## 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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