## The Annals of Probability

- Ann. Probab.
- Volume 15, Number 2 (1987), 478-514.

### Gradient Dynamics of Infinite Point Systems

#### Abstract

Nonequilibrium gradient dynamics of $d$-dimensional particle systems is investigated. The interaction is given by a superstable pair potential of finite range. Solutions are constructed in the well-defined set of locally finite configurations with a logarithmic order of energy fluctuations. If the system is deterministic and $d \leq 2$, then singular potentials are also allowed. For stochastic models with a smooth interaction we need $d \leq 4$. In order to develop some prerequisites for the theory of hydrodynamical fluctuations in equilibrium, we investigate smoothness of the Markov semigroup and describe some properties of its generator.

#### Article information

**Source**

Ann. Probab., Volume 15, Number 2 (1987), 478-514.

**Dates**

First available in Project Euclid: 19 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aop/1176992156

**Digital Object Identifier**

doi:10.1214/aop/1176992156

**Mathematical Reviews number (MathSciNet)**

MR885128

**Zentralblatt MATH identifier**

0623.60119

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07]

**Keywords**

Interacting Brownian particles superstable potentials generators of semigroups cores and essential self-adjointness

#### Citation

Fritz, J. Gradient Dynamics of Infinite Point Systems. Ann. Probab. 15 (1987), no. 2, 478--514. doi:10.1214/aop/1176992156. https://projecteuclid.org/euclid.aop/1176992156