Tagged particle processes and their non-explosion criteria. (English) Zbl 1196.60166

Summary: We give a derivation of tagged particle processes from unlabeled interacting Brownian motions. We give a criteria of the non-explosion property of tagged particle processes. We prove the quasi-regularity of Dirichlet forms describing the environment seen from the tagged particle, which were used in previous papers to prove the invariance principle of tagged particles of interacting Brownian motions.


60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J60 Diffusion processes
82B21 Continuum models (systems of particles, etc.) arising in equilibrium statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
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[1] S. Albeverio, Y. G. Kondratiev and M. Röckner, Analysis and geometry on configuration spaces: the Gibbsian case, J. Funct. Anal., 157 (1998), 242-291. · Zbl 0931.58019
[2] A. De Masi, P. A. Ferrari, S. Goldstein and W. D. Wick, An invariance principle for reversible Markov processes, Applications to random motions in random environments, J. Statist. Phys., 55 (1989), 787-855. · Zbl 0713.60041
[3] T. Fattler and M. Grothaus, Tagged particle process in continuum with singular interactions, preprint, available at arXiv: · Zbl 1267.82089
[4] M. Fukushima, Y. Oshima and M. Takeda, Dirichlet forms and symmetric Markov processes, Walter de Gruyter, 1994. · Zbl 0838.31001
[5] J. Fritz, Gradient dynamics of infinite point systems, Ann. Probab., 15 (1987), 478-514. · Zbl 0623.60119
[6] M. Z. Guo, Limit theorems for interacting particle systems, Thesis, Dept. of Mathematics, New York University, 1984.
[7] M. Z. Guo and G. C. Papanicolaou, Self-diffusion of interacting Brownian particles, in “Probabilistic Method in Mathematical Physics”, Proc. Taniguch International Sympo. at Katata and Kyoto, 1985 (eds. K. Ito and N. Ikeda), Kinokuniya, 1987, pp.,113-152. · Zbl 0656.60109
[8] K. Inukai, Collision or non-collision problem for interacting Brownian particles, Proc. Japan Acad. Ser. A Math. Sci., 82 (2006), 66-70. · Zbl 1107.60041
[9] C. Kipnis and S. R. S. Varadhan, Central limit theorems for additive functional of reversible Markov process and applications to simple exclusions, Comm. Math. Phys., 104 (1986), 1-19. · Zbl 0588.60058
[10] R. Lang, Unendlich-dimensionale Wienerprocesse mit Wechselwirkung I, Z. Wahrsch. Verw. Gebiete, 38 (1977), 55-72. · Zbl 0349.60103
[11] R. Lang, Unendlich-dimensionale Wienerprocesse mit Wechselwirkung II, Z. Wahrsch. Verw. Gebiete, 39 (1978), 277-299. · Zbl 0342.60067
[12] Z.-M. Ma and M. Röckner, Introduction to the theory of (non-symmetric) Dirichlet forms, Springer-Verlag, Berlin, 1992. · Zbl 0826.31001
[13] M. Mehta, Radom matrices, Third Edition, Elsevier, 2004.
[14] H. Osada, Dirichlet form approach to infinitely dimensional Wiener processes with singular interactions, Comm. Math. Phys., 176 (1996), 117-131. · Zbl 0837.60073
[15] H. Osada, An invariance principle for Markov processes and Brownian particles with singular interaction, Ann. Inst. H. Poincaré, 34 (1998), 217-248. · Zbl 0914.60041
[16] H. Osada, Positivity of the self-diffusion matrix of interacting Brownian particles with hard core, Probab. Theory Related Fields, 112 (1998), 53-90. · Zbl 0920.60056
[17] H. Osada, Non-collision and collision properties of Dyson’s model in infinite dimensions and other stochastic dynamics whose equilibrium states are determinantal random point fields, in Stochastic Analysis on Large Scale Interacting Systems, (eds. T. Funaki and H. Osada), Adv. Stud. Pure Math., 39 , Math. Soc. Japan, Tokyo, 2004, pp.,325-343. · Zbl 1061.60109
[18] H. Osada, Interacting Brownian motions in infinite dimensions with logarithmic interaction potentials, preprint, available at arXiv: · Zbl 1271.60105
[19] H. Osada, Infinite-dimensional stochastic differential equations related to random matrices, preprint, available at arXiv: · Zbl 1253.82061
[20] S. Resnick, Extreme values, regular variation, and point processes, Springer, 2000. · Zbl 1136.60004
[21] D. Ruelle, Superstable interactions in classical statistical mechanics, Comm. Math. Phys., 18 (1970), 127-159. · Zbl 0198.31101
[22] T. Shiga, A remark on infinite-dimensional Wiener processes with interactions, Z. Wahrsch. Verw. Gebiete, 47 (1979), 299-304. · Zbl 0407.60098
[23] A. Soshnikov, Determinantal random point fields, Russian Math. Surveys, 55 (2000), 923-975. · Zbl 0991.60038
[24] H. Tanemura, A system of infinitely many mutually reflecting Brownian balls in \(\R^d\), Probab. Theory Related Fields, 104 (1996), 399-426. · Zbl 0849.60087
[25] H. Tanemura, Uniqueness of Dirichlet forms associated with systems of infinitely many Brownian balls in \(\Rd\), Probab. Theory Related Fields, 109 (1997), 275-299. · Zbl 0888.60075
[26] H. J. Yoo, Dirichlet forms and diffusion processes for fermion random point fields, J. Funct. Anal., 219 (2005), 143-160. · Zbl 1104.60065
[27] M. W. Yoshida, Construction of infinite-dimensional interacting diffusion processes through Dirichlet forms, Probab. Theory Related Fields, 106 (1996), 265-297. · Zbl 0859.60068
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