Issue |
ESAIM: PS
Volume 6, 2002
|
|
---|---|---|
Page(s) | 147 - 155 | |
DOI | https://doi.org/10.1051/ps:2002008 | |
Published online | 15 November 2002 |
Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems
Facultad de Matemáticas,
Pontificia Universidad Católica de Chile, Casilla 306, Correo 22,
Santiago, Chile; aramirez@mat.puc.cl.
Received:
15
December
2001
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
. 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
and that
.
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
,
where
is the configuration obtained from η
altering only the coordinate at site x. Assume that
are of finite range, bounded and that
.
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.
Mathematics Subject Classification: 82C20 / 82C22 / 60H07 / 60K35
Key words: Infinite dimensional diffusions / Malliavin calculus / Interacting particles systems.
© EDP Sciences, SMAI, 2002
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