First-Hitting Location Laws as Boundary Observables of Drift-Diffusion Processes

Abstract

First-passage theory usually emphasizes when absorption occurs. Here, we instead treat where absorption occurs (the first-hitting location, or FHL) as a primary boundary observable of drift-diffusion processes. We formulate its law as the exit measure induced by the diffusion generator and recover its density from the normal derivative of an elliptic Green function. This yields exact half-space kernels in arbitrary ambient dimension $d$ for constant drift, with the two- and three-dimensional cases validated by Monte Carlo simulations. In the zero-drift limit, the kernels reduce to scale-free Cauchy-type laws with algebraic tails; drift introduces exponential screening and the characteristic length $\ell_u=\sigma^2/|\mathbf{v}|$, where $\sigma$ is the noise strength and $\mathbf{v}$ is the constant drift vector, thereby localizing the boundary footprint. An entropy-based effective width provides a finite diagnostic of this crossover. To test the geometric reach of the planar theory, we also analyze drift-free exterior hitting of a circle. Its exact Poisson kernel recovers the planar Cauchy law as a local near-boundary limit, admits an exact Cauchy-Poisson composition through an intermediate line, and yields a finite-time supporting-line bound. These results organize geometry, drift, and pathwise constraints within a unified description of first-hitting location laws.