## The Annals of Probability

- Ann. Probab.
- Volume 23, Number 4 (1995), 1853-1874.

### Intermittency-Type Estimates for Some Nondegenerate SPDE'S

#### Abstract

In this paper we prove some intermittency-type estimates for the stochastic partial differential equation $du = \mathscr{L}u dt + \mathscr{M}_lu\circ dW^l_t$, where $\mathscr{L}$ is a strongly elliptic second-order partial differential operator and the $\mathscr{M}_l$'s are first-order partial differential operators. Here the $W^l$'s are standard Wiener processes and $\circ$ denotes Stratonovich integration. We assume for simplicity that $u(0,\cdot) \equiv 1$. Our interest here is the behavior of $\mathbb{E}\lbrack|u(t,x)|^p\rbrack$ for large time and large $p$; more specifically, our interest is the growth of $(p^2t)^{-1}\log\mathbb{E}\lbrack|u(t,x)|^p\rbrack$ as $t$, then $p$, become large.

#### Article information

**Source**

Ann. Probab., Volume 23, Number 4 (1995), 1853-1874.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176987806

**Digital Object Identifier**

doi:10.1214/aop/1176987806

**Mathematical Reviews number (MathSciNet)**

MR1379171

**Zentralblatt MATH identifier**

0852.60070

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60H15: Stochastic partial differential equations [See also 35R60]

Secondary: 47D07: Markov semigroups and applications to diffusion processes {For Markov processes, see 60Jxx} 60G60: Random fields 76W05: Magnetohydrodynamics and electrohydrodynamics 93E11: Filtering [See also 60G35]

**Keywords**

Intermittency moment Lyapunov functions stochastic partial differential equations

#### Citation

Sowers, Richard B. Intermittency-Type Estimates for Some Nondegenerate SPDE'S. Ann. Probab. 23 (1995), no. 4, 1853--1874. doi:10.1214/aop/1176987806. https://projecteuclid.org/euclid.aop/1176987806