Equations with random coefficients: Convergence to deterministic or stochastic limits and theory of correctors
Speaker: Guillaume Bal, Columbia University
Abstract: The theory of homogenization for equations with random coefficients is now rather well-developed. What is less studied is the theory for the correctors to homogenization, which asymptotically characterize the randomness in the solution of the equation and as such are important to quantify in many areas of applied sciences. I will present some results in the theory of correctors for elliptic and parabolic problems. Homogenized (deterministic effective medium) solutions are not the only possible limits for solutions of equations with highly oscillatory random coefficients as the correlation length in the medium converges to zero. When fluctuations are sufficiently large, the limit may take the form of a stochastic equation and stochastic partial differential equations (SPDE) are routinely used to model small scale random forcing. In the specific setting of a parabolic equation with large, Gaussian, random potential, I will show the following result: in low spatial dimensions, the solution to the parabolic equation indeed converges to the solution of a SPDE with multiplicative noise, which needs to be written in a Stratonovich form; in high spatial dimension, the solution to the parabolic equation converges to a homogenized (hence deterministic) equation and randomness appears as a central limit- type corrector solution of a SPDE with additive noise. One of the possible corollaries for this result is that SPDE models may indeed be appropriate in low spatial dimensions but not necessarily in higher spatial dimensions.
Room Reservation Information
Room Number: 113 McAllister
Time: 1:00pm - 2:00pm