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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.
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