaliaguet and P. E. Souganidis which guarantees the existence of sublinear correctors outside “flat parts” of the effective Hamiltonians associated with the convex functions from which G is built. We derive derivative estimates for these correctors which allow us to use them as correctors for the
original G. This is a joint work with Andrea Davini, Sapienza University of Rome.

Presentation (PDF File)

Back to Workshop IV: Stochastic Analysis Related to Hamilton-Jacobi PDEs