A NOTE ON “A NOTE ON THE EQUILIBRIUM

POTENTIAL OF CERTAIN DIRICHLET SPACES”

P. J. Fitzsimmons

University of California, San Diego

We give a probabilistic proof of the main result of [1]. Let X = (X_{t}, P^{x}) be a symmetric Markov process. We employ the usual notation.

Let B be a Borel subset of E with finite capacity, and let π denote the equilibrium measure for B. Thus, the hitting probability φ(x) := P^{x}(T_{B} < ∞) (x ∈ E) is an excessive m-version of the equilibrium potential U(π). In fact,

(1)  P^{x}(f(X_{L_{B}−}); L_{B} > 0) = U(f · π)(x)

for q.e. x ∈ E, for each positive Borel function f. Here L_{B} is the last exit time from B, so that {L_{B} > 0} = {T_{B} < ∞}.

Lemma. Let G ∈ E^{e} be finely open, and define g(x) := E^{x}(exp(−T_{G^{c}})). Then g is finely continuous, and

{t ≥ 0 : X_{t} ∈ G} = {t ≥ 0 : g(X_{t}) < 1}

P^{x}-a.s. for q.e. x ∈ E.

Proof. The function g is 1-excessive, hence finely continuous. Evidently G ⊂ {g < 1}. Moreover, {g < 1} \ G is the set of points of G^{c} that are irregular for G^{c}. This set is semipolar, hence m-polar.

Suppose that f in (1) vanishes outside the fine interior B^{of} of B. Let g be the function in the Lemma corresponding to G = B^{of}. If X_{L_{B}−} ∈ B^{of} and X_{L_{B}−} = X_{L_{B}} then g(X_{L_{B}}) < 1, so g(X) < 1 on [L_{B}, L_{B} + ϵ[ for some (random) ϵ > 0, because t ↦ g(X_{t}) is almost surely right continuous. But this implies that X is in B^{of} ⊂ B after the last exit time from B, which is absurd. It follows that if X_{L_{B}−} ∈ B^{of} then X_{L_{B}} ∉ B^{of}; in particular, L_{B} is a jump time of X on {X_{L_{B}−} ∈ B^{of}}. Thus, for q.e. x ∈ E,

(2) \begin{matrix}P^{x}( & f(X_{L_{B}−}); L_{B} > 0) \\ = P^{x} ∑_{t∈J,t<ζ} f(X_{t−})1_{\{X_{t−}∈B^{of} \}}1_{\{T_{B}∘θ_{t}=∞\}} + P^{x}(f(X_{ζ−})1_{\{X_{ζ−}∈B^{of} \}}) \\ = P^{x} ∫_{0}^{∞} f(X_{t})1_{B^{of}} (X_{t})N(X_{t}, 1 − φ) dH_{t} + ∫_{0}^{∞} f(X_{t})1_{B^{of}} (X_{t}) dK_{t} \\ = U(1_{B^{of}} f · [N(1 − φ)ν_{H} + ν_{K}])(x) \end{matrix}

where J is the set of jump times for X, (N, H) is a Lévy system for J, the PCAF K is the dual predictable projection of 1_{{X_{ζ−}∈E}}ϵ_{ζ}, and ν_{H} and ν_{K} are the Revuz measures of H and K respectively. By the uniqueness of charges

(3)  π(dy) = h(y) ν_{H}(dy) + ν_{K}(dy)  on B^{of},

where

h(y) = N(1 − φ) = ∫_{E}[1 − φ(z)] N(y, dz).

In the context of [1] (symmetric Lévy processes on R), one can take H_{t} = t; moreover K_{t} = Ct for some constant C ∈ [0, ∞[. Thus, in this case, π admits a density with respect to m:

\frac{π(dy)}{m(dy)} = h(y) + C.

Formula (3) also sharpens the main result of [1] in that B^{of} (the fine interior of B) appears in place of the interior of B.

References

[1] A. Dunkels, A note on the equilibrium potential of certain Dirichlet spaces, Potenital Analysis 6 (1997), 99–104.

Department of Mathematics, University of California San Diego, 9500 Gilman Drive, La Jolla, CA 92093–0112

   E-mail address: pfitz@euclid.ucsd.edu