SOJOURN TIME OF SOME REFLECTED BROWNIAN MOTION IN THE
UNIT DISK

Madalina Deaconu

Mihai Gradinaru

Jean Rodolphe Roche

Abstract: 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 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.

1991 AMS Mathematics Subject Classification: 60J65, 60H30, 35K20, 35J25, 30C35,
30E25.

Key words and phrases: reflected Brownian motion, boundary value problems,
fractional linear transformation, numerical computations.