A multidimensional process involving local time. (English) Zbl 0613.60050

Sznitman, A. S.; Varadhan, S. R. S.

Probab. Theory Relat. Fields 71, 553-579 (1986).

This paper is devoted to the study of a multidimensional stochastic differential equation involving the local times of the unknown process on a finite number of hyperplanes \(H_ i\). The construction of the solution answers a question of H. P. McKean [Lectures differential. Equat. 2 (USA 1966-1967), 177-194 (1969; Zbl 0181.444)] concerning the modelization of a system of n interacting particles on \({\mathbb{R}}\). A key step in the construction is the observation that the process does not visit the singular set \(\cup_{i\neq j}(H_ i\cap H_ j)\), provided it starts outside of this set.
This property makes it possible to consider the sequence \((T_ n)\) of successive times of visit to different hyperplanes \(H_ i\) and to construct the process by induction of the time intervals \((T_ n;T_{n+1})\).

Reviewer: J.-F.Le Gall

MSC:

60H10	Stochastic ordinary differential equations (aspects of stochastic analysis)
60J55	Local time and additive functionals
60J60	Diffusion processes
60J65	Brownian motion

Keywords:

multidimensional stochastic differential equation; local times; interacting particles

Citations:

Zbl 0181.444

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