A New Approach to the Martin Boundary Via Diffusions Conditioned to Hit a Compact Set

Abstract

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 $\partial D$. Define $X^h(t)$ to be the process $X(t)$ conditioned to hit $\partial D$. 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 $\partial D$, define the harmonic measure for $X^h(t)$ on $\partial D$ starting from $x \in D$ by $\mu^h_x(dy) = P_x^h(X^h(\tau_D) \in dy)$. Let $\{x_n\}^\infty_{n = 1} \subset D$ be a sequence satisfying $\lim_{n \rightarrow \infty}|x_n| = \infty$ for which $\mu^h_{x_n}$ converges weakly. Call two such sequences $\{x_n\}^\infty_{n = 1}$ and $\{x'_n\}^\infty_{n = 1}$ equivalent if $\lim_{n \rightarrow \infty}\mu^h_{x_n} = \lim_{n \rightarrow \infty} \mu^h_{x'_n}$. 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 $\tilde{L}$ 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 $\tilde{L}$ 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 $\tilde{L}$ for several classes of diffusion processes.