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

ESAIM: Probability and Statistics (2010)

- Volume: 6, page 147-155
- ISSN: 1292-8100

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topRamírez, Alejandro F.. "Uniqueness of invariant product measures for elliptic infinite dimensional diffusions and particle spin systems." ESAIM: Probability and Statistics 6 (2010): 147-155. <http://eudml.org/doc/104284>.

@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 -

## References

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- T.M. Liggett, Interacting Particle Systems. Springer-Verlag, New York (1985). Zbl0559.60078
- T.S. Mountford, A Coupling of Infinite Particle Systems. J. Math. Kyoto Univ.35 (1995) 43-52. Zbl0840.60097
- A.F. Ramírez, An elementary proof of the uniqueness of invariant product measures for some infinite dimensional diffusions. C. R. Acad. Sci. Paris Sér. I Math. (to appear). Zbl0998.60093
- A.F. Ramírez, Relative Entropy and Mixing Properties of Infinite Dimensional Diffusions. Probab. Theory Related Fields110 (1998) 369-395. Zbl0929.60081
- A.F. Ramírez and S.R.S. Varadhan, Relative Entropy and Mixing Properties of Interacting Particle Systems. J. Math. Kyoto Univ.36 (1996) 869-875. Zbl0884.60094

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