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
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
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