A NEW APPROACH TO THE MARTIN BOUNDARY VIA DIFFUSIONS CONDITIONED TO HIT A COMPACT SET

• 成果类型: Article
• 署名作者: PINSKY, RG
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989411
• 发表日期: 1993
• 页码: 453-481
• 关键词: harmonic-functions; asymptotic-behavior; brownian-motion; MANIFOLDS

摘要

Let L generate a transient diffusion X(t) on R(d) and let D be an exterior domain. Let h be the smallest positive solution of Lh = 0 in D and h = 1 on partial derivative D. Define X(h)(t) to be the process X(t) conditioned to hit partial derivative D. By Doob's h-transform theory, X(h)(t) is also a Markov diffusion and its generator L(h) is defined by L(h)f = (1/h)L(hf). Letting tau(D) be the hitting time of partial derivative D, define the harmonic measure for X(h)(t) on partial derivative D starting from x is-an-element-of D by mu(x)h(dy) = p(x)h(X(h)(tau(d)) is-an-element-of dy). Let {x(n)}n=1 infinity subset-of D be a sequence satisfying lim(n-->infinity)\x(n)\ = infinity for which mu(xn)h converges weakly. Call two such sequences {x(n)}n=1 infinity and {x(n)'}n=1 infinity equivalent if lim(n-->infinity)mu(xn)h = lim(n-->infinity)mu(xn)'h. We call the set of equivalence classes thus generated the harmonic measure boundary at infinity for L(h). This boundary is independent of the particular exterior domain D. We prove that the harmonic measure boundary at infinity for L(h) coincides with the Martin boundary for L on R(d), the formal adjoint of the operator L on R(d). In the case that L generates a reversible diffusion, the Martin boundaries of L and L coincide and hence the harmonic measure boundary of L(h) coincides with the Martin boundary for L on R(d). A similar probabilistic description of the Martin boundary for L on R(d) can be given in the nonreversible case. These results are then used to give explicit representations of the Martin boundaries of L and L for several classes of diffusion processes.