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We investigate the statistics of encounters of a diffusing particle withdifferent subsets of the boundary of a confining domain. The encounters witheach subset are characterized by the boundary local time on that subset. Weextend a recently proposed approach to express the joint probability density ofthe particle position and of its multiple boundary local times via amulti-dimensional Laplace transform of the conventional propagator satisfyingthe diffusion equation with mixed Robin boundary conditions. In the particularcases of an interval, a circular annulus and a spherical shell, thisrepresentation can be explicitly inverted to access the statistics of twoboundary local times. We provide the exact solutions and their probabilisticinterpretation for the case of an interval and sketch their derivation for twoother cases. We also obtain the distributions of various associatedfirst-passage times and discuss their applications.

Joint distribution of multiple boundary local times and related first-passage time problems with multiple targets