Scaling limit of stochastic dynamics in classical continuous systems

Grothaus M, Kondratiev Y, Lytvynov E, Röckner M (2003)
ANNALS OF PROBABILITY 31(3): 1494-1532.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Autor*in
Grothaus, Martin; Kondratiev, YuriUniBi; Lytvynov, Eugene; Röckner, MichaelUniBi
Abstract / Bemerkung
We investigate a scaling limit of gradient Stochastic dynamics associated with Gibbs states in classical continuous systems on R-d, d greater than or equal to 1. The aim is to derive macroscopic quantities from a given microscopic or mesoscopic system. The scaling we consider has been investigated by Brox (in 1980), Rost (in 1981), Spohn (in 1986) and Guo and Papanicolaou (in 1985), under the assumption that the underlying potential is in C-0(3) and positive. We prove that the Dirichlet forms of the scaled stochastic dynamics converge on a core of functions to the Dirichlet form of a generalized Ornstein-Uhlenbeck process. The proof is based on the analysis and geometry on the configuration space which was developed by Albeverio, Kondratiev and Rockner (in 1998), and works for general Gibbs measures of Ruelle type. Hence, the underlying potential may have a singularity at the origin, only has to be bounded from below and may not be compactly supported. Therefore, singular interactions of physical interest are covered, as, for example, the one given by the Lennard-Jones potential, which is studied in the theory of fluids. Furthermore, using the Lyons-Zheng decomposition we give a simple proof for the tightness of the scaled processes. We also prove that the corresponding generators, however, do not converge in the L-2-sense. This settles a conjecture formulated by Brox, by Rost and by Spohn.
Stichworte
diffusion processes; interacting particle systems; limit theorems
Erscheinungsjahr
2003
Zeitschriftentitel
ANNALS OF PROBABILITY
Band
31
Ausgabe
3
Seite(n)
1494-1532
ISSN
0091-1798
Page URI
https://pub.uni-bielefeld.de/record/1611030

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Grothaus M, Kondratiev Y, Lytvynov E, Röckner M. Scaling limit of stochastic dynamics in classical continuous systems. ANNALS OF PROBABILITY. 2003;31(3):1494-1532.
Grothaus, M., Kondratiev, Y., Lytvynov, E., & Röckner, M. (2003). Scaling limit of stochastic dynamics in classical continuous systems. ANNALS OF PROBABILITY, 31(3), 1494-1532.
Grothaus, M., Kondratiev, Y., Lytvynov, E., and Röckner, M. (2003). Scaling limit of stochastic dynamics in classical continuous systems. ANNALS OF PROBABILITY 31, 1494-1532.
Grothaus, M., et al., 2003. Scaling limit of stochastic dynamics in classical continuous systems. ANNALS OF PROBABILITY, 31(3), p 1494-1532.
M. Grothaus, et al., “Scaling limit of stochastic dynamics in classical continuous systems”, ANNALS OF PROBABILITY, vol. 31, 2003, pp. 1494-1532.
Grothaus, M., Kondratiev, Y., Lytvynov, E., Röckner, M.: Scaling limit of stochastic dynamics in classical continuous systems. ANNALS OF PROBABILITY. 31, 1494-1532 (2003).
Grothaus, Martin, Kondratiev, Yuri, Lytvynov, Eugene, and Röckner, Michael. “Scaling limit of stochastic dynamics in classical continuous systems”. ANNALS OF PROBABILITY 31.3 (2003): 1494-1532.

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