Infinite interacting diffusion particles I: Equilibrium process and its scaling limit

Kondratiev Y, Lytvynov E, Röckner M (2006)
FORUM MATHEMATICUM 18(1): 9-43.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
A stochastic dynamics (X(t))(t >= 0) of a classical continuous system is a stochastic process which takes values in the space Gamma of all locally finite subsets (configurations) in R-d and which has a Gibbs measure mu as an invariant measure. We assume that It corresponds to a symmetric pair potential phi(x - y). An important class of stochastic dynamics of a classical continuous system is formed by diffusions. Till now, only one type of such dynamics the so-called gradient stochastic dynamics, or interacting Brownian particles-has been investigated. By using the theory of Dirichlet forms from [27], we construct and investigate a new type of stochastic dynamics, which we call infinite interacting diffusion particles. We introduce a Dirichlet form E-mu(Gamma) on L-2(Gamma;mu) and under general conditions on the potential phi, prove its closability. For a potential phi having a "weak" singularity at zero, we also write down an explicit form of the generator of E-mu(Gamma) on the set of smooth cylinder functions. We then show that, for any Dirichlct form E-mu(Gamma), there exists a diffusion process that is properly associated with it. Finally, in a way parallel to [17], we study a scaling limit of interacting diffusions in terms of convergence of the corresponding Dirichlet forms, and we also show that these scaled processes are tight in C([0, infinity), D'), where D' is the dual space of D : = C-0(infinity) (R-d).
Erscheinungsjahr
2006
Zeitschriftentitel
FORUM MATHEMATICUM
Band
18
Ausgabe
1
Seite(n)
9-43
ISSN
0933-7741
Page URI
https://pub.uni-bielefeld.de/record/1599748

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Kondratiev Y, Lytvynov E, Röckner M. Infinite interacting diffusion particles I: Equilibrium process and its scaling limit. FORUM MATHEMATICUM. 2006;18(1):9-43.
Kondratiev, Y., Lytvynov, E., & Röckner, M. (2006). Infinite interacting diffusion particles I: Equilibrium process and its scaling limit. FORUM MATHEMATICUM, 18(1), 9-43. https://doi.org/10.1515/FORUM.2006.002
Kondratiev, Y., Lytvynov, E., and Röckner, M. (2006). Infinite interacting diffusion particles I: Equilibrium process and its scaling limit. FORUM MATHEMATICUM 18, 9-43.
Kondratiev, Y., Lytvynov, E., & Röckner, M., 2006. Infinite interacting diffusion particles I: Equilibrium process and its scaling limit. FORUM MATHEMATICUM, 18(1), p 9-43.
Y. Kondratiev, E. Lytvynov, and M. Röckner, “Infinite interacting diffusion particles I: Equilibrium process and its scaling limit”, FORUM MATHEMATICUM, vol. 18, 2006, pp. 9-43.
Kondratiev, Y., Lytvynov, E., Röckner, M.: Infinite interacting diffusion particles I: Equilibrium process and its scaling limit. FORUM MATHEMATICUM. 18, 9-43 (2006).
Kondratiev, Yuri, Lytvynov, Eugene, and Röckner, Michael. “Infinite interacting diffusion particles I: Equilibrium process and its scaling limit”. FORUM MATHEMATICUM 18.1 (2006): 9-43.

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