Killed Random Processes and Heat Kernels

Let V(x) ≥ 0 be a given function tending to a constant at infinity. It is well known that the density of the Brownian motion B_t killed at the infinitesimal rate V is a Green's function for the heat operator with such a potential. With an appropriate generalization, its Laplace transform also gives the density of ∫ ₀ ^t V(B_s)ds. We construct such a Green's function via spectral analysis of the classical one-dimensional stationary Schrodinger operator. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 423–432, August, 2005. An erratum to this article is available at .