Remarks on Harmonic Functions and Invariant Measures of Markov Processes

Abstract

SummaryAssuming duality and certain analytic conditions on the potential density u(x,y), we show that any harmonic function is a constant and the invariant measure, if one exists, is unique.We will assume that Xt is a standard process in duality with another standard process X t ′ relative to some fixed reference measure m on the state space E. The reader is refered to [l, VI, Sec 1] for the precise definition of duality and to either [1] or [2] for other usual definitions. We will also assume that both X and X′ are transient; for any compact K, TKo θt → ∞ as t → ∞ a.s.. However, only the transience of X is used in Proposition 1 and only that of X′ is used in the rest of the discussion.For simplicity, any subset A of E and any function f defined on E is automatically assumed to be measurable with respect to the Borel field of E.There is a common potential density u(x,y) which is excessive in x and co-excessive in y and satisfies:Uf(x) = ∫u(x,y)f(y)m(dy) and U′f(y) = ∫m(dx)f(x)u(x,y) for any f ≥ 0, where U and U′ are, repectively, the potential operators of X and X′.A function h ≥ 0 on E is said to be harmonic if for any compact subset K of E, PKch = h.For Brownian motion in Rn with n ≥ 3, any harmonic function is a constant. Our first proposition generalizes this classical result under a general setting, which includes as special cases, not only Brownian motion, but also transient symmetric stable processes and one sided stable processes. Our proof which uses the general theory of Markov processes is quite short, compared with the complicated analytic proof for symmetric stable processes given in [3].