The distribution of Brownian motion in Rn at a natural stopping time

Abstract

For a measure μ on R n let (( B t , P μ ) be Brownian motion in R n with initial distribution μ. Let D be an open subset of R n with exit time ζ ≡ inf { t > 0: B t ∉ D}. In the case where D is a Green region with Green function G and μ is a measure in D such that G μ is not identically infinite on any component of D, we have given necessary and sufficient conditions for a measure ν in D to be of the form ν( dx) = P μ ( B T ∉ dx, T < ζ), where T is some natural stopping time for ( B t ), and we have applied this characterization to show that a measure ν in D satisfies G ν ⩽ G μ iff ν is of the form ν( dx) = P α ( B T ∉ dx, T < ζ) + β( dx), where T is some natural stopping time for ( B t ) and α and β are measures in D such that α + β = μ and β lives on a polar set. We have proved analogous results in the case where D = R 2 and μ is a finite measure on R 2 such that ∫ log + ∥ x∥ du( x) < ∞, and applied this to give a characterization of the stopping times T for Brownian motion in R 2 such that (log + ∥ B T∧ t ∥) 0< t<∞ is P μ-uniformly integrable.