Generalized Images for Brownian First Passage to Monomial Boundaries
Gerardo Hernández-del-Valle
Source abstract
We study the first passage of Brownian motion to a monomial moving boundary. For every integer degree at least two, we construct a time-independent signed image measure whose heat potential is the killed Brownian transition density on a nonzero interval of the natural Brownian time scale. The image density is determined by a second-kind Volterra equation in the image coordinate, not by the classical Volterra equations in time for the hitting density. Its kernel is an explicit Wright function, and the first-passage density is recovered afterwards from the boundary flux. Exponential bounds establish convergence, while a beta-moment transform connects the Wright kernel with the finite-order differential hierarchy previously obtained from transported moving zeros. A cubic numerical experiment independently checks the resulting density against simulated Brownian paths.
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