Abstract:
We study the Kardar-Parisi-Zhang equation in dimension $d\geq 3$ with space-time white noise which is smoothed in space. When the intensity of the noise is small, the approximating solution converges to a random limit. We show that the rate of this convergence is polynomial and the finite dimensional distributions of the rescaled deviations from this random limit converge to a Gaussian field. The latter limit is a solution of the non noisy heat equation but with a random initial condition given by the Gaussian free field. Joint work with F. Comets (Paris/Shanghai) and C. Cosco (Paris).
The solution of the related stochastic heat equation carries a natural Gaussian mutiplicative noise (GMC) on the Wiener space. As a natural counterpart of the small noise case, we also show that when the disorder is large, the total mass of this GMC loses uniform integrability and converges to zero, while the endpoint distribution of a Brownian path under the GMC measure enters the so-called glassy phase and produces only purely atomic states. Joint work with Yannic Bröker (Münster).
Biography:
Chiranjib Mukherjee is a professor of Mathematics specializing in probability theory at the University of Muenster, Germany. Prior to his appointment at Muenster, he was a visiting assistant professor of mathematics at the Courant Institute, NYU and he held postdoc positions at Weierstrass Institute Berlin and TU Munich. He completed his PhD from Max-Planck Institute for Mathematics and University of Leipzig. A major thrust of his work comes from studying stochastic partial differential equations, directed polymers, random walks in random environments and applications in quantum physics and statistical mechanics.
Seminar by the NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai