Homogenization of Heat Equation with Large Time-dependent Random Potential

Stochastic Processes and their Applications (Impact Factor: 1.06). 01/2014; 125(1). DOI: 10.1016/
Source: arXiv

ABSTRACT This paper concerns the homogenization problem of heat equation with large,
time-dependent, random potentials in high dimensions $d\geq 3$. Depending on
the competition between temporal and spatial mixing of the randomness, the
homogenization procedure turns to be different. We characterize the difference
by proving the corresponding weak convergence of Brownian motion in random
scenery. When the potential depends on the spatial variable macroscopically, we
prove a convergence to SPDE.

4 Reads
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: In this paper, we establish a version of the Feynman-Kac formula for multidimensional stochastic heat equation driven by a general semimartingale. This Feynman-Kac formula is then applied to study some nonlinear stochastic heat equations driven by nonhomogenous Gaussian noise: First, it is obtained an explicit expression for the Malliavin derivatives of the solutions. Based on the representation we obtain the smooth property of the density of the law of the solution. On the other hand, we also obtain the H\"older continuity of the solutions.
    Stochastic Processes and their Applications 10/2011; 123(3). DOI:10.1016/ · 1.06 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: This paper deals with the homogenization problem for a one-dimensional parabolic PDE with random stationary mixing coefficients in the presence of a large zero order term. We show that under a proper choice of the scaling factor for the said zero order terms, the family of solutions of the studied problem converges in law, and describe the limit process. It should be noted that the limit dynamics remain random.
    Annales de l Institut Henri Poincaré Probabilités et Statistiques 07/2008; 44(3). DOI:10.1214/07-AIHP134 · 1.06 Impact Factor
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: We consider the homogenization of parabolic equations with large spatially-dependent potentials modeled as Gaussian random fields. We derive the homogenized equations in the limit of vanishing correlation length of the random potential. We characterize the leading effect in the random fluctuations and show that their spatial moments converge in law to Gaussian random variables. Both results hold for sufficiently small times and in sufficiently large spatial dimensions $d\geq\m$, where $\m$ is the order of the spatial pseudo-differential operator in the parabolic equation. In dimension $d<\m$, the solution to the parabolic equation is shown to converge to the (non-deterministic) solution of a stochastic equation in the companion paper [2]. The results are then extended to cover the case of long range random potentials, which generate larger, but still asymptotically Gaussian, random fluctuations.
    SIAM Journal on Multiscale Modeling and Simulation 10/2008; 8(4). DOI:10.1137/090754066 · 1.63 Impact Factor
Show more


4 Reads
Available from