Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems

Ramírez, Alejandro F.. "Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems." ESAIM: Probability and Statistics 6 (2002): 147-155. <http://eudml.org/doc/245644>.

@article{Ramírez2002,
abstract = {Consider an infinite dimensional diffusion process process on $T^\{\{\bf Z\}^d\}$, where $T$ is the circle, defined by the action of its generator $L$ on $C^2(T^\{\{\bf Z\}^d\})$ local functions as $Lf(\eta )=\sum _\{i\in \{\bf Z\}^d\}\left(\frac\{1\}\{2\}a_i \frac\{\partial ^2 f\}\{\partial \eta _i^2\}+b_i\frac\{\partial f\}\{\partial \eta _i\}\right)$. Assume that the coefficients, $a_i$ and $b_i$ are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that $a_i$ is only a function of $\eta _i$ and that $\inf _\{i,\eta \}a_i(\eta )&gt;0$. Suppose $\nu $ is an invariant product measure. Then, if $\nu $ is the Lebesgue measure or if $d=1,2$, it is the unique invariant measure. Furthermore, if $\nu $ is translation invariant, then it is the unique invariant, translation invariant measure. Now, consider an infinite particle spin system, with state space $\lbrace 0,1\rbrace ^\{\{\bf Z\}^d\}$, defined by the action of its generator on local functions $f$ by $Lf(\eta )=\sum _\{x\in \{\bf Z\}^d\}c(x,\eta )(f(\eta ^x)-f(\eta ))$, where $\eta ^x$ is the configuration obtained from $\eta $ altering only the coordinate at site $x$. Assume that $c(x,\eta )$ are of finite range, bounded and that $\inf _\{x,\eta \}c(x,\eta )&gt;0$. Then, if $\nu $ is an invariant product measure for this process, $\nu $ is unique when $d=1,2$. Furthermore, if $\nu $ is translation invariant, it is the unique invariant, translation invariant measure. The proofs of these results show how elementary methods can give interesting information for general processes.},
author = {Ramírez, Alejandro F.},
journal = {ESAIM: Probability and Statistics},
keywords = {infinite dimensional diffusions; Malliavin calculus; interacting particles systems},
language = {eng},
pages = {147-155},
publisher = {EDP-Sciences},
title = {Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems},
url = {http://eudml.org/doc/245644},
volume = {6},
year = {2002},
}

TY - JOUR
AU - Ramírez, Alejandro F.
TI - Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems
JO - ESAIM: Probability and Statistics
PY - 2002
PB - EDP-Sciences
VL - 6
SP - 147
EP - 155
AB - Consider an infinite dimensional diffusion process process on $T^{{\bf Z}^d}$, where $T$ is the circle, defined by the action of its generator $L$ on $C^2(T^{{\bf Z}^d})$ local functions as $Lf(\eta )=\sum _{i\in {\bf Z}^d}\left(\frac{1}{2}a_i \frac{\partial ^2 f}{\partial \eta _i^2}+b_i\frac{\partial f}{\partial \eta _i}\right)$. Assume that the coefficients, $a_i$ and $b_i$ are smooth, bounded, finite range with uniformly bounded second order partial derivatives, that $a_i$ is only a function of $\eta _i$ and that $\inf _{i,\eta }a_i(\eta )&gt;0$. Suppose $\nu $ is an invariant product measure. Then, if $\nu $ is the Lebesgue measure or if $d=1,2$, it is the unique invariant measure. Furthermore, if $\nu $ is translation invariant, then it is the unique invariant, translation invariant measure. Now, consider an infinite particle spin system, with state space $\lbrace 0,1\rbrace ^{{\bf Z}^d}$, defined by the action of its generator on local functions $f$ by $Lf(\eta )=\sum _{x\in {\bf Z}^d}c(x,\eta )(f(\eta ^x)-f(\eta ))$, where $\eta ^x$ is the configuration obtained from $\eta $ altering only the coordinate at site $x$. Assume that $c(x,\eta )$ are of finite range, bounded and that $\inf _{x,\eta }c(x,\eta )&gt;0$. Then, if $\nu $ is an invariant product measure for this process, $\nu $ is unique when $d=1,2$. Furthermore, if $\nu $ is translation invariant, it is the unique invariant, translation invariant measure. The proofs of these results show how elementary methods can give interesting information for general processes.
LA - eng
KW - infinite dimensional diffusions; Malliavin calculus; interacting particles systems
UR - http://eudml.org/doc/245644
ER -