A Diffusion First Passage Problem

Abstract

Let (Xt)t≥0 be a diffusion in natural scale on I ⊆ [0,∞), with speed measure m and initial distribution µ. Suppose that 0 ∈ I, and 0 is a regular boundary point (without loss of generality we may suppose the diffusion is reflected instantaneously at 0). Let τ ≡ inf|t; Xt =0}. Suppose that X’ is another diffusion on I, again in natural scale and with speed measure m, but this time with initial distribution µ’.