## The Annals of Probability

- Ann. Probab.
- Volume 45, Number 1 (2017), 82-99.

### Dissipation and high disorder

Le Chen, Michael Cranston, Davar Khoshnevisan, and Kunwoo Kim

#### Abstract

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 $t\to\infty$.

#### Article information

**Source**

Ann. Probab., Volume 45, Number 1 (2017), 82-99.

**Dates**

Received: November 2014

Revised: June 2015

First available in Project Euclid: 26 January 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1485421329

**Digital Object Identifier**

doi:10.1214/15-AOP1040

**Mathematical Reviews number (MathSciNet)**

MR3601646

**Zentralblatt MATH identifier**

1382.60103

**Subjects**

Primary: 60J60: Diffusion processes [See also 58J65] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K37: Processes in random environments

Secondary: 47B80: Random operators [See also 47H40, 60H25] 60H25: Random operators and equations [See also 47B80]

**Keywords**

Parabolic Anderson model strong disorder stochastic partial differential equations

#### Citation

Chen, Le; Cranston, Michael; Khoshnevisan, Davar; Kim, Kunwoo. Dissipation and high disorder. Ann. Probab. 45 (2017), no. 1, 82--99. doi:10.1214/15-AOP1040. https://projecteuclid.org/euclid.aop/1485421329