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

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

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## Abstract

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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 $Lf\left(\eta \right)={\sum }_{i\in {𝐙}^{d}}\left(\frac{1}{2}{a}_{i}\frac{{\partial }^{2}f}{\partial {\eta }_{i}^{2}}+{b}_{i}\frac{\partial f}{\partial {\eta }_{i}}\right)$. 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 ${inf}_{i,\eta }{a}_{i}\left(\eta \right)>0$. 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 $Lf\left(\eta \right)={\sum }_{x\in {𝐙}^{d}}c\left(x,\eta \right)\left(f\left({\eta }^{x}\right)-f\left(\eta \right)\right)$, where ${\eta }^{x}$ is the configuration obtained from η altering only the coordinate at site x. Assume that $c\left(x,\eta \right)$ are of finite range, bounded and that ${inf}_{x,\eta }c\left(x,\eta \right)>0$. 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.

## How to cite

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Ramírez, 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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6. 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
7. A.F. Ramírez, Relative Entropy and Mixing Properties of Infinite Dimensional Diffusions. Probab. Theory Related Fields110 (1998) 369-395.  Zbl0929.60081
8. 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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