NON-SYMMETRIC PERTURBATIONS FOR SYMMETRIC DIRICHLET FORMS

P.J. Fitzsimmons and K. Kuwae UCSD and Yokohama City University

1. Setting

X = (Ω, X_{t}, P^{x}) is a symmetric diffusion process with state space E, symmetry measure m, Dirichlet form (E, D). The transition semigroup of X is denoted (P_{t}), the resolvent (R_{α}), the L^{2}(m)-infinitesimal generator (L, D(L)).

2. Kato Class

A (positive) measure μ on E is said to be of Kato class (notation μ ∈ S_{K}) provided an estimate of the form

|μR_{α}(f)| ≤ C_{α}∥f∥_{1}, ∀f ∈ L^{1}(m) (1)

obtains, with lim_{α→∞} C_{α} = 0. If (1) holds for even a single α > 0, then μ is necessarily a smooth measure. A smooth measure μ, with associated PCAF A^{μ}, is of the Kato class if and only if the function a_{t}(x) := E^{x}[A_{t}^{μ}] satisfies

lim_{t→∞} ∥a_{t}∥_{∞} = 0,

where the norm ∥ · ∥_{∞} is that of L^{∞}(m). A signed measure μ on E, formally the difference μ^{+} − μ^{−} of a pair (μ^{+}, μ^{−}) of positive measures, is said to be of Kato class provided |μ| := μ^{+} + μ^{−} is an element of S_{K}; equivalently, provided both μ^{+} and μ^{−} are elements of S_{K}.

3. Hardy Class

Let μ ∈ S_{K} be a positive measure. It is well-known (Kato, Aizenman-Simon[1], Stollmann-Voigt[15]) that for each δ > 0 there is a finite constant γ = γ(δ) ≥ 0 such that

∫_{E} ũ^{2}dμ ≤ δE(u, u) + γ ∫_{E} u^{2}dm, ∀u ∈ D. (2)

Suppose now that μ is a measure on E such that for some δ > 0 there exists γ ≥ 0 such that (2) holds. Then μ is a smooth measure. We shall say that a smooth measure μ is of Hardy class (notation μ ∈ S_{H}) if there exists δ ∈ (0, ∞) and γ ∈ [0, ∞) such that (2) holds. A wellknown sufficient condition for μ ∈ S_{H} is that for some δ > 0, β ≥ 0 the β-potential U^{β}μ is bounded above q.e. by δ, in which case γ = δβ does the job.

4. Non-symmetric perturbed form

We now fix local martingale CAFs M and M̂ from M_{loc} with energy measure μ_{⟨M⟩} and μ_{⟨M̂⟩} from S_{H}, and a signed measure ν ∈ S_{H}. Here δ(M), δ(M̂) and δ(|ν|) are three strictly positive constants. Let γ(M), γ(M̂), and γ(|ν|) denote the coefficient of ∥u∥_{2}^{2} in the estimate (2) for M, M̂, and |ν|, respectively. We assume that

δ_{0} := [2δ(M)]^{1/2} + [2δ(M̂)]^{1/2} + δ(|ν|) < 1. (3)

Given these elements, we define a quadratic form Q on D by

Q(u, v) := E(u, v) − ∫_{E} ṽdμ_{⟨M^{[u]},M⟩} − ∫_{E} ũdμ_{⟨M^{[v]},M̂⟩} − ∫_{E} ũṽdν, u, v ∈ D.

It is easy to check that there is a constant C > 0 that

|Q(u, v)| ≤ CE_{1}(u, u)^{1/2}E_{1}(v, v)^{1/2}, u, v ∈ D. (4)

Moreover,

Q_{α}(u, u) ≥ (1 − δ_{0})E(u, u) + (α − α_{0})∥u∥_{2}^{2}, u ∈ D, (5)

where α_{0} := γ(M)\sqrt{2/δ(M)} + γ(M̂)\sqrt{2/δ(M̂)} + γ(|ν|). The (nonsymmetric) quadratic form (Q, D) is therefore closed as a form on L^{2}(m). Standard resolvent theory now yields the existence of a strongly continuous semigroup (Q_{t})_{t≥0} of operators on L^{2}(m) with ∥Q_{t}∥_{L^{2}→L^{2}} ≤ e^{α_{0}t} for all t ≥ 0, whose generator (L^{Q}, D(L^{Q})) is related Q by

(−L^{Q}u, v) = Q(u, v), ∀u ∈ D(L^{Q}), v ∈ D. (6)

5. Main Result

Let M, M̂, and ν be as above. Define a multiplicative function Z = (Z_{t}) by

Z_{t} := exp(M_{t} + M̂_{t} ∘ r_{t} − \frac{1}{2}⟨M − M̂⟩_{t} + A_{t}^{ν}), (7)

where A^{ν} is the (signed)CAF of locally bounded variation associated with the smooth signed measure ν and r_{t} : Ω ∩ {t < ζ} → Ω is the reverse operator defined by r_{t}(ω)(s) := ω(t−s) if 0 ≤ s ≤ t, and r_{t}(ω)(s) := ω(0) if t < s.

Theorem 5.1. Define

\overline{P}_{t}f(x) := E^{x}[Z_{t}f(X_{t})], (8)

whenever the integral makes sense. Then

(a) \overline{P}_{t} is a well-defined mapping of L^{2}(m) into itself, and (\overline{P}_{t})_{t≥0} is a strongly continuous semigroup of bounded operator on L^{2}(m), with ∥\overline{P}_{t}∥_{L^{2}→L^{2}} ≤ e^{α_{0}t} for all t > 0.

(b) For each f ∈ L^{2}(m), Q_{t}f = \overline{P}_{t}f, m-a.e.

References

[1] M. Aizenman and B. Simon, Brownian motion and Harnack inequality for Schrödinger operators, Comm. Pure Appl. Math. 35 (1982), 209–273.

[2] Z.Q. Chen, Gaugeability and conditional gaugeability, 2001, preprint.

[3] Z.Q. Chen and T.-S. Zhang, Girsanov and Feynman-Kac type trnasformations for symmetric Markov processes, to appear in Ann. Inst. Henri Poincare, 2001, preprint.

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

[5] P.J. Fitzsimmons, Hardy’s inequality for Dirichlet forms. (English. English summary) J. Math. Anal. Appl. 250 (2000), 548–560.

[6] P.J. Fitzsimmons, On the quasi regularity of semi-Dirichlet forms, Potential Anal., 15 (2001), 151–185.

[7] M. Fukushima, Y. Oshima and M. Takeda, “Dirichlet Forms and Symmetric Markov Processes,”de Gruyter Studies in Mathematics, 19 Walter de Gruyter & Co., Berlin, 1994.

[8] K. Kuwae, Maximum principles for subharmonic functions via local semi-Dirichlet forms, preprint 2001.

[9] J. Lunt, T.J. Lyons and T.S. Zhang, Integrability of functionals of Dirichlet processes, probabilistic representations of semigroups, and estimates of heat kernels, J. Funct. Anal. 153 (1998), 320–342.

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

[11] S. Nakao, Girsanov transformation in Dirichlet space theory, in preparation.

[12] M. Röckner and T.S. Zhang, Convergence of operators semigroups generated by elliptic operators, Osaka J. Math. 34 (1997), 923–932.

[13] M. Röckner and T.S. Zhang, Probabilistic representations and hyperbound estimates for semigroups, (English. English summary) Infin. Dimens. Anal. Quantum Probab. Relat. Top. 2 (1999), 337–358.

[14] M. Röckner and T.S. Zhang, Lower order perturbations of Dirichlet processes, to appear in Forum Math. 2001 preprint.

[15] P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures. Potential Anal. 5 (1996), 109–138.