Probability Theory and Related Fields

, Volume 114, Issue 3, pp 279–289

On the long time behavior of the stochastic heat equation

Authors

  • Lorenzo Bertini
    • Dipartimento di Matematica, Università di Roma La Sapienza, P.le A. Moro 2, I-00185 Roma, Italy. e-mail: bertini@mat.uniroma1.it
  • Giambattista Giacomin
    • Institut für Angewandte Mathematik der Universität Zürich-Irchel, Winterthurerstr. 190, CH-8057 Zürich, Switzerland

DOI: 10.1007/s004400050226

Cite this article as:
Bertini, L. & Giacomin, G. Probab Theory Relat Fields (1999) 114: 279. doi:10.1007/s004400050226

Abstract

We consider the stochastic heat equation in one space dimension and compute – for a particular choice of the initial datum – the exact long time asymptotic. In the Carmona-Molchanov approach to intermittence in non stationary random media this corresponds to the identification of the sample Lyapunov exponent. Equivalently, by interpreting the solution as the partition function of a directed polymer in a random environment, we obtain a weak law of large numbers for the quenched free energy. The result agrees with the one obtained in the physical literature via the replica method. The proof is based on a representation of the solution in terms of the weakly asymmetric exclusion process.

Mathematics Subject Classification (1991): Primary 60H15, 60K35; secondary 82B44
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© Springer-Verlag Berlin Heidelberg 1999