Open Access
January 2017 Dissipation and high disorder
Le Chen, Michael Cranston, Davar Khoshnevisan, Kunwoo Kim
Ann. Probab. 45(1): 82-99 (January 2017). DOI: 10.1214/15-AOP1040


Given a field $\{B(x)\}_{x\in\mathbf{Z}^{d}}$ of independent standard Brownian motions, indexed by $\mathbf{Z}^{d}$, the generator of a suitable Markov process on $\mathbf{Z}^{d},\mathcal{G}$, and sufficiently nice function $\sigma:[0,\infty)\mapsto [0,\infty)$, we consider the influence of the parameter $\lambda$ on the behavior of the system, \begin{eqnarray*}\mathrm{d}u_{t}(x)&=&(\mathcal{G}u_{t})(x)\,\mathrm{d}t+\lambda\sigma(u_{t}(x))\,\mathrm{d}B_{t}(x)\qquad [t>0,\ x\in\mathbf{Z}^{d}],\\u_{0}(x)&=&c_{0}\delta_{0}(x).\end{eqnarray*} We show that for any $\lambda>0$ in dimensions one and two the total mass $\sum_{x\in\mathbf{Z}^{d}}u_{t}(x)$ converges to zero as $t\to\infty$ while for dimensions greater than two there is a phase transition point $\lambda_{c}\in(0,\infty)$ such that for $\lambda>\lambda_{c},\sum_{x\in \mathbf{Z}^{d}}u_{t}(x)\to0$ as $t\to\infty$ while for $\lambda<\lambda_{c},\sum_{x\in \mathbf{Z}^{d}}u_{t}(x)\not\to0$ as $t\to\infty$.


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Le Chen. Michael Cranston. Davar Khoshnevisan. Kunwoo Kim. "Dissipation and high disorder." Ann. Probab. 45 (1) 82 - 99, January 2017.


Received: 1 November 2014; Revised: 1 June 2015; Published: January 2017
First available in Project Euclid: 26 January 2017

zbMATH: 1382.60103
MathSciNet: MR3601646
Digital Object Identifier: 10.1214/15-AOP1040

Primary: 60J60 , 60K35 , 60K37
Secondary: 47B80 , 60H25

Keywords: Parabolic Anderson model , Stochastic partial differential equations , Strong disorder

Rights: Copyright © 2017 Institute of Mathematical Statistics

Vol.45 • No. 1 • January 2017
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