Bertini, Lorenzo; Giacomin, Giambattista On the long time behavior of the stochastic heat equation. (English) Zbl 0934.60058 Probab. Theory Relat. Fields 114, No. 3, 279-289 (1999). Summary: 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 nonstationary 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. Cited in 14 Documents MSC: 60H15 Stochastic partial differential equations (aspects of stochastic analysis) 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics Keywords:stochastic heat equation; sample Lyapunov exponent; directed polymers in random environment; simple exclusion processes PDFBibTeX XMLCite \textit{L. Bertini} and \textit{G. Giacomin}, Probab. Theory Relat. Fields 114, No. 3, 279--289 (1999; Zbl 0934.60058) Full Text: DOI