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Boundary behavior of a constrained brownian motion between reflecting-repellent walls

Lépingle D.

Probability and Mathematical Statistics

2010

Vol. 30, Fasc. 2

273--287

Stochastic variational inequalities provide a unified treatment for stochastic differential equations living in a closed domain with normal reflection and/or singular repellent drift. When the domain is a convex polyhedron, we prove that the reflected-repelled Brownian motion does not hit the non-smooth part of the boundary. A sufficient condition for nonhitting a face of the polyhedron is derived from the one-dimensional situation. A full answer to the question of attainability of the walls of the Weyl chamber may be given for a radial Dunkl process.

Sojourn time of some reflected Brownian motion in the unit disk

Deaconu M., Gradinaru M., Roche J. R.

Probability and Mathematical Statistics

2000

Vol. 20, Fasc. 1

19--38

We study the heat diffusion in a domain with an obstacle inside. More precisely, we are interested in the quantity of heat in so far as a function of the position of the heat source at time 0. This quantity is also equal to the expectation of the sojourn time of the Brownian motion, reflected on the boundary of a small disk contained in the unit disk, and killed on the unit circle. We give the explicit expression of this expectation. This allows us to make some numerical estimates and thus to illustrate the behaviour of this expectation as a function of starting point of the Brownian motion.