On some relations between the Martin boundary and the Feller boundary
Hisao Watanabe · Journal of the Mathematical Society of Japan · 1961
In this paper we shall consider the integral representation of bounded harmonic functions by means of a regular Borel measure on the Feller boundary $\ovalbox{\tt\small REJECT}(\mathfrak{S})$ (cf.Section 9).For this purpose we investigate mutual relations between the family of bounded harmonic functions, a function lattice on the Martin boundary and a function lattice on the Feller boundary, by use of the Martin representation theorem of harmonic functions (cf.J. L. Doob [3] and T. Wata- nabe [12], [13]).This subject is closely related to some results of D. G. Kendall [9] which we shall prove here by a different method.2. Let $X$ be a countable state space with the discrete topology.Let $XU\{\rho\}$ be denoted by $\tilde{X}$ in which $\{\rho\}$ is added to $X$ as an isolated point.Let $W$ be the totality of $\tilde{X}$ -valued right-continuous functions $w$ on the inter- val $T=[0, \infty]$ .The value of $w$ at time $t$ is denoted by $w(t)$ or $x_{c}(u')$ .Let $M$ $=\{X, W, P_{x}, x\in\tilde{X}\}$ be a minimal Markov process1) where $X$ is the state space, $W$ is the sample space and $P_{x}$ is the probability measure on the Borel field $\mathscr{Z}(W)$ generated by the sets $\{w;x_{t}(w)\in A\}$ ( $A$ : a Borel set on $\tilde{X}$ ).DefineFor $x,y\in\tilde{X}$ , we set $\Pi(x, y)=P_{x}\{w;x_{r_{x}}(w)=y, \tau_{x}<+\infty\}$ .Then $\Pi(x, \rho)=1-$ $\sum_{y\in X}\Pi(x, y)$ and $\Pi(\rho, \rho)=1$ .In this paper, a finite real valued function $u(\cdot)$ over $X$ will be called $x_{t^{-}}$ harmonic if it satisfies $u(x)=\sum_{y\in X}\Pi(x, y)u(y)$ (in the sense of absolute convergence) for any $x$ in $X$ .1) The term ' minimal process' is $\dot{u}$ scd in thesense of W. Fellcr [6,.Also a precise definition of such process is seen in [13, Chapter 1].2) We denote $\tau_{x}$ in case $A=\{x\}$ .