2001 | OriginalPaper | Buchkapitel
Brownian Motion on the Martin Space
verfasst von : Joseph L. Doob
Erschienen in: Classical Potential Theory and Its Probabilistic Counterpart
Verlag: Springer Berlin Heidelberg
Enthalten in: Professional Book Archive
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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 {wξh(·), ℱξh(·)} be an h-Brownian motion in D from ξ with lifetime Sξh. For A a subset of D let SξhA and LξhA, respectively, be the hitting and last hitting times of A by wξh(·). 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 wξh(Sξh−) 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 wξh(Sξh−) = ζ almost surely. With this choice of h we shall sometimes write wξζ(·), SξζA, LξζA, respectively, for wξh(·), SξhA, LξhA.