# zbMATH — the first resource for mathematics

Sojourn time of some reflected Brownian motion in the unit disk. (English) Zbl 0993.60084
Let $$D$$ be a unit disk and $$O$$ a small disk in $$D$$. Assume that a heat source is placed in $$D$$ and the heat is reflected on $$\partial O$$ and absorbed on $$\partial D$$. The problem in this article is fo find the place of heat source $$z$$ such that the quantity of the heat $Q(z)=\int_{D\setminus O} dw \int_0^\infty dt u(t,z,w) = E_z(\tau)$ becomes maximum, where $$u$$ is the fundamental solution of the associated Brownian motion and $$\tau$$ is the hitting time of $$\partial D$$. In the case of concentric circle, an explicit expression of $$Q(z)$$ is first given. Then, by using the fractional linear transformation which maps the concentric circle to the general domain, $$Q(z)$$ for general domain is calculated. Since the direct computation of the maximum of $$Q(z)$$ is not easy, some numerical computations of $$Q(z)$$ are illustrated.
##### MSC:
 60J65 Brownian motion 60H30 Applications of stochastic analysis (to PDEs, etc.) 35K20 Initial-boundary value problems for second-order parabolic equations 30E25 Boundary value problems in the complex plane