Uniqueness and regularity for a system of interacting Bessel processes via the Muckenhoupt condition
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- by Sebastian Andres and Max-K. von Renesse
- Trans. Amer. Math. Soc. 364 (2012), 1413-1426
- DOI: https://doi.org/10.1090/S0002-9947-2011-05457-7
- Published electronically: October 11, 2011
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Abstract:
We study the regularity of a diffusion on a simplex with singular drift and reflecting boundary condition which describes a finite system of particles on an interval with Coulomb interaction and reflection between nearest neighbors.
As our main result we establish the strong Feller property for the process in both cases of repulsion and attraction. In particular, the system can be started from any initial state, including multiple point configurations. Moreover, we show that the process is a Euclidean semi-martingale if and only if the interaction is repulsive. Hence, contrary to classical results about reflecting Brownian motion in smooth domains, in the attractive regime a construction via a system of Skorokhod SDEs is impossible. Finally, we establish exponential heat kernel gradient estimates in the repulsive regime.
The main proof for the attractive case is based on potential theory in Sobolev spaces with Muckenhoupt weights.
References
- Sergio Albeverio and Michael Röckner, Classical Dirichlet forms on topological vector spaces—closability and a Cameron-Martin formula, J. Funct. Anal. 88 (1990), no. 2, 395–436. MR 1038449, DOI 10.1016/0022-1236(90)90113-Y
- Luigi Ambrosio, Giuseppe Savaré, and Lorenzo Zambotti, Existence and stability for Fokker-Planck equations with log-concave reference measure, Probab. Theory Related Fields 145 (2009), no. 3-4, 517–564. MR 2529438, DOI 10.1007/s00440-008-0177-3
- Sebastian Andres and Max-K. von Renesse, Particle approximation of the Wasserstein diffusion, J. Funct. Anal. 258 (2010), no. 11, 3879–3905. MR 2606878, DOI 10.1016/j.jfa.2009.10.029
- Richard F. Bass, Krzysztof Burdzy, and Zhen-Qing Chen, Uniqueness for reflecting Brownian motion in lip domains, Ann. Inst. H. Poincaré Probab. Statist. 41 (2005), no. 2, 197–235 (English, with English and French summaries). MR 2124641, DOI 10.1016/j.anihpb.2004.06.001
- Richard F. Bass and Pei Hsu, The semimartingale structure of reflecting Brownian motion, Proc. Amer. Math. Soc. 108 (1990), no. 4, 1007–1010. MR 1007487, DOI 10.1090/S0002-9939-1990-1007487-8
- Richard F. Bass and Pei Hsu, Some potential theory for reflecting Brownian motion in Hölder and Lipschitz domains, Ann. Probab. 19 (1991), no. 2, 486–508. MR 1106272
- Richard F. Bass and Edwin A. Perkins, Degenerate stochastic differential equations with Hölder continuous coefficients and super-Markov chains, Trans. Amer. Math. Soc. 355 (2003), no. 1, 373–405. MR 1928092, DOI 10.1090/S0002-9947-02-03120-3
- Jean Bertoin, Excursions of a $\textrm {BES}_0(d)$ and its drift term $(0<d<1)$, Probab. Theory Related Fields 84 (1990), no. 2, 231–250. MR 1030728, DOI 10.1007/BF01197846
- Emmanuel Cépa and Dominique Lépingle, Diffusing particles with electrostatic repulsion, Probab. Theory Related Fields 107 (1997), no. 4, 429–449. MR 1440140, DOI 10.1007/s004400050092
- Andreas Eberle, Uniqueness and non-uniqueness of semigroups generated by singular diffusion operators, Lecture Notes in Mathematics, vol. 1718, Springer-Verlag, Berlin, 1999. MR 1734956, DOI 10.1007/BFb0103045
- Eugene B. Fabes, Carlos E. Kenig, and Raul P. Serapioni, The local regularity of solutions of degenerate elliptic equations, Comm. Partial Differential Equations 7 (1982), no. 1, 77–116. MR 643158, DOI 10.1080/03605308208820218
- Torben Fattler and Martin Grothaus, Strong Feller properties for distorted Brownian motion with reflecting boundary condition and an application to continuous $N$-particle systems with singular interactions, J. Funct. Anal. 246 (2007), no. 2, 217–241. MR 2321042, DOI 10.1016/j.jfa.2007.01.014
- J. Fritz and R. L. Dobrushin, Non-equilibrium dynamics of two-dimensional infinite particle systems with a singular interaction, Comm. Math. Phys. 65 (1979), no. 1, 96. MR 1552623, DOI 10.1007/BF01940963
- Masatoshi Fukushima, A construction of reflecting barrier Brownian motions for bounded domains, Osaka Math. J. 4 (1967), 183–215. MR 231444
- Masatoshi Fukushima, On semi-martingale characterizations of functionals of symmetric Markov processes, Electron. J. Probab. 4 (1999), no. 18, 32. MR 1741537, DOI 10.1214/EJP.v4-55
- David J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices, Ann. Inst. H. Poincaré Probab. Statist. 35 (1999), no. 2, 177–204 (English, with English and French summaries). MR 1678525, DOI 10.1016/S0246-0203(99)80010-7
- Tero Kilpeläinen, Weighted Sobolev spaces and capacity, Ann. Acad. Sci. Fenn. Ser. A I Math. 19 (1994), no. 1, 95–113. MR 1246890
- P.-L. Lions and A.-S. Sznitman, Stochastic differential equations with reflecting boundary conditions, Comm. Pure Appl. Math. 37 (1984), no. 4, 511–537. MR 745330, DOI 10.1002/cpa.3160370408
- Hirofumi Osada, Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions, Comm. Math. Phys. 176 (1996), no. 1, 117–131. MR 1372820
- Daniel Revuz and Marc Yor, Continuous martingales and Brownian motion, 3rd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 293, Springer-Verlag, Berlin, 1999. MR 1725357, DOI 10.1007/978-3-662-06400-9
- Herbert Spohn, Interacting Brownian particles: a study of Dyson’s model, Hydrodynamic behavior and interacting particle systems (Minneapolis, Minn., 1986) IMA Vol. Math. Appl., vol. 9, Springer, New York, 1987, pp. 151–179. MR 914993, DOI 10.1007/978-1-4684-6347-7_{1}3
- Daniel W. Stroock, Diffusion semigroups corresponding to uniformly elliptic divergence form operators, Séminaire de Probabilités, XXII, Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 316–347. MR 960535, DOI 10.1007/BFb0084145
- K. T. Sturm, Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality, J. Math. Pures Appl. (9) 75 (1996), no. 3, 273–297. MR 1387522
- Hiroshi Tanaka, Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9 (1979), no. 1, 163–177. MR 529332
- Alberto Torchinsky, Real-variable methods in harmonic analysis, Pure and Applied Mathematics, vol. 123, Academic Press, Inc., Orlando, FL, 1986. MR 869816
- Gerald Trutnau, Skorokhod decomposition of reflected diffusions on bounded Lipschitz domains with singular non-reflection part, Probab. Theory Related Fields 127 (2003), no. 4, 455–495. MR 2021192, DOI 10.1007/s00440-003-0296-9
- Bengt Ove Turesson, Nonlinear potential theory and weighted Sobolev spaces, Lecture Notes in Mathematics, vol. 1736, Springer-Verlag, Berlin, 2000. MR 1774162, DOI 10.1007/BFb0103908
- Max-K. von Renesse and Karl-Theodor Sturm, Entropic measure and Wasserstein diffusion, Ann. Probab. 37 (2009), no. 3, 1114–1191. MR 2537551, DOI 10.1214/08-AOP430
- Feng Yu Wang, Application of coupling methods to the Neumann eigenvalue problem, Probab. Theory Related Fields 98 (1994), no. 3, 299–306. MR 1262968, DOI 10.1007/BF01192256
Bibliographic Information
- Sebastian Andres
- Affiliation: Department of Mathematics, Technische Universität Berlin, Strasse des 17, Juni 136, 10623 Berlin, Germany
- Email: andres@math.tu-berlin.de
- Max-K. von Renesse
- Affiliation: Department of Mathematics, Technische Universität Berlin, Strasse des 17, Juni 136, 10623 Berlin, Germany
- Email: mrenesse@math.tu-berlin.de
- Received by editor(s): November 20, 2009
- Received by editor(s) in revised form: May 21, 2010
- Published electronically: October 11, 2011
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 364 (2012), 1413-1426
- MSC (2010): Primary 60J60, 42B37
- DOI: https://doi.org/10.1090/S0002-9947-2011-05457-7
- MathSciNet review: 2869181