文章基本信息

标题：Weak and strong disorder for the stochastic heat equation and continuous directed polymers in $d\geq 3$
本地全文：下载
作者：Chiranjib Mukherjee ; Alexander Shamov ; Ofer Zeitouni 等
期刊名称：Electronic Communications in Probability
印刷版ISSN：1083-589X
出版年度：2016
卷号：21
DOI：10.1214/16-ECP18
语种：English
出版社：Electronic Communications in Probability
摘要：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.