Translator Disclaimer
2016 Weak and strong disorder for the stochastic heat equation and continuous directed polymers in $d\geq 3$
Chiranjib Mukherjee, Alexander Shamov, Ofer Zeitouni
Electron. Commun. Probab. 21: 1-12 (2016). DOI: 10.1214/16-ECP18

Abstract

We consider the smoothed multiplicative noise stochastic heat equation \[\mathrm{d} u_{\varepsilon ,t}= \frac 12 \Delta u_{\varepsilon ,t} \mathrm{d} t+ \beta \varepsilon ^{\frac{d-2} {2}}\, \, u_{\varepsilon , t} \, \mathrm{d} B_{\varepsilon ,t} , \;\;u_{\varepsilon ,0}=1,\] in dimension $d\geq 3$, where $B_{\varepsilon ,t}$ is a spatially smoothed (at scale $\varepsilon $) space-time white noise, and $\beta >0$ is a parameter. We show the existence of a $\bar \beta \in (0,\infty )$ so that the solution exhibits weak disorder when $\beta <\bar \beta $ and strong disorder when $\beta > \bar \beta $. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.

Citation

Download Citation

Chiranjib Mukherjee. Alexander Shamov. Ofer Zeitouni. "Weak and strong disorder for the stochastic heat equation and continuous directed polymers in $d\geq 3$." Electron. Commun. Probab. 21 1 - 12, 2016. https://doi.org/10.1214/16-ECP18

Information

Received: 19 January 2016; Accepted: 21 August 2016; Published: 2016
First available in Project Euclid: 12 September 2016

zbMATH: 1348.60094
MathSciNet: MR3548773
Digital Object Identifier: 10.1214/16-ECP18

Subjects:
Primary: 60F10, 60J55, 60J65

JOURNAL ARTICLE
12 PAGES


SHARE
Back to Top