Collision or non-collision problem for interacting Brownian particles. (English) Zbl 1107.60041

Inukai, Kiyokazu

Proc. Japan Acad., Ser. A 82, No. 4, 66-70 (2006).

Summary: The purpose of this paper is to study the collision or non-collision problem for interacting Brownian particles in the framework of theory of Dirichlet forms. The result is closely related to a question on existence and uniqueness of strong solutions for stochastic differential equations with singular drifts.

MSC:

60J45	Probabilistic potential theory
31C15	Potentials and capacities on other spaces
31C25	Dirichlet forms
60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J60	Diffusion processes

Keywords:

diffusion; Dirichlet form; capacity

References:

[1]	M.-F. Bru, Diffusions of perturbed principal component analysis, J. Multivariate Anal. 29 (1989), no. 1, 127-136. · Zbl 0687.62048
[2]	E. Cépa and D. Lépingle, Diffusing particles with electrostatic repulsion, Probab. Theory Related Fields 107 (1997), no. 4, 429-449. · Zbl 0883.60089
[3]	F. J. Dyson, A Brownian-motion model for the eigenvalues of a random matrix, J. Mathematical Phys. 3 (1962), 1191-1198. · Zbl 0111.32703
[4]	M. Fukushima, Y. Ōshima, and M. Takeda, Dirichlet forms and symmetric Markov processes , de Gruyter, Berlin, 1994.
[5]	K. Itô and H. P. McKean, Jr., Diffusion processes and their sample paths , Springer, Berlin, 1974.
[6]	H. Osada, Dirichlet form approach to infinite-dimensional Wiener processes with singular interactions, Comm. Math. Phys. 176 (1996), no. 1, 117-131. · Zbl 0837.60073
[7]	L. C. G. Rogers and Z. Shi, Interacting Brownian particles and the Wigner law, Probab. Theory Related Fields 95 (1993), no. 4, 555-570. · Zbl 0794.60100
[8]	H. Spohn, Interacting Brownian particles: a study of Dyson’s model, in Hydrodynamic behavior and interacting particle systems , ed. by G. C. Papanicolaou ( Minneapolis , Minn ., 1986), 151-179, IMA Vol. Math. Appl., 9, Springer, New York, 1987. · Zbl 0674.60096