@article{Ramírez2010,
abstract = {
Consider an infinite dimensional
diffusion process process on TZd, where
T is the circle, defined by the action of its generator
L on C2(TZd) 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, ai and bi are
smooth,
bounded, finite range with uniformly bounded second order
partial derivatives, that ai is only a function of $\eta_i$
and that $\inf_\{i,\eta\}a_i(\eta)>0$.
Suppose ν is an invariant product measure. Then,
if ν is the Lebesgue measure or if d=1,2, it is the
unique invariant measure. Furthermore, if ν is
translation invariant, then it is the unique invariant, translation
invariant measure. Now, consider an infinite particle
spin system, with state space \{0,1\}Zd, 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 η
altering only the coordinate at site x. Assume that $c(x,\eta)$
are of finite range, bounded and that $\inf_\{x,\eta\}c(x,\eta)>0$.
Then, if ν is an invariant product measure for this process,
ν is unique when d=1,2. Furthermore, if ν 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.; infinite dimensional diffusions; interacting particles systems},
language = {eng},
month = {3},
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/104284},
volume = {6},
year = {2010},
}
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
DA - 2010/3//
PB - EDP Sciences
VL - 6
SP - 147
EP - 155
AB -
Consider an infinite dimensional
diffusion process process on TZd, where
T is the circle, defined by the action of its generator
L on C2(TZd) 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, ai and bi are
smooth,
bounded, finite range with uniformly bounded second order
partial derivatives, that ai is only a function of $\eta_i$
and that $\inf_{i,\eta}a_i(\eta)>0$.
Suppose ν is an invariant product measure. Then,
if ν is the Lebesgue measure or if d=1,2, it is the
unique invariant measure. Furthermore, if ν is
translation invariant, then it is the unique invariant, translation
invariant measure. Now, consider an infinite particle
spin system, with state space {0,1}Zd, 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 η
altering only the coordinate at site x. Assume that $c(x,\eta)$
are of finite range, bounded and that $\inf_{x,\eta}c(x,\eta)>0$.
Then, if ν is an invariant product measure for this process,
ν is unique when d=1,2. Furthermore, if ν 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.; infinite dimensional diffusions; interacting particles systems
UR - http://eudml.org/doc/104284
ER -
