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How long does a diffusing molecule spend in a close vicinity of a confiningboundary or a catalytic surface? This quantity is determined by the boundarylocal time, which plays thus a crucial role in the description of varioussurface-mediated phenomena such as heterogeneous catalysis, permeation throughsemi-permeable membranes, or surface relaxation in nuclear magnetic resonance.In this paper, we obtain the probability distribution of the boundary localtime in terms of the spectral properties of the Dirichlet-to-Neumann operator.We investigate the short-time and long-time asymptotic behaviors of this randomvariable for both bounded and unbounded domains. This analysis providescomplementary insights onto the dynamics of diffusing molecules near partiallyreactive boundaries.

Probability distribution of the boundary local time of reflected Brownian motion in Euclidean domains