Brownian Motion on the Martin Space

Let D be a connected Greenian subset of ℝN, let K be a Martin function for D, let h be a strictly positive superbarmonic function on D, and let {w_ξh(·), ℱ_ξh(·)} be an h-Brownian motion in D from ξ with lifetime S_ξh. For A a subset of D let S_ξhA and L_ξhA, respectively, be the hitting and last hitting times of A by w_ξh(·). According to Theorem l.XII.10, if h is harmonic, the Martin boundary is h-resolutive and μ _D h(ξ, dζ) = K(ζ, ξ)M _h (dζ)/h(ξ), where M _h is the Martin representing measure of h corresponding to K. According to Theorem II.2, the left limit w_ξh(S_ξh−) exists almost surely and has distribution μ _D h(ξ, ·) supported (Section l.XII.7) by the minimal Martin boundary ∂₁mD. In particular, if ζ is a minimal Martin boundary point and if h = K(ζ, ·), then μ _D h(·, {ζ}) = 1; so w_ξh(S_ξh−) = ζ almost surely. With this choice of h we shall sometimes write w_ξζ(·), S_ξζA, L_ξζA, respectively, for w_ξh(·), S_ξhA, L_ξhA.