Abstract
The paper is concerned with the weak convergence of n-particle processes to deterministic stationary paths as \({n \rightarrow \infty}\). A Mosco type convergence of a class of bilinear forms is introduced. The Mosco type convergence of bilinear forms results in a certain convergence of the resolvents of the n-particle systems. Based on this convergence a criterion in order to verify weak convergence of invariant measures is established. Under additional conditions weak convergence of stationary n-particle processes to stationary deterministic paths is proved. The method is applied to the particle approximation of a Ginzburg-Landau type diffusion.
The present paper is in close relation to the paper [9]. Different definitions of bilinear forms and versions of Mosco type convergence are introduced. Both papers demonstrate that the choice of the form and the type of convergence relates to the particular particle system.
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References
E. B. Dynkin, Markov processes, Vol I and II, Springer 1965.
S. N. Ethier, T. Kurtz, Markov processes, Characterization and convergence, John Wiley 1986.
M. Hino,Convergence of non-symmetric forms. J. Math. Kyoto Univ. 38 No. 2 (1998), 329–341.
O. Kallenberg, Foundations of Modern Probability, 2nd Edition, Springer 2002.
A. V. Kolesnikov,Convergence of Dirichlet forms with changing speed measures on \({\mathbb{R}^d}\) . Forum Math. 17 No. 2 (2005), 225–259.
A. V. Kolesnikov, Mosco convergence of Dirichlet forms in infinite dimensions with changing reference measures. J. Funct. Anal. 230 No. 2 (2006), 382–418.
K. Kuwae, T. Shioya, Convergence of spectral structures: a functional analytic theory and its applcations to spectral geometry. Comm. Anal. Geom. 11 No. 4 (2003), 599-673.
J.-U. Löbus, A stationary Fleming-Viot type Brownian particle system. Math. Z. 263 No. 3 (2009), 541-581.
J.-U. Löbus, Mosco type convergence of bilinear forms and weak convergence of nparticle systems. Preprint arXiv:1209.4766 (2012).
S. Lu, Equilibrium fluctuations of a one-dimensional nongradient Ginzburg-Landau model. Ann. Probab. 22 No. 3 (1994), 1252–1272.
S. Lu, Hydrodynamic scaling limits with deterministic initial configurations. Ann. Probab. 23 No. 4 (1995), 1831–1852.
P. Mörters, Y. Peres,Brownian motion. With an appendix by O. Schramm and W. Werner, Cambridge University Press 2010.
U. Mosco, Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 No. 2 (1994), 368–421.
Z.-M. Ma, M. Röckner Introduction to the Theory of (Non-Symmetric) Dirichlet Forms, Springer 1992.
S. Olla, S. R. S. Varadhan Scaling limit for interacting Ornstein-Uhlenbeck processes. Comm. Math. Phys. 135 No. 2 (1991), 355–378.
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer 1983.
W. Sun, Weak convergence of Dirichlet processes. Sci. China Ser. A 41 No. 1 (1998), 8–21.
K. Uchiyama, Scaling limits of interacting diffusion with arbitrary limits. Probab. Theory Relat. Fields 99 No. 1 (1994), 97–110.
S. R. S. Varadhan, Scaling limits for interacting diffusions. Comm. Math. Phys. 135 No. 2 (1991), 313–353.
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Löbus, JU. Weak Convergence of n-Particle Systems Using Bilinear Forms. Milan J. Math. 81, 37–77 (2013). https://doi.org/10.1007/s00032-013-0200-8
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DOI: https://doi.org/10.1007/s00032-013-0200-8