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

ESAIM: Probability and Statistics (2002)

- 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 (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 $\inf _\{i,\eta \}a_i(\eta )>0$. 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 $\inf _\{x,\eta \}c(x,\eta )>0$. 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 $\inf _{i,\eta }a_i(\eta )>0$. 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 $\inf _{x,\eta }c(x,\eta )>0$. 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 -

## References

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- [7] A.F. Ramírez, Relative Entropy and Mixing Properties of Infinite Dimensional Diffusions. Probab. Theory Related Fields 110 (1998) 369-395. Zbl0929.60081MR1616555
- [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. Zbl0884.60094MR1443753

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