Abstract
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.
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Bertini, L., Cancrini, N. The stochastic heat equation: Feynman-Kac formula and intermittence. J Stat Phys 78, 1377–1401 (1995). https://doi.org/10.1007/BF02180136
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DOI: https://doi.org/10.1007/BF02180136