DIRICHLET FUNCTIONS

P. J. Fitzsimmons

University of California, San Diego

Abstract. We provide a short proof that if X is a symmetric diffusion and u(X_{t}) is a Dirichlet process (basically, a process of finite quadratic variation) then u is locally in the Dirichlet space of X.

The following result was proved (in a global form) for elliptic divergence form diffusions in R^{d} by Lyons & Zheng [LZa], and (in a local form) by Chitashvili & Mania [CM] for one-dimensional Brownian motion. See also[EW]. The proof presented here covers a very general case, and seems to be simpler that that used in the earlier work cited. It is based on the idea of even and odd additive functionals as developed in [FITa] and [FITb]. See also [FPS].

The context is a symmetric Markov diffusion process X = (X_{t})_{t≥0} in a general state space E.

Theorem. Let u : E → R be a Borel function such that A_{t}^{u} := ũ(X_{t}) − ũ(X_{0}) is a Dirichlet process; that is, decomposable as M_{t} + N_{t} where M is continuous local martingale and N is locally of zero energy. Then u ∈ D_{loc}.

Proof. The decomposition cited is necessarily unique. This uniqueness and the additivity of A^{u} imply that M and N are CAFs of X. But we know from [FITa] that a CAF locally of zero energy is necessarily even. Thus, A^{u} is the odd part of M. Lemma 3.15 of [FITb] now implies that u ∈ D_{loc}. □

Proceeding somewhat speculatively, let A be an arbitrary CAF. By a result of Oshima and Yamada [OY] we have

A_{t} = f(X_{t}) − f(X_{0}) + M_{t},

where f is quasi-continuous and M is a local martingale CAF. If A is even, then it is the even part of M, namely −Λ(M), so f ∈ D_{loc} and M = −M^{f} in this case. [The linear functional M ↦ Λ(M), from martingale CAFs to zero-energy CAFs, was introduced by S. Nakao in [NAK], and further studied in [FITa]; Nakao uses the notation Γ(M) rather than Λ(M).] If A is odd, then it coincides with

A^{f} + M + Λ(M).

1991 Mathematics Subject Classification. .

Key words and phrases. .

In general,

A = [−Λ(M)] + [A^{f} + M + Λ(M)]

is the decomposition of A into even and odd parts. Of course A has finite quadratic variation if and only if A^{f} does.

References

[CM] Chitashvili, R. and Mania, M., On functions transforming Brownian motion into a Dirichlet process, Probability theory and mathematical statistics (Tokyo, 1995), World Scientific, River Edge, NJ, 1996, pp. 20–27.

[EW] Engelbert, H.-J. and Wolf, J., Dirichlet functions of reflected Brownian motion, Preprint (1999).

[FITa] Fitzsimmons, P. J., Even and odd continuous additive functionals, Dirichlet forms and stochastic processes, (Beijing, 1993), de Gruyter, Berlin, 1995, pp. 139–154.

[FITb] Fitzsimmons, P. J., Absolute continuity of symmetric diffusions, Ann. Probab. 25 (1997), 230–258.

[FPS] Föllmer, H., Protter, P. and Shiryayev, A.N., Quadratic covariation and an extension of It’s formula, Bernoulli 1 (1995), 149–169.

[LZa] Lyons, T.J. and Zheng, W.-A., Diffusion processes with nonsmooth diffusion coefficients and their density functions, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 231–242.

[LZb] Lyons, T.J. and Zheng, W.-A., A crossing estimate for the canonical process on a Dirichlet space and a tightness result, Astérisque 157-158 (1988), 249–271.

[NAK] Nakao, S., Stochastic calculus for continuous additive functionals of zero energy, Z. Wahrsch. Verw. Gebiete 68 (1985), 557–578.

[OY] Ōshima, Y. and Yamada, T, On some representations of continuous additive functionals locally of zero energy, J. Math. Soc. Japan 36 (1984), 315–339.

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

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