Convergence to Homogenized or Stochastic Partial Differential Equations

Abstract

We consider the behavior of solutions to parabolic equations with large, highly oscillatory, possibly time dependent, random potential with Gaussian statistics. The Gaussian potential fluctuates in the spatial variables and possibly in the temporal variable. We seek the limit of the solution to the parabolic equation as the scale ε at which the random medium oscillates converges to zero. Depending on spatial dimension and on the decorrelation properties of the Gaussian potential, we show that the solution converges, as ε tends to 0, either to the solution of a deterministic, homogenized, equation with negative effective medium potential or to the solution of a stochastic partial differential equation with multiplicative noise that should be interpreted as a Stratonovich integral. The transition between the deterministic and stochastic limits depends on the elliptic operator in the parabolic equation and on the decorrelation properties of the random potential. In the setting of convergence to a deterministic solution, we characterize the random corrector, which asymptotically captures the stochasticity in the solution. Such models can be used to calibrate upscaling schemes that aim at understanding the influence of microscopic structures in macroscopic calculations.