Theoretical analysis reveals a limit semigroup with Robin-type boundary conditions in perturbed Brownian motion, highlighting a novel approach to calculate local time distributions upon exit.
Within the framework of the theory of semigroups of operators, we consider a local singular perturbation of a one-dimensional Laplace operator. The Feller semigroups we study describe Brownian motions on an interval with killing that is more and more concentrated around a chosen point. As a result, the description of the limit semigroup involves a Robin-type boundary condition that reflects the killing mechanism that applies to this point and to this point only. As an application we provide a semigroup-theoretical approach to calculating the distribution of the local time for Brownian motion when it exits an interval.
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Bobrowski et al. (2026) studied this question.
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