Brownian Motion on the Martin Space

Abstract

Let D be a connected Greenian subset of ℝ^N, let K be a Martin function for D, let h be a strictly positive superharmonic function on D, and let $\left\{ {w_{\xi }^{h}( \bullet )} \right.$, ℱ$F_{\xi }^{h}(\bullet )\}$ be an h-Brownian motion in D from ξ with lifetime $S_{\xi }^{h}$. For A a subset of D let $S_{\xi }^{{hA}}$ and $L_{\xi }^{{hA}}$, respectively, be the hitting and last hitting times of A by $w_{\xi }^{h}( \bullet )$. According to Theorem 1.XII.10, if h is harmonic, the Martin boundary is h-resolutive and $\mu _{D}^{h}(\xi ,d\zeta ) = K(\zeta ,\xi ){{M}_{h}}(d\zeta )/h(\xi )$, where M_h is the Martin representing measure of h corresponding to K. According to Theorem II.2, the left limit $w_{\xi }^{h}(S_{\xi }^{h} - )$ exists almost surely and has distribution $\mu _{D}^{h}(\xi , \bullet )$ supported (Section 1.XII.7) by the minimal Martin boundary $\partial _{1}^{M}D$. In particular, if ζ is a minimal Martin boundary point and if h = K(ζ, •), then $\mu _{D}^{h}( \bullet ,\left\{ \zeta \right\}$ = 1; so $w_{\xi }^{h}(S_{\xi }^{h} - ) = \zeta$ almost surely. With this choice of h we shall sometimes write $w_{\xi }^{\zeta }( \bullet ),S_{\xi }^{{\zeta A}},L_{\xi }^{{\zeta A}}$, respectively, for $w_{\xi }^{\zeta }( \bullet ),S_{\xi }^{{\zeta A}},L_{\xi }^{{\zeta A}}$.