an:01553923
Zbl 1080.60508
Bertini, Lorenzo; Cancrini, Nicoletta
The stochastic heat equation: Feynman-Kac formula and intermittence
EN
J. Stat. Phys. 78, No. 5-6, 1377-1401 (1995).
00053517
1995
j
60H15 60J50 82C31
stochastic partial differential equations; random media; moment Lyapunov exponents; local times
Summary: We study, in one space dimension, the heat equation with a random potential that is a white noise in space and time. This equation is a linearized model for the evolution of a scalar field in a space-time-dependent random medium. It has also been related to the distribution of two-dimensional directed polymers in a random environment, to the KPZ model of growing interfaces, and to the Burgers equation with conservative noise. We show how the solution can be expressed via a generalized Feynman-Kac formula. We then investigate the statistical properties: the two-point correlation function is explicitly computed and the intermittence of the solution is proven. This analysis is carried out showing how the statistical moments can be expressed through local times of independent Brownian motions.