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Weak and strong disorder for the stochastic heat equation and continuous directed polymers in \(d\geq 3\). (English) Zbl 1348.60094

Summary: We consider the smoothed multiplicative noise stochastic heat equation \[ \text{d}u_{\varepsilon, t}= \frac{1}{2} \Delta u_{\varepsilon ,t} \text{d}t+\beta \varepsilon^{\frac{d-2}{2}} u_{\varepsilon, t}\text{d}B_{\varepsilon, t}, \;\;u_{\varepsilon, 0}=1, \] 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.

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
60G15 Gaussian processes
82D60 Statistical mechanics of polymers
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