Probability Theory and Related Fields

, Volume 114, Issue 3, pp 279–289

On the long time behavior of the stochastic heat equation

  • Lorenzo Bertini
  • Giambattista Giacomin


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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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Lorenzo Bertini
    • 1
  • Giambattista Giacomin
    • 2
  1. 1.Dipartimento di Matematica, Università di Roma La Sapienza, P.le A. Moro 2, I-00185 Roma, Italy. e-mail: bertini@mat.uniroma1.itIT
  2. 2.Institut für Angewandte Mathematik der Universität Zürich-Irchel, Winterthurerstr. 190, CH-8057 Zürich, SwitzerlandCH

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