Probabilistic Approaches to Partial Differential Equations with Large Random Potentials

The thesis is devoted to an analysis of the heat equation with large random potentials in high dimensions. The size of the potential is chosen so that the large, highly oscillatory, random field is producing non-trivial effects in the asymptotic limit. We prove either homogenization, i.e., the random potential is replaced by some deterministic constant, or convergence to a stochastic partial differential equation, i.e., the random potential is replaced by some stochastic noise, depending on the correlation property. When the limit is deterministic, we provide estimates of the error between the heterogeneous and homogenized solutions when certain mixing assumption of the random potential is satisfied. We also prove a central limit type of result when the random potential is Gaussian or Poissonian. Lower dimensional and time-dependent cases are also treated. Most of the ingredients in the analysis are probabilistic, including a Feynman-Kac representation, a Brownian motion in random scenery, the Kipnis-Varadhan's method, and a quantitative martingale central limit theorem.