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

Alejandro F. Ramírez

ESAIM: Probability and Statistics (2002)

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

Abstract

top
Consider an infinite dimensional diffusion process process on T 𝐙 d , where T is the circle, defined by the action of its generator L on C 2 ( T 𝐙 d ) local functions as L f ( η ) = i 𝐙 d 1 2 a i 2 f η i 2 + b i f η i . 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 η i and that inf i , η a i ( η ) > 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 } 𝐙 d , defined by the action of its generator on local functions f by L f ( η ) = x 𝐙 d c ( x , η ) ( f ( η x ) - f ( η ) ) , where η x is the configuration obtained from η altering only the coordinate at site x . Assume that c ( x , η ) are of finite range, bounded and that inf x , η c ( x , η ) > 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

top

Ramírez, 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 $\inf _\{i,\eta \}a_i(\eta )&gt;0$. 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 $\inf _\{x,\eta \}c(x,\eta )&gt;0$. 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 $\inf _{i,\eta }a_i(\eta )&gt;0$. 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 $\inf _{x,\eta }c(x,\eta )&gt;0$. 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

top
  1. [1] R. Holley and D. Stroock, In one and two dimensions, every stationary measure for a stochastic Ising model is a Gibbs state. Commun. Math. Phys. 55 (1977) 37-45. MR451455
  2. [2] R. Holley and D. Stroock, Diffusions on an Infinite Dimensional Torus. J. Funct. Anal. 42 (1981) 29-63. Zbl0501.58039MR620579
  3. [3] H. Kunsch, Non reversible stationary measures for infinite interacting particle systems. Z. Wahrsch. Verw. Gebiete 66 (1984) 407-424. Zbl0541.60098MR751579
  4. [4] T.M. Liggett, Interacting Particle Systems. Springer-Verlag, New York (1985). Zbl0559.60078MR776231
  5. [5] T.S. Mountford, A Coupling of Infinite Particle Systems. J. Math. Kyoto Univ. 35 (1995) 43-52. Zbl0840.60097MR1317272
  6. [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. [7] A.F. Ramírez, Relative Entropy and Mixing Properties of Infinite Dimensional Diffusions. Probab. Theory Related Fields 110 (1998) 369-395. Zbl0929.60081MR1616555
  8. [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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.