Convergence to SPDEs for second order evolutionary equation with singular short–range correlated potential

Abstract

The random homogenization for second order evolutionary equation with singular short–range correlated potential is derived. Comparing with the first order evolutionary equation, more difficulty need to be overcome in the moment estimation of the solution due to non–symmetrical semigroup of second order evolutionary equation. In our approach the solution is written out by the Duhamel's formula and the moment estimation of the solution is obtained by some analytic methods. Then by means of diagrammatic expansions and chaos expansions, the solution is shown to converge in distribution to the solution of stochastic partial differential equations (SPDEs) in Stratonovich form driven by spatial white noise.