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Convergence of Brownian motion with a scaled Dirac delta potential

Part of: Other generalizations Groups and semigroups of linear operators, their generalizations and applications Markov processes

Published online by Cambridge University Press:  23 February 2012

Florian Conrad ,
Martin Grothaus ,
Janna Lierl  and
Olaf Wittich
Florian Conrad
Affiliation:
Mathematics Department, University of Kaiserslautern, PO Box 3049, 67653 Kaiserslautern, Germany (grothaus@mathematik.uni-kl.de) Mathematics Department, Bielefeld University, PO Box 100131, 33501 Bielefeld, Germany (fconrad@math.uni-bielefeld.de)
Martin Grothaus
Affiliation:
Mathematics Department, University of Kaiserslautern, PO Box 3049, 67653 Kaiserslautern, Germany (grothaus@mathematik.uni-kl.de)
Janna Lierl
Affiliation:
Department of Mathematics, Cornell University, 120 Malott Hall, Ithaca, NY 14853, USA (jlierl@math.cornell.edu)
Olaf Wittich
Affiliation:
Lehrstuhl A für Mathematik, RWTH Aachen, 52056 Aachen, Germany (olaf.wittich@matha.rwth-aachen.de)
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Abstract

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The method of deriving scaling limits using Dirichlet-form techniques has already been successfully applied to a number of infinite-dimensional problems. However, extracting the key tools from these papers is a rather difficult task for non-experts. This paper meets the need for a simple presentation of the method by applying it to a basic example, namely the convergence of Brownian motions with potentials given by n multiplied by the Dirac delta at 0 to Brownian motion with absorption at 0.

Keywords

Dirichlet formslimit theoremsstrong Feller property

MSC classification

Secondary: 47D07: Markov semigroups and applications to diffusion processes 60J60: Diffusion processes 31C25: Dirichlet spaces
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 55 , Issue 2 , June 2012 , pp. 403 - 427
Copyright
Copyright © Edinburgh Mathematical Society 2012

References

1.Adams, R. A., Sobolev spaces, Pure and Applied Mathematics, Volume 65 (Academic Press, New York, 1975).Google Scholar
2.Albeverio, S., Kondratiev, Yu. G. and Röckner, M., Strong Feller properties for distorted Brownian motion and applications to finite particle systems with singular interactions, in Finite and infinite dimensional analysis in honor of Leonard Gross, Contemporary Mathematics, Volume 317 (American Mathematical Society, Providence, RI, 2003).Google Scholar
3.Attouch, H., Variational convergence for functions and operators, Applicable Mathematics Series (Pitman Advanced Publishing Program, Boston, MA, 1984).Google Scholar
4.Maso, G. Dal, An introduction to gamma-convergence (Birkhäuser, Boston, MA, 1993).Google Scholar
5.Prato, G. Da and Röckner, M., Singular dissipative stochastic equations in Hilbert spaces, Prob. Theory Relat. Fields 124(2) (2002), 261–303.Google Scholar
6.de Masi, A., Ianiro, N., Pellegrinotti, A. and Presutti, E., A survey of the hydrodynamical behavior of many-particle systems, in Nonequilibrium phenomena II: from stochastics to hydrodynamics, Studies in Statistical Mechanics, Volume 11, pp. 193–294 (North-Holland, Amsterdam, 1984).Google Scholar
7.Durrett, R., Stochastic calculus: a practical introduction (CRC Press, New York, 1996).Google Scholar
8.Ethier, S. and Kurtz, T., Markov processes: characterization and convergence (Wiley, 1986).Google Scholar
9.Fukushima, M., Oshima, Y. and Takeda, M., Dirichlet forms and symmetric Markov processes (Walter de Gruyter, Berlin, 1994).Google Scholar
10.Grothaus, M., Scaling limit of fluctuations for the equilibrium Glauber dynamics in continuum, J. Funct. Analysis 239 (2006), 414–445.Google Scholar
11.Grothaus, M., Kondratiev, Yu. G., Lytvynov, E. and Röckner, M., Scaling limit of stochastic dynamics in classical continuous systems, Annals Probab. 31 (2003), 1494–1532.Google Scholar
12.Grothaus, M., Kondratiev, Yu. G. and Röckner, M., N/V-limit for stochastic dynamics in continuous particle systems, Prob. Theory Relat. Fields 137 (2007), 121–160.Google Scholar
13.Kato, T., Perturbation theory for linear operators (Springer, 1966).Google Scholar
14.Liskevich, V. and Semenov, Y. A., Some problems on Markov semigroups, Mathematical Topics, Volume 11 (Akademie, Berlin, 1996).Google Scholar
15.Lyons, T. J. and Zhang, T. S., Decomposition of Dirichlet processes and its applications, Annals Probab. 22 (1994), 494–524.Google Scholar
16.Lyons, T. J. and Zheng, W. A., A crossing estimate for the canonical process on a Dirichlet space and a tightness result, Astérisque 157/158 (1988), pp. 249–272.Google Scholar
17.Ma, Z.-M. and Röckner, M., Introduction to the theory of (non-symmetric) Dirichlet forms (Springer, 1992).Google Scholar
18.Mosco, U., Composite media and asymptotic Dirichlet forms, J. Funct. Analysis 123 (1994), 368–421.Google Scholar
19.Reed, M. and Simon, B., Methods of modern mathematical physics, I, Functional analysis (Academic Press, New York, 1980).Google Scholar
20.Simon, B., A canonical decomposition for quadratic forms with applications to monotone convergence theorems, J. Funct. Analysis 28 (1978), 377–385.Google Scholar