Quenched central limit theorem for the stochastic heat equation in weak disorder

Abstract

We continue with the study of the mollified stochastic heat equation in $d\ge 3$ given by $d{u}_{\epsilon ,t}=\frac{1}{2}\Delta u_{\epsilon ,t}+\beta {\epsilon }^{(d-2)/2}{u}_{\epsilon ,t}d{B}_{\epsilon ,t}$ with spatially smoothened cylindrical Wiener process $B$, whose (renormalized) Feynman-Kac solution describes the partition function of the continuous directed polymer. In an earlier work (\cite{MSZ16}), a phase transition was obtained, depending on the value of $\beta >0$ in the limiting object of the smoothened solution $u_{\epsilon }$ as the smoothing parameter $\epsilon \to 0$ This partition function naturally defines a quenched polymer path measure and we prove that as long as $\beta >0$ stays small enough while $u_{\epsilon }$ converges to a strictly positive non-degenerate random variable, the distribution of the diffusively rescaled Brownian path converges under the aforementioned polymer path measure to standard Gaussian distribution.

Minor revision

References in corpus (1)

• Brownian Polymers in Poissonian Environment: a survey

Cited by in corpus (2)

• A PDE hierarchy for directed polymers in random environments
• Geometry of the Gaussian multiplicative chaos in the Wiener space