NON-SYMMETRIC PERTURBATIONS OF SYMMETRIC DIRICHLET FORMS

PATRICK J. FITZSIMMONS AND KAZUHIRO KUWAE

Abstract. We provide a path-space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local Dirichlet form. The representation is a combination of Feynman-Kac and Girsanov formulas, and extends previously known results in the framework of symmetric diffusion processes through the use of the Hardy class of smooth measures, which contains the Kato class of smooth measures.

1. Introduction

Let X = (Ω, F, F_{t}, X_{t}, P^{x}) be a symmetric Markov diffusion processes with state space E and symmetry measure m. Let (E, D) be the associated (strongly local, quasi-regular) Dirichlet form. We examine a class of lower order perturbations of (E, D), obtaining path-space integral representations of the associated semigroups. These representations are a combination of Feynman-Kac and Girsanov formulas.

Let M and M̂ be local martingale continuous additive functionals (CAFs) of X (denoted M, M̂ ∈ M_{loc}), and let C be a CAF of X with paths locally of finite variation. The ordered triple (M, M̂, C) is a convenient way of parametrizing the perturbations we have in mind.

The quadratic variation processes ⟨M⟩ and ⟨M̂⟩ are PCAFs of X, with corresponding smooth Revuz measures μ_{⟨M⟩} and μ_{⟨M̂⟩}, their so-called energy measures. Similarly, C admits a smooth signed measure ν as Revuz measure.

We say that a smooth measure μ is of the Hardy class (and write μ ∈ S_{H}) provided there are constants δ = δ(μ) ∈]0, ∞[ and γ = γ(μ) ∈ [0, ∞[ such that

∫_{E} ũ^{2} dμ ≤ δ · E(u) + γ · ∥u∥_{2}^{2}, ∀u ∈ D. (1.1)

Here ũ denotes a quasi-continuous m-version of u, E(u) := E(u, u), and ∥ · ∥_{2} is the norm in L^{2}(m). (In the terminology of [22], (1.1) amounts to the fact that the quadratic form u ↦ ∫_{E} ũ^{2} dμ (densely defined in L^{2}(m)) is E_{1}-bounded.)

Decompose ν as the difference ν = ν^{+} − ν^{−} of mutually singular smooth measures ν^{+} and ν^{−}, and let C^{+} (resp. C^{−}) be the PCAF corresponding to ν^{+} (resp. ν^{−}). Let |C| denote the total variation process of the signed CAF C. Then |C| is a PCAF whose Revuz measure |ν| is (at least locally) the total variation of the signed measure ν. Our basic hypothesis is that each of the measures μ_{⟨M⟩}, μ_{⟨M̂⟩}, |ν| is of the Hardy class, and that

Date: November 5, 2002.

1991 Mathematics Subject Classification. Primary 31C25, secondary 60J57, 60J60, 60J45.

Key words and phrases. Dirichlet form, positivity preserving form, smooth measure, Kato class, Hardy class, Hardy inequality, local martingale, continuous additive functional, zero energy, even and odd continuous additive functionals, Fukushima decomposition, Nakao divergence, Feynman-Kac formula, Girsanov formula,

The second author was partially supported by a Grant-in-Aid for Scientific Research (C) No. 13640220 from the Japanese Ministry of Education, Culture, Sports, Science and Technology.

δ_{0} := \sqrt{2δ(M)} + \sqrt{2δ(M̂)} + δ(C^{+}) < 1, (1.2)

where δ(M) = δ(μ_{⟨M⟩}) is as in (1.1) for the measure μ_{⟨M⟩}, etc. Under these conditions, we can define a quadratic form Q on D by

\begin{matrix}Q(u, v) := E(u, v) − ∫_{E} ṽ dμ_{⟨M^{[u]},M⟩} \\ − ∫_{E} ũ dμ_{⟨M^{[v]},M̂⟩} − ∫_{E} ũṽ dν, u, v ∈ D, \end{matrix} (1.3)

where M^{[u]}, M^{[v]} are the square integrable martingale CAFs appearing in the Fukushima decomposition (2.5) below. (The covariation process ⟨M^{[u]}, M⟩ is a CAF with paths locally of finite variation and μ_{⟨M^{[u]},M⟩} is the associated smooth signed measure, etc.) It is easy to check that there is a constant k ∈]0, ∞[ such that

|Q(u, v)| ≤ k · E_{1}(u)^{1/2}E_{1}(v)^{1/2}, u, v ∈ D. (1.4)

Moreover,

Q_{α}(u, u) ≥ (1 − δ_{0})E(u) + (α − α_{0})∥u∥_{2}^{2}, u ∈ D, α > α_{0}, (1.5)

where Q_{α}(u, v) := Q(u, v) + α(u, v)_{m} and α_{0} := γ(M)\sqrt{2/δ(M)} + γ(M̂)\sqrt{2/δ(M̂)} + γ(C^{+}). The (non-symmetric) quadratic form (Q, D) is a closed form on L^{2}(m). Standard theory now yields the existence of a strongly continuous semigroup (Q_{t})_{t≥0} of bounded operators on L^{2}(m) (with ∥Q_{t}∥_{2→2} ≤ e^{α_{0}t} for all t ≥ 0) whose generator (L^{Q}, D(L^{Q})) is related to Q by

(−L^{Q}u, v)_{m} = Q(u, v), ∀u ∈ D(L^{Q}), v ∈ D. (1.6)

Notice that if u ∈ D then u^{+} ∈ D as well. Using the functional calculus for energy measures [17, Thm. 5.6.2] we obtain μ_{⟨M^{[u^{+}]}⟩} = I_{{ũ>0}}μ_{⟨M^{[u]}⟩} for u ∈ D; the estimate (1.1) now makes it is easy to see that for α > α_{0},

Q_{α}(u^{+}, u) ≥ 0, ∀u ∈ D. (1.7)

It follows that the semigroup (Q_{t})_{t≥0} is positivity preserving: If u ≥ 0 is an element of L^{2}(m), then Q_{t}u ≥ 0 m-a.e. This assertion also follows immediately from Theorem 1.1 below.

The statement of our main result requires the time reversal operators r_{t}, t > 0, acting on the sample space Ω by [r_{t}ω](s) := ω(t − s), 0 ≤ s ≤ t.

Theorem 1.1. Define a multiplicative functional Z by

Z_{t} := exp[M_{t} + M̂_{t} ∘ r_{t} − \frac{1}{2}⟨M − M̂⟩_{t} + C_{t}], t ≥ 0, (1.8)

and a family (\overline{P}_{t})_{t≥0} of operators by

\overline{P}_{t}f(x) := E^{x}[Z_{t}f(X_{t})]. (1.9)

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 operators on L^{2}(m), with ∥\overline{P}_{t}∥_{2→2} ≤ e^{α_{0}t} for t > 0.

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

It is clear from (1.3) that the adjoint semigroup (Q_{t}^{∗}), which is associated with the quadratic form Q^{∗}(u, v) := Q(v, u), is the perturbation of (P_{t}) corresponding to the triple (M̂, M, C). This is consistent with Theorem 1.1, because (2.13) below implies that the adjoint \overline{P} _{t}^{∗} of \overline{P}_{t} satisfies (1.9) with Z_{t} replaced by Z_{t} ∘ r_{t}; in view of (1.8) and Proposition 2.1(c), the functional Z_{t} ∘ r_{t} satisfies (1.8) with the roles of M and M̂ reversed.

Theorem 1.1 improves all results of its type of which we are aware, including [21, Prop. 3.4], [30, Thm. 3.4] and [43, Thm. 6.1], [44, Thm. 2.3], [47, Thm. II.6.2], and [52, Thm. 5.1]. We impose no structural assumptions on the diffusion X, and condition (1.2) is weaker than conditions imposed by earlier authors. (It is customary in this area to work under conditions implying that μ_{⟨M⟩}, μ_{⟨M̂⟩}, |ν| are measures of the Kato class.) Our method is a refinement of that used in [30], made possible by the use of even and odd CAFs [12] and Nakao’s divergence [37].

Example 1.1. Suppose that X is standard Brownian motion in ℝ^{d}. The local martingale CAF M then admits a stochastic integral representation

M_{t} = ∫_{0}^{t} b(X_{s}) dX_{s}, (1.10)

where the measurable vector field b : ℝ^{d} → ℝ^{d} is pathwise locally square integrable, in the sense that

∫_{0}^{t} |b(X_{s})|^{2} ds < ∞, ∀t > 0, (1.11)

P^{m}-a.s. A similar representation, with vector field b̂, holds for M̂. Suppose, for simplicity, that C_{t} := ∫_{0}^{t} c(X_{s}) ds for some c : ℝ^{d} → ℝ. A formal integration by parts in (1.3) shows that the infinitesimal generator L^{Q} can be expressed as

L^{Q}u = \frac{1}{2}Δu + b · ∇u − div(u · b̂) + cu

= \frac{1}{2}Δu + (b − b̂) · ∇u + (c − div b̂)u. (1.12)

The perturbation implied by the first expression for L^{Q} in (1.12) is clearly reflected in the formula (1.8) for Z_{t}. The second expression for L^{Q} in (1.12) is of technical importance in the proof of Theorem 1.1; the work of Nakao [37] provides a version of this expression that is valid in the general case.

Under condition (1.2) the semigroup defined by (1.9) operates in L^{p}(m) for an interval [p_{−}, p_{+}] ⊂ [1, ∞] of values of p. In the context of elliptic operators in ℝ^{d}, special cases of this result can be found in [5] (M = M̂ = 0) and [27] (M̂ = 0); see also [28], [23]. In the same context, a further refinement of the result presented below can be found in [48] and [29].

For p ∈ [1, ∞] with conjugate exponent q := p/(p − 1) define

δ_{0}(p) := \frac{2}{p} \sqrt{2δ(M)} + \frac{2}{q} \sqrt{2δ(M̂)} + δ(C^{+}) (1.13)

and

α_{0}(p) := \frac{2}{p}γ(M)\sqrt{2/δ(M)} + \frac{2}{q}γ(M̂)\sqrt{2/δ(M̂)} + γ(C^{+}), (1.14)

with the understanding that \frac{2}{∞} = 0. Notice that δ_{0} = δ_{0}(2) and α_{0} = α_{0}(2). Now define

p_{±} := \frac{4}{2 − κ + κ̂ ∓ \sqrt{(κ − κ̂ − 2)^{2} − 4(2κ̂ + λ)}}, (1.15)

where κ := \sqrt{2δ(M)}, κ̂ := \sqrt{2δ(M̂)} and λ := δ(C^{+}). It is easy to check that p_{−} < 2 < p_{+} provided δ_{0}(2) < 1. Moreover, p ∈ [p_{−}, p_{+}] if and only if δ_{0}(p) ≤ \frac{2}{p} · \frac{2}{q}.

Theorem 1.2. As before, suppose that δ_{0}(2) < 1. If p ∈ [p_{−}, p_{+}] then \overline{P}_{t} maps L^{p}(m) into itself and

∥\overline{P}_{t}∥_{p→p} ≤ e^{α_{0}(p)t}. (1.16)

Furthermore, if p is finite then (\overline{P}_{t})_{t≥0} is a strongly continuous semigroup on L^{p}(m).

In section 2 we describe precisely the setting in which we shall be working, and we recall necessary facts about symmetric diffusions and their Dirichlet forms. Section 3 contains the proof of Theorem 1.1 while section 4 contains the proof of Theorem 1.2. A number of examples are discussed in section 5.

2. Preliminaries

As stated in the first section, our interest is in Girsanov/Feynman-Kac transformations of a symmetric Markov diffusion process X = (Ω, F, F_{t}, X_{t}, θ_{t}, P^{x}). For general background on such diffusions we refer the reader to [17].

Getting down to specifies, we assume that the state space (E, B(E)) of X is homeomorphic to a Borel subset of some compact metric space; here B(E) is the class of Borel subsets of E. Furthermore, we assume that the transition semigroup of X, defined by

P_{t}f(x) := E^{x}[f(X_{t})], t ≥ 0, f ∈ bB(E), (2.1)

maps bB(E) (the class of bounded real-valued B(E)-measurable functions on E) into itself. In view of the symmetry of X (discussed below) this assumption of Borel measurability entails no loss of generality; see [16, §3]. To allow for the possibility that P_{t}I_{E}(x) < 1 for some states x ∈ E, we adjoin a cemetery state Δ to E; the process X is sent to Δ at its lifetime ζ. By convention any function (resp. measure) defined on E (resp. E^{∗}) is extended to Δ by declaring its value at Δ (resp. {Δ}) to be 0.

By diffusion we mean that X is a strong Markov process such that (i) the paths [0, ζ(ω)[∋ t ↦ X_{t}(ω) are continuous, and (ii) the lifetime ζ is a predictable stopping time. For this reason we can (and do) take the sample space Ω to be the space of paths ω from [0, ∞[ to E ∪{Δ} that are E-valued and continuous on [0, ζ(ω)[ and hold the value Δ after ζ(ω) := inf{t| ω(t) = Δ}.

In saying that X is symmetric, with symmetry measure m, we mean that m is a σ-finite measure on (E, B(E)) such that

(f, P_{t}g)_{m} = (P_{t}f, g)_{m} ∀f, g ∈ pB(E), (2.2)

where (u, v)_{m} := ∫_{E} uv dm. In this case (P_{t}) restricted to bB(E) ∩ L^{2}(m) extends uniquely to a strongly continuous contraction semigroup in L^{2}(m). Of course, (2.2) implies that this semigroup is self-adjoint. The Dirichlet form associated with X is the bilinear form

E(u, v) := lim_{t↓0} t^{−1}(u, v − P_{t}v)_{m} (2.3)

defined on the vector space

D := {u ∈ L^{2}(m) | sup_{t>0} t^{−1}(u, u − P_{t}u)_{m} < ∞}. (2.4)

From [15, §2] and the work of Albeverio, Ma, and Röckner [3, 4] we know that the Dirichlet form (E, D) is quasi-regular. See [33] and [17] for discussions of quasi-regularity. Because X is a diffusion, the Dirichlet form (E, D) is strongly local, in the sense that E(u, v) = 0 whenever u, v ∈ D and u is constant m-a.e. on a neighborhood of the support of the measure |v| · m; see [17] or [25].

The first exit time inf{t > 0 | X_{t} ∉ B} of a Borel subset of E is denoted τ(B). An increasing sequence {B_{n}} of Borel subsets of E is a nest provided P^{m}(lim_{n} τ(B_{n}) < ζ) = 0. The reader is referred to [33, Lemma IV. 4.5] for a characterization of this notion in terms of the Dirichlet form (E, D). A Borel set N ⊂ E is said to be X-exceptional provided P^{m}(τ(E \ N) < ζ) = 0. It can be shown that N ∈ B(E) is X-exceptional if and only if there is a nest of compacts {K_{n}} such that N ⊂ ∩_{n}K_{n}^{c}.

Each element u ∈ D admits an m-modification ũ such that t ↦ ũ(X_{t}) is continuous on [0, ∞[, P^{m}-a.s. The function ũ is quasi-continuous in the sense that there is a nest (K_{n}) of compact subsets of E such that ũ|_{K_{n}} ∈ C(K_{n}) for every n. Moreover, the CAF A_{t}^{[u]} := ũ(X_{t}) − ũ(X_{0}) admits a Fukushima decomposition

ũ(X_{t}) − ũ(X_{0}) = M_{t}^{[u]} + N_{t}^{[u]}, ∀t ≥ 0, (2.5)

P^{x}-a.s. for q.e. x ∈ E, in which M^{[u]} and N^{[u]} are CAFs of X, M^{[u]} is a martingale such that sup_{t>0} t^{−1}E^{m}[(M^{[u]})^{2}] < ∞, and lim_{t→0} E^{m}[(N^{[u]})^{2}] = 0. This decomposition is unique, and its components M^{[u]} and N^{[u]} are referred to as the martingale and zero-energy parts, respectively, of A^{[u]}.

We use M (resp. N) to denote the class of martingale CAFs of finite energy (resp. CAFs of zero energy). To define these classes precisely, introduce the mutual energy e(M, N) of two CAFs by

e(M, N) := lim_{t↓0}(2t)^{−1}E^{m}[M_{t}N_{t}] (2.6)

whenever the limits exists. Then, writing e(M) for e(M, M),

M := {M | M is a CAF of X; for each t > 0, E^{x}[M_{t}^{2}] < ∞ and

E^{x}[M_{t}] = 0 for q.e. x ∈ E; e(M) < ∞}, (2.7)

and

N := {N | N is a CAF of X; for each t > 0, E^{x}[|N_{t}|] < ∞

for q.e. x ∈ E; e(N) = 0}. (2.8)

When equipped with the inner product (M, M′) ↦ e(M, M′), the class M is a Hilbert space.

A function u : E → ℝ is locally in D (notation: u ∈ D_{loc}) provided there is a nest {E_{n}} of finely open sets and a sequence {u_{n}} of elements of D such that u = u_{n}, m-a.e. on E_{n} for each n ∈ ℕ. Each u ∈ D_{loc} admits a quasi-continuous modification ũ such that (2.5) holds for t ∈ [0, ζ[, where now M^{[u]} ∈ M_{loc} and N^{[u]} ∈ N_{loc}. [The localizations M_{loc} and N_{loc} of M and N are defined in the obvious way; for example, N ∈ N_{loc} if and only if there is a nest {E_{n}} of finely open sets and a sequence {N^{(n)}} of elements of N such that N_{t} = N_{t}^{(n)} for all 0 ≤ t < τ(E_{n}), P^{m}-a.s. for each n ∈ ℕ. The elements of M_{loc} and N_{loc} are local additive functionals as discussed on page 226 in [17].]

If M ∈ M_{loc}, then the quadratic variation process ⟨M⟩ is a PCAF of X. The associated Revuz measure μ_{⟨M⟩} is a smooth measure, and M ∈ M if and only if μ_{⟨M⟩}(E) < ∞, in which case e(M) = \frac{1}{2}μ_{⟨M⟩}(E). As a matter of notation, μ_{⟨u⟩} := μ_{⟨M^{[u]}⟩} for u ∈ D_{loc}. More generally, given M, M′ ∈ M_{loc}, the covariation process ⟨M, M′⟩ is a CAF of X locally of finite variation; as such it admits a smooth signed Revuz measure μ_{⟨M,M′⟩}, and if M and M′ both lie in M then e(M, M′) = \frac{1}{2}μ_{⟨M,M′⟩}(E). If f is a quasi-continuous function, then the Itô integral

(f • M)_{t} := ∫_{0}^{t} f(X_{s}) dM_{s} (2.9)

defines an element of M_{loc}. Evidently f • M ∈ M if and only if f ∈ L^{2}(μ_{⟨M⟩}), in which case e(f • M) = \frac{1}{2} ∫_{E} f^{2} dμ_{⟨M⟩}. In consonance with the Cauchy-Schwarz inequality satisfied by e, we have, by the Kunita-Watanabe inequality [11, VII. 54],

|∫_{E} fg dμ_{⟨M,M′⟩}| ≤ (∫_{E} f^{2} dμ_{⟨M⟩})^{1/2}(∫_{E} g^{2} dμ_{⟨M′⟩})^{1/2}. (2.10)

Also, because μ_{⟨M±M′⟩} = μ_{⟨M⟩} ± 2μ_{⟨M,M′⟩} + μ_{⟨M′⟩} is a positive measure, we have

μ_{⟨M+M′⟩} ≤ 2μ_{⟨M⟩} + 2μ_{⟨M′⟩}. (2.11)

Given t > 0 and a path ω with ζ(ω) > t, define the reversed (at time t) path r_{t}ω by

r_{t}(ω)(s) := \{ \begin{matrix}ω(t − s), 0 ≤ s ≤ t, \\ ω(0), s > t. \end{matrix} (2.12)

Actually, r_{t} is a mapping of equivalence classes of paths that coincide up to time t; thus F ∘r_{t} is well defined on {t < ζ} provided F is F_{t}-measurable. See [50] for more details. The importance of r_{t} stems from the following amplification of the symmetry property (2.2): For every t > 0 and every positive F_{t}-measurable function F,

E^{m}[F ∘ r_{t} : t < ζ] = E^{m}[F : t < ζ]. (2.13)

Given a CAF A of X, the even and odd parts of A are defined by the formulae

A_{t}^{even} := [A_{t} + A_{t} ∘ r_{t}]/2, A_{t}^{odd} := [A_{t} − A_{t} ∘ r_{t}]/2, 0 ≤ t < ζ. (2.14)

We say that A is even (resp. odd) provided A_{t} = A_{t}^{even} (resp. A_{t} = A_{t}^{odd}) P^{m}-a.s. on {t < ζ} for each t > 0. It is easy to check to that A_{t+s}^{even} = A_{t}^{even} + A_{s}^{even} ∘ θ_{t} P^{m}-a.s. on {t + s < ζ}, and similarly for A^{odd}.

For proofs of the facts asserted in the following proposition, see [12, 13].

Proposition 2.1. (a) Every element of N_{loc} is an even CAF.

(b) If u ∈ D_{loc}, then the even part of the local martingale CAF M^{[u]} is −N^{[u]}, while the odd part is A^{[u]}.

(c) Every CAF locally of finite variation is even.

The odd part of a general martingale CAF can be expressed in terms of a divergence-like functional defined by S. Nakao: according to [37, Section 3], there is a linear map Λ : M → N, such that

lim_{t↓0} t^{−1}E^{m}[h(X_{0})Λ(M)_{t}] = −e(M^{[h]}, M), ∀M ∈ M, h ∈ D. (2.15)

In particular, Λ(M^{[u]}) = N^{[u]} for all u ∈ D, because of [17, Thm. 5.2.4]. By [37, Thm. 2.2], the identity (2.15) uniquely determines Λ(M) ∈ N. Localization arguments allow one to extend Λ to a linear mapping of M_{loc} into N_{loc} such that M^{odd} = M + Λ(M) and M^{even} = −Λ(M) for all M ∈ M_{loc}. It should be noted that Λ(M) ∈ N_{loc} for M ∈ M_{loc} is uniquely determined in the sense that for M^{(1)}, M^{(2)} ∈ M_{loc}, M_{t}^{(1)} = M_{t}^{(2)} ∀t ∈ [0, ζ[ P^{x}-a.s. for q.e. x ∈ E implies Λ(M^{(1)})_{t} = Λ(M^{(2)})_{t} ∀t ∈ [0, ζ[ P^{x}-a.s. for q.e. x ∈ E.

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

|μU_{α}(f)| ≤ c_{α}∥f∥_{1}, ∀f ∈ L^{1}(m), (2.16)

obtains, with lim_{α→∞} c_{α} = 0. If (2.16) holds for even a single α > 0 with c_{α} ∈ [0, ∞[, then μ is said to be of Dynkin class (notation μ ∈ S_{D}). Any element of S_{D} is a smooth measure, by [16, (4.15)]. Alternatively, a smooth measure μ, with associated PCAF A^{μ}, is of the Kato class (resp. Dynkin class) if and only if the function a_{t}(x) := E^{x}[A_{t}^{μ}] satisfies

(2.17)  lim_{t→0} ∥a_{t}∥_{∞} = 0, (resp. a_{t} ∈ L^{∞}(m) for some (or all) t > 0),

where the norm ∥ · ∥_{∞} is that of L^{∞}(m).

Given μ ∈ S_{D}, [14, Example (1.17)] implies that (1.1) holds with δ = c_{α} and γ = αc_{α}; cf. [22, Lemma VI. 4.8a] and [49, Thm. 3.1]. Thus S_{K} ⊂ S_{D} ⊂ S_{H}, where the Hardy class S_{H} is as defined in section 1.

By combining [49, Proposition 1.9] with the proof of [14, Thm. 4.22], we obtain the following link between S_{K} and the class of smooth measures.

Proposition 2.2. If μ is a positive smooth measure, then there exists a strictly positive quasicontinuous φ ∈ D such that the measure φ · μ is of the Kato class.

3. Proof of Theorem 1.1

We now fix a triple (M, M̂, C) as described in section 1; especially, condition (1.2) holds. The bilinear form Q, its associated semigroup (Q_{t}), and the multiplicative functional Z are also as in section 1. Let (V_{α})_{α>α_{0}} be the resolvent associated with (Q, D). It follows immediately from (1.4) and (1.5) that (Q, D) is a quasi-regular positivity preserving form, and that the quasi notions (nest, quasi-continuity, etc.) associated with (Q, D) coincide with the analogous notions for (E, D). The same remarks apply to the restricted forms (Q^{(n)}, D^{(n)}) considered below.

Before embarking on the proof of Theorem 1.1, we establish some properties of Z. To this end, the following expression for Z_{t} will be useful. Because Λ(M̂) is even and M̂ + Λ(M̂) is odd,

M̂_{t} ∘ r_{t} = [M̂_{t} + Λ(M̂)_{t}] ∘ r_{t} − Λ(M̂)_{t} = −M̂_{t} − 2Λ(M̂)_{t} (3.1)

P^{m}-a.s. on {t < ζ} for each t > 0. Therefore

(3.2) Z_{t} = exp(K_{t} − \frac{1}{2}⟨K⟩_{t} + C_{t} − 2Λ(M̂)_{t}), P^{m}-a.s. on \{t < ζ\}, ∀t > 0,

where K := M − M̂.

With regard to condition (ii) in part (b) of the lemma to follow, notice that if M̂ = M^{o}+M^{[u]} where M^{o} is odd, then Λ(M̂) = N^{[u]}. For general M̂, the representation Λ(M̂) = N^{[u]} is locally true, by an important result of Ōshima and Yamada [41].

Lemma 3.1. (a) If μ_{⟨M⟩} + μ_{⟨M̂⟩} + |ν| ∈ S_{K}, then \overline{P}_{t} defined by (1.9) maps L^{2}(m) into itself and there is a constant c_{1} such that ∥\overline{P}_{t}∥_{2→2} ≤ c_{1} · e^{c_{1}t} for all t > 0.

(b) Suppose that (i) μ_{⟨M⟩} + μ_{⟨M̂⟩} + |ν| ∈ S_{K}, (ii) 2Λ(M̂) = N^{[ρ]} for some ρ ∈ D_{loc}, (iii) μ_{⟨ρ⟩} ∈ S_{K}, and (iv) ρ̃ is bounded. Then there is a constant c_{2} such that for m-a.e. x ∈ E,

E^{x}[ sup_{0≤s<t∧ζ} Z_{s}] ≤ c_{2}e^{c_{2}t}, ∀t > 0. (3.3)

Proof. (a) Decompose the CAF C as the difference C^{+} − C^{−} of PCAFs with mutually singular Revuz measures ν^{+} and ν^{−}, respectively. Then ν^{+} ≤ |ν|, and because μ_{⟨M⟩} + μ_{⟨M̂⟩} + |ν| ∈ S_{K}, so also

η := (9/2)[μ_{⟨M⟩} + μ_{⟨M̂⟩}] + 3ν^{+} ∈ S_{K}. (3.4)

Let f and g be non-negative elements of L^{2}(m). Then by Hölder’s inequality and the expression (1.8) for Z_{t},

\begin{matrix}E^{m}[f(X_{0})Z_{t}g(X_{t})] ≤ E^{m}[f(X_{0})^{2}e^{3M_{t}−(9/2)⟨M⟩_{t}}]^{1/3} \\ ×E^{m}[g(X_{t})^{2}e^{3M̂_{t}∘r_{t}−(9/2)⟨M̂⟩_{t}}]^{1/3}E^{m}[f(X_{0})e^{B_{t}}g(X_{t})]^{1/3}, \end{matrix} (3.5)

where B_{t} := (9/2)⟨M⟩_{t}+(9/2)⟨M̂⟩_{t}+3C_{t}^{+} is the PCAF with Revuz measure η. Now e^{3M_{t}−(9/2)⟨M⟩_{t}} is a positive supermartingale, so E^{x}[e^{3M_{t}−(9/2)⟨M⟩_{t}}] ≤ 1 for q.e. x ∈ E. Thus the first factor on the right side of (3.5) is no bigger than ∥f∥_{2}^{2/3}. Because ⟨M̂⟩ is even, the middle factor on the right side of (3.5) is equal to

E^{m}[g(X_{t})^{2}e^{3M̂_{t}∘r_{t}−(9/2)⟨M̂⟩_{t}∘r_{t}}]^{1/3} = E^{m}[g(X_{0})^{2}e^{3M̂_{t}−(9/2)⟨M̂⟩_{t}}]^{1/3} ≤ ∥g∥_{2}^{2/3}, (3.6)

because e^{3M̂_{t}−(9/2)⟨M̂⟩_{t}} is also a positive supermartingale. Finally, by [19, (3.14)], (3.4) implies the existence of a constant c_{1} such that the third factor on the right side of (3.5) is no bigger than c_{1}e^{c_{1}t}∥f∥_{2}^{1/3}∥g∥_{2}^{1/3}. Combining these estimates we find that

E^{m}[f(X_{0})Z_{t}g(X_{t})] ≤ c_{1} · e^{c_{1}t}∥f∥_{2}∥g∥_{2}, (3.7)

which proves the assertion.

(b) By hypothesis there exists r > 0 such that |ρ̃| ≤ r for q.e. x ∈ E. Using (3.2) and the notation C^{+} from the proof of part (a), we see that

(3.8)  sup_{0≤s<t∧ζ} Z_{s} ≤ e^{2r} sup_{0≤s≤t} exp(J_{s} − \frac{1}{2}⟨J⟩_{s}) exp(\frac{1}{2}⟨J⟩_{t} + C_{t}^{+}),

P^{m}-a.s., where J := K + M^{[ρ]}. Now μ_{⟨J⟩} ≤ 4μ_{⟨M⟩} + 4μ_{⟨M̂⟩} + 2μ_{⟨ρ⟩} by (2.11), so the Revuz measures of ⟨J⟩ and C^{+} are of the Kato class; thus there is a constant k ∈]0, ∞[ such that for q.e. x ∈ E,

E^{x}[exp(⟨J⟩_{t} + 2C_{t}^{+})] ≤ ke^{kt} (3.9)

and

E^{x}[exp(6⟨J⟩_{t})] ≤ ke^{kt} (3.10)

for all t > 0; see for example [16, (3.11)]. Applying Doob’s inequality to the martingale exp(J − \frac{1}{2}⟨J⟩) we obtain

E^{x}[| sup_{0≤s≤t} exp(J_{s} − \frac{1}{2}⟨J⟩_{s})|^{2}] ≤ 4E^{x}[exp(2J_{t} − ⟨J⟩_{t})]
≤ 4E^{x}[exp(4J_{t} − 8⟨J⟩_{t})]^{1/2}E^{x}[exp(6⟨J⟩_{t})]^{1/2}
≤ 4\sqrt{k}e^{(k/2)t},

(3.11) because exp(4J_{t} − 8⟨J⟩_{t}) is a martingale. In view of (3.9) and (3.11), the assertion follows from (3.8) and the Cauchy-Schwarz inequality.

Let L^{Q}, with domain D(L^{Q}), denote the infinitesimal generator of the semigroup (Q_{t}). Fix u ∈ D(L^{Q}). Then by (1.3), for all v ∈ D

\begin{matrix}E(u, v)= Q(u, v) + ∫_{E} ṽ dμ_{⟨M^{[u]},M⟩} + ∫_{E} ũ dμ_{⟨M^{[v]},M̂⟩} + ∫_{E} ũṽ dν \\ = (−L^{Q}u, v)_{m} + ∫_{E} ṽ dμ_{⟨M^{[u]},M−M̂⟩} \\ +2e(ũ • M^{[v]} + ṽ • M^{[u]}, M̂) + ∫_{E} ũṽ dν. \end{matrix} (3.12)

From this it follows that

\begin{matrix}N_{t}^{[u]} = ∫_{0}^{t} L^{Q}u(X_{s}) ds − ⟨M^{[u]}, K⟩_{t} \\ +2 ∫_{0}^{t} ũ(X_{s}) ∘ dΛ(M̂)_{s} − ∫_{0}^{t} ũ(X_{s}) dC_{s}, \end{matrix} (3.13)

where the Stratonovich-type integral ∫_{0}^{t} ũ(X_{s}) ∘ dΛ(M̂)_{s} is defined by

∫_{0}^{t} ũ(X_{s}) ∘ dΛ(M̂)_{s} := Λ(∫_{0}^{·} ũ(X_{s}) dM̂_{s})_{t} − \frac{1}{2}⟨M^{[u]}, M̂⟩_{t}. (3.14)

That is, the Fukushima decomposition associated with u is

\begin{aligned}ũ(X_{t}) − ũ(X_{0}) & = M_{t}^{[u]} + ∫_{0}^{t} L^{Q}u(X_{s}) ds − ⟨K, M^{[u]}⟩_{t} \\ & +2 ∫_{0}^{t} ũ(X_{s}) ∘ dΛ(M̂)_{s} − ∫_{0}^{t} ũ(X_{s}) dC_{s}. \end{aligned} (3.15)

Lemma 3.2. Suppose, in addition to the other hypothesis of Theorem 1.1, that 2Λ(M̂) = N^{[ρ]} for some ρ ∈ D_{loc}. Then for each t > 0,

ũ(X_{t})Z_{t} = ũ(X_{0}) + ∫_{0}^{t} Z_{s} dM_{s}^{[u]} + ∫_{0}^{t} Z_{s}ũ(X_{s}) dK_{s} + ∫_{0}^{t} Z_{s}L^{Q}u(X_{s}) ds, (3.16)

P^{m}-a.s. on {t < ζ}.

Proof. Because of the assumed form of Λ(M̂), we have M̂ ∘ r_{t} = −M̂_{t} − N_{t}^{[ρ]}. Using this and (3.2) we factor Z as

(3.17) Z_{t} = exp(K_{t} − \frac{1}{2}⟨K⟩_{t} + C_{t} + ρ̃(X_{0}) + M_{t}^{[ρ]}) exp(−ρ̃(X_{t})), P^{m}-a.s. on \{t < ζ\}.

By [37, §4] or [31, Thm. 2.4.1] we have

\begin{matrix}ũ(X_{t})e^{−ρ̃(X_{t})} − ũ(X_{0})e^{−ρ̃(X_{0})} \\ = − ∫_{0}^{t} ũ(X_{s})e^{−ρ̃(X_{s})} dM_{s}^{[ρ]} + ∫_{0}^{t} e^{−ρ̃(X_{s})} dM_{s}^{[u]} \\ − ∫_{0}^{t} ũ(X_{s})e^{−ρ̃(X_{s})} ∘ dN_{s}^{[ρ]} + ∫_{0}^{t} e^{−ρ̃(X_{s})} ∘ dN_{s}^{[u]} \\ +\frac{1}{2} ∫_{0}^{t} ũ(X_{s})e^{−ρ̃(X_{s})} d⟨M^{[ρ]}⟩_{s} − ∫_{0}^{t} e^{−ρ̃(X_{s})} d⟨M^{[u]}, M^{[ρ]}⟩_{s}. \end{matrix} (3.18)

We focus on the terms

− ∫_{0}^{t} ũ(X_{s})e^{−ρ̃(X_{s})} ∘ dN_{s}^{[ρ]} + ∫_{0}^{t} e^{−ρ̃(X_{s})} ∘ dN_{s}^{[u]}. (3.19)

In view of (3.13),

N_{t}^{[u]} = ∫_{0}^{t} L^{Q}u(X_{s}) ds − ⟨M^{[u]}, K⟩_{t} + ∫_{0}^{t} ũ(X_{s}) ∘ dN_{s}^{[ρ]} − ∫_{0}^{t} ũ(X_{s}) dC_{s}. (3.20)

By the injectivity of Λ : M_{loc} → N_{loc}, we have

∫_{0}^{t} ũ(X_{s}) ∘ dN_{s}^{[ρ]} = Λ(∫_{0}^{·} ũ(X_{s}) dM_{s}^{[ρ]})_{t} − \frac{1}{2}⟨M^{[u]}, M^{[ρ]}⟩_{t}. (3.21)

Consequently,

\begin{matrix}∫_{0}^{t} e^{−ρ̃(X_{s})} ∘ d(∫_{0}^{·} ũ(X_{r}) ∘ N_{r}^{[ρ]} & ) & s \\ = ∫_{0}^{t} e^{−ρ̃(X_{s})} ∘ dΛ(∫_{0}^{·} ũ(X_{r}) dM_{r}^{[ρ]})_{s} − \frac{1}{2} ∫_{0}^{t} e^{−ρ̃(X_{s})} d⟨M^{[u]}, M^{[ρ]}⟩_{s} \\ = Λ(∫_{0}^{·} e^{−ρ̃(X_{s})}ũ(X_{s}) dM_{s}^{[ρ]})_{t} − \frac{1}{2}⟨M^{[e^{−ρ}]}, u • M^{[ρ]}⟩_{t} \\ −\frac{1}{2} ∫_{0}^{t} e^{−ρ̃(X_{s})} d⟨M^{[u]}, M^{[ρ]}⟩_{s} \\ = Λ(∫_{0}^{·} e^{−ρ̃(X_{s})}ũ(X_{s}) dM_{s}^{[ρ]})_{t} − \frac{1}{2}⟨M^{[ue^{−ρ}]}, M^{[ρ]}⟩_{s} \\ = ∫_{0}^{t}[ũ(X_{s})e^{−ρ̃(X_{s})}] ∘ dN_{s}^{[ρ]}. \end{matrix} (3.22)

Thus, (3.19) is equal to

∫_{0}^{t} e^{−ρ̃(X_{s})}L^{Q}u(X_{s}) ds − ∫_{0}^{t} e^{−ρ̃(X_{s})} d⟨K, M^{[u]}⟩_{s} − ∫_{0}^{t} e^{−ρ̃(X_{s})}ũ(X_{s}) dC_{s}, (3.23)

which is evidently a continuous semimartingale. Feeding (3.23) into (3.18), then using Itô’s formula, we obtain (after a straightforward, if tedious, calculation) the assertion of the lemma.

Proof of Theorem 1.1. We would like to take expectations of both sides of (3.16), but we must deal with the fact that the local martingale terms on the right side need not be martingales. To this end let {f_{j}}_{j∈ℕ} ⊂ L^{2}(m) be total in L^{2}(m) and such that f_{j} ≥ 0 for each j ∈ ℕ. Fix α > α_{0} and notice that the linear span of {V_{α}f_{j} | j ∈ ℕ} is contained in D(L^{Q}) and is dense therein with respect to the graph norm: If u ∈ D(L^{Q}) then there is a sequence {g_{k}} of elements of the linear span of {f_{j}} such that lim_{k} ∥V_{α}g_{k} − u∥_{2} = 0 and lim_{k} ∥L^{Q}(V_{α}g_{k}) − L^{Q}u∥_{2} = 0. Each V_{α}f_{j}, being an element of D, admits a quasi-continuous m-version { \atop V_{α}f_{j}}; therefore, by a standard construction [17, Thm. 2.1.2], there is an E-nest of compacts \{K_{n}^{1}\} such that { \atop V_{α}f_{j}}|_{K_{n}^{1}} ∈ C(K_{n}^{1}) for each n and each j. Next, by (2.2), there is a strictly positive quasi-continuous function φ such that the measure φ·(μ_{⟨M⟩}+μ_{⟨M̂⟩}+|ν|) is an element of S_{K}. Because φ is quasi-continuous, there is a nest {K_{n}^{2}} such that φ|_{K_{n}^{2}} ∈ C(K_{n}^{2}) for each n. By [41, Thm. 2.1], there is a nest {K_{n}^{3}} and a sequence ρ_{n} ∈ D_{loc} such that 2Λ(M̂)_{t} = N_{t}^{[ρ_{n}]} for P^{m}-a.s. on 0 ≤ t < τ(K_{n}^{3}) for each n. Finally, since μ_{⟨ρ_{n}⟩} is a smooth measure and ρ̃_{n} is quasi-continuous, there is a nest {K_{n,j}}_{j∈ℕ} such that I_{K_{n,j}} · μ_{⟨ρ_{n}⟩} ∈ S_{K} and ρ̃_{n}|_{K_{n,j}} ∈ C(K_{n,j}) for each n, j ∈ ℕ. Let π be a probability measure equivalent to m; then lim_{j} E^{π}[exp(−τ(K̂_{n} ∩ K_{n,j})) − exp(−τ(K̂_{n}))] = 0 so we can choose j_{n} so large that E^{π}[exp(−τ(K̂_{n} ∩ K_{n,j_{n}})) − exp(−τ(K̂_{n}))] < 1/n^{2}, where K̂_{n} := K_{n}^{1} ∩ K_{n}^{2} ∩ K_{n}^{3}, Clearly

K_{n} := K̂_{n} ∩ K_{n,j_{n}} ∩ {φ ≥ 1/n}, n ∈ ℕ (3.24)

defines a nest of compact sets. Let E_{n} denote the fine interior of K_{n} with respect to X. Then E_{n} is a finely open nearly Borel set and is quasi-open.

Define D^{(n)} := {u ∈ D | ũ = 0 q.e. on E_{n}^{c}}, and let Q^{(n)} denote the restriction of Q to D^{(n)}. Clearly (Q^{(n)}, D^{(n)}) is a quasi-regular positivity preserving coercive closed form on L^{2}(E_{n}; m) satisfying the same hypothesis as (Q, D). In fact, (Q^{(n)}, D^{(n)}) is related to the restriction of E to D^{(n)} (which is the Dirichlet form of the part process X^{E_{n}}) in exactly the same way that (Q, D) is related to (E, D). In addition, the conditions of Lemma 3.1 and Lemma 3.2 are satisfied by X^{E_{n}} and (Q^{(n)}, D^{(n)}). Let (Q_{t}^{(n)})_{t>0} and (V_{α}^{(n)})_{α>α_{0}} be the semigroup and resolvent on L^{2}(E_{n}; m) associated with (Q^{(n)}, D^{(n)}). Let (L^{Q,(n)}, D(L^{Q,(n)})) denote the infinitesimal generator of (Q_{t}^{(n)}). Consider a bounded u ∈ D(L^{Q,(n)}). Let τ_{n} be the first exit time of X from E_{n}. Then by Lemma 3.2 (applied to X^{E_{n}} and (Q^{(n)}, D^{(n)})),

\begin{aligned}ũ(X_{t∧τ_{n}})Z_{t∧τ_{n}} & = ũ(X_{0}) + ∫_{0}^{t∧τ_{n}} Z_{s} dM_{s}^{[u],(n)} \\ & + ∫_{0}^{t∧τ_{n}} Z_{s}ũ(X_{s}) dK_{s} + ∫_{0}^{t∧τ_{n}} Z_{s}L^{Q,(n)}u(X_{s}) ds, \end{aligned} (3.25)

because u(X_{τ_{n}}) = 0, P^{m}-a.s. Here was used the fact that Z^{(n)} (resp. K^{(n)}), the analog of Z (resp. K) with respect to X^{E_{n}} and (Q^{(n)}, D^{(n)}), coincides with Z (resp. K) on [0, τ_{n}[. Let {T_{k}} be an increasing sequence of stopping times with T_{k} ↑ ζ as k → ∞ reducing the local martingale terms on the right hand side of (3.25). Replacing t by t ∧ T_{k} in (3.25) and taking expectations we obtain

E^{x}[ũ(X_{t∧T_{k}∧τ_{n}})Z_{t∧T_{k}∧τ_{n}}] = ũ(x) + E^{x}[∫_{0}^{t∧T_{k}∧τ_{n}} Z_{s}(L^{Q,(n)}u)(X_{s}) ds], (3.26)

for m-a.e. x ∈ E_{n}. Because u is bounded, Lemma 3.1(b) permits us to conclude that, as k → ∞, the left side of (3.26) converges to E^{x}[ũ(X_{t∧τ_{n}})] for m-a.e. x ∈ E_{n}. On the other hand

|E^{x}[∫_{0}^{t∧τ_{n}} Z_{s}(L^{Q,(n)}u)(X_{s}) ds]| ≤ ∫_{0}^{t} \overline{P} _{s}^{(n)}|L^{Q,(n)}u|(x) ds < ∞ (3.27)

for m-a.e. x ∈ E_{n}, where

\overline{P} _{s}^{(n)}f(x) := E^{x}[f(X_{s})Z_{s} : s < τ_{n}] (3.28)

and the final inequality on the right hand side of (3.27) follows from Lemma 3.1(a). Passing to the limit in (3.25) as k → ∞, we obtain

E^{x}[ũ(X_{t∧τ_{n}})Z_{t∧τ_{n}}] = u(x) + E^{x}[∫_{0}^{t∧τ_{n}} Z_{s}(L^{Q,(n)}u)(X_{s}) ds] (3.29)

for m-a.e. x ∈ E_{n}, provided u ∈ D(L^{Q,(n)}) is bounded. For u ∈ D(L^{Q,(n)}) of the form V_{α}^{(n)}g (0 ≤ g ∈ L^{2}(E_{n}; m)), by the construction of the nest {K_{n}}, there is a sequence {g_{k}} of nonnegative elements of L^{2}(m) such that u_{k} := V_{α}^{(n)}g_{k} is in L^{∞}(E_{n}; m), u_{k} converges in L^{2}(E_{n}; m) to u and L^{Q,(n)}u_{k} = αu_{k}−g_{k} converges to L^{Q,(n)}u in L^{2}(E_{n}; m). [The boundedness of u_{k} comes from the inequality V_{α}^{(n)}f ≤ 1_{K_{n}} · V_{α}f for all non-negative f ∈ L^{2}(m), which is a consequence of the fact that these resolvents are associated with quasi-regular positivity preserving forms. Indeed, the inequality is easy to prove if (V_{α})_{α>α_{0}} is subMarkov, hence the resolvent of a right process; the h-transform technique of section 5 of [34] allows us to reduce the general case to the subMarkov case.] Substituting u_{k} for u in (3.29) and then passing the limit as k → ∞, we see that (3.29) is valid for all u ∈ D(L^{Q,(n)}), since any such u can be written as V_{α}^{(n)}g_{1} −V_{α}^{(n)}g_{2} for non-negative g_{1}, g_{2} ∈ L^{2}(E_{n}; m). That is, we have,

\overline{P} _{t}^{(n)}u(x) = u(x) + ∫_{0}^{t} \overline{P} _{s}^{(n)}(L^{Q,(n)}u)(x) ds, m-a.e. x ∈ E_{n}, (3.30)

for all u ∈ D(L^{Q,(n)}). This implies the strong continuity of \overline{P} _{t}^{(n)} on D(L^{Q,(n)}), hence on L^{2}(E_{n}; m). Note that \overline{P} _{t}^{(n)} maps L^{2}(E_{n}; m) into itself by Lemma 3.1(a), because I_{E_{n}}(μ_{⟨M⟩} + μ_{⟨M̂⟩} + |ν|) is a Kato class smooth measure with respect to X^{E_{n}}. Hence

lim_{t↓0} \frac{\overline{P} _{t}^{(n)}u − u}{t} = L^{Q,(n)}u. (3.31)

Thus, using (\overline{L}^{(n)}, D(\overline{L}^{(n)})) to denote the infinitesimal generator of (\overline{P} _{t}^{(n)})_{t>0}, we have shown

D(L^{Q,(n)}) ⊂ D(\overline{L}^{(n)}) and L^{Q,(n)} = \overline{L}^{(n)} on D(L^{Q,(n)}). (3.32)

Let \overline{U}_{α}^{(n)} := ∫_{0}^{∞} e^{−αt}\overline{P} _{t}^{(n)} dt denote the resolvent operator associated with \overline{L}^{(n)}. Fix f ∈ L^{2}(E_{n}; m) and define, for α > α_{0}, v := V_{α}^{(n)}f. Then v ∈ D(L^{Q,(n)}) ⊂ D(\overline{L}^{(n)}) and \overline{L}^{(n)}v = L^{Q,(n)}v = αv − f. But D(\overline{L}^{(n)}) coincides with \overline{U}_{α}^{(n)}(L^{2}(E_{n}; m)), so there exists g ∈ L^{2}(E_{n}; m) such that v = \overline{U}_{α}^{(n)}g, in which case \overline{L}^{(n)}v = αv − g. It follows that f = g, and then that \overline{U}_{α}^{(n)}f = V_{α}^{(n)}f, for all f ∈ L^{2}(E_{n}; m). This identity of resolvents implies that the associated semigroups (\overline{P} _{t}^{(n)}) and (Q_{t}^{(n)}) coincide.

It remains to let n → ∞. Since lim_{n} τ_{n} = ζ, P^{m}-a.s., dominated convergence tells us that lim_{n} \overline{U}_{α}^{(n)}f(x) = \overline{U}_{α}f(x) for m-a.e. x ∈ E and each f ∈ L^{2}(m). On the other hand, by Lemma 3.6 in [25], lim_{n} V_{α}^{(n)}f = V_{α}f in a norm stronger than that of L^{2}(m) for f in a dense subset of L^{2}(m). It follows that the resolvents of the semigroups (\overline{P}_{t}) and (Q_{t}) coincide, hence so do the semigroups. Our theorem is proved.

4. Proof of Theorem 1.2

The proof of Theorem 1.2 is adapted from [27] but the method is well known; see [9, § 2.2]. By duality it suffices to consider the case p ∈]2, p_{+}]. Fix a non-negative f ∈ L^{p}(m) ∩ L^{2}(m). Then u := \overline{P}_{t}f is an element of D. By considering the nest \{G_{n}\} defined by G_{n} := \{ũ < n\} and the associated part space (Q^{(n)}, D^{(n)}) (as in section 3), we can (and do) assume that u is bounded. Then u ∈ L^{p}(m), and both u^{p−1} and u^{p/2} are elements of D, because |u(y)^{r}−u(x)^{r}| ≤ ∥u∥_{∞}^{r−1}|u(y) − u(x)| and |u(x)^{r}| ≤ ∥u∥_{∞}^{r−1}|u(x)| for r ≥ 1. We now compute

\begin{aligned}\frac{d}{dt} ∫_{E}[e^{−α_{0}(p)t}\overline{P}_{t}f]^{p} dm & = pe^{−α_{0}(p)pt} ∫_{E} [u^{p−1}L^{Q}u − α_{0}(p)u^{p}] dm \\ & = −pe^{−α_{0}(p)pt} [Q(u, u^{p−1}) + α_{0}(p)∥u∥_{p}^{p}], \end{aligned} (4.1)

to see that it suffices to check that

Q(u, u^{p−1}) + α_{0}(p)∥u∥_{p}^{p} ≥ 0. (4.2)

Noting that ∫_{E} u^{p−2} dμ_{⟨u⟩} = 8p^{−2}E(u^{p/2}, u^{p/2}) = 2(p − 1)^{−1}E(u, u^{p−1}), we see that

\begin{aligned}Q(u, u^{p−1}) & = 4(p − 1)p^{−2}E(u^{p/2}, u^{p/2}) − ∫_{E} ũ^{p−1} dμ_{⟨M^{[u]},M⟩} \\ & − ∫_{E} ũ dμ_{⟨M^{[u^{p−1}]},M̂⟩} − ∫_{E} ũ^{p} dν \\ & ≥ (4(pq)^{−1} − δ_{0}(p)) E(u^{p/2}, u^{p/2}) − α_{0}(p)∥u∥_{p}^{p} \\ & ≥ −α_{0}(p)∥u∥_{p}^{p}. \end{aligned} (4.3)

Thus, for non-negative f ∈ L^{p}(m) ∩ L^{2}(m) we have

\frac{d}{dt} ∫_{E} |e^{−α_{0}(p)t}\overline{P}_{t}f|^{p} dm ≤ 0, t ≥ 0,

which implies that ∥e^{−α_{0}(p)t}\overline{P}_{t}f∥_{p} ≤ ∥f∥_{p} for such f. By an obvious approximation argument this inequality extends to all f ∈ L^{p}(m). Thus each \overline{P}_{t} maps L^{p}(m) into itself and (1.16) holds.

For finite p, to prove the strong continuity it is enough, by [51, IX.1], to establish weak continuity; that is, the continuity of t ↦ (g, \overline{P}_{t}f)_{m} for all f ∈ L^{p}(m) and g ∈ L^{q}(m). In view of the norm estimate proved in the preceding paragraph, we need only consider f and g in L^{1}(m) ∩ L^{∞}(m). But L^{1}(m) ∩ L^{∞}(m) ⊂ L^{2}(m), so the required weak continuity follows from the strong continuity of (\overline{P}_{t}) on L^{2}(m) asserted in Theorem 1.1.

5. Examples

Example 5.1 (Sobolev inequality, cf. [30, 44]). Fix d > 2. Assume that a quasi-regular strongly local Dirichlet form (E, D) on L^{2}(m) satisfies the following Sobolev inequality: There is a constant C_{d} ∈]0, ∞[ such that D ⊂ L^{\frac{2d}{d−2}} (m) and for u ∈ D

∥u∥2_{\frac{2d}{d−2}} ≤ C_{d} E(u, u). (5.1)

More concretely, let E be a smooth complete d-dimensional Riemannian manifold (M, g) without boundary, with m the volume element of (M, g). It is known that if the Ricci curvature is bounded below and if the injectivity radius of (M, g) is positive, then the following global Sobolev inequality holds (see [20]): H^{1}(M) ⊂ L^{\frac{2d}{d−2}} (M; m) and for u ∈ H^{1}(M)

∥u∥2_{\frac{2d}{d−2}} ≤ C_{d} ∫_{M} |∇u|_{g}^{2} dm, (5.2)

which is (5.1) for the Dirichlet form E(u, v) := \frac{1}{2} ∫_{E}⟨∇u, ∇v⟩_{g} dm naturally associated with (M, g). The Sobolev inequality (5.1) yields the following estimate (see Theorem 2.4.2 and its proof in [9]): for α ≥ 1, there exists a constant C_{α,d} ∈]0, ∞[ such that

∥P_{t}∥_{α→∞} ≤ C_{α,d}t^{− \frac{d}{2α}} , ∀t > 0, (5.3)

where ∥P_{t}∥_{α→∞} denotes the operator norm of the semigroup (P_{t})_{t>0} from L^{α}(m) to L^{∞}(m). Then, for f ∈ L^{p}(m) with p ≥ d, f^{2}m is of the Hardy class and δ(f^{2}m) can be taken to be arbitrarily small. Indeed, when p > d, f^{2}m is of the Kato class because

\begin{aligned}|⟨f^{2}mU_{α}, h⟩| & ≤ ∫_{0}^{∞} e^{−αt}∥P_{t}f^{2}∥_{∞}∥h∥_{1} dt \\ & ≤ ∫_{0}^{∞} e^{−αt}t^{− \frac{d}{p}} ∥f∥_{p}^{2}∥h∥_{1} dt \\ & ≤ α^{\frac{d}{p} −1} ∫_{0}^{∞} e^{−s}s^{− \frac{d}{p}} ds∥f∥_{p}^{2}∥h∥_{1} for h ∈ L^{1}(m). \end{aligned} (5.4)

When p = d, f^{2}m is not of the Kato class in general; however, for u ∈ D,

\begin{aligned}∫_{E} u^{2}f^{2} dm & = ∫_{\{|f|≤N\}} u^{2}f^{2} dm + ∫_{\{|f|>N\}} u^{2}f^{2} dm \\ & ≤ N^{2}∥u∥_{2}^{2} + ∥I_{\{|f|>N\}}f∥_{d}^{2}∥u∥2_{\frac{2d}{d−2}} \\ & ≤ N^{2}∥u∥_{2}^{2} + C_{d}∥I_{\{|f|>N\}}f∥_{d}^{2}E(u, u), \end{aligned} (5.5)

which implies that δ(f^{2}m) := C_{d}∥I_{{|f|>N}}f∥_{d}^{2} can be made as small as we please by choosing N sufficiently large.

Suppose that (E, D) has the following expression:

E(u, v) = \frac{1}{2} ∫_{E} Γ(u, v) dm for u, v ∈ D; (5.6)

that is, the energy measure μ_{⟨u,v⟩}, u, v ∈ D has a density Γ(u, v) ∈ L^{1}(m). Then, for each M ∈ M_{loc}, the Revuz measure μ_{⟨M⟩} associated with the quadratic variation process ⟨M⟩_{t} has the expression Γ(M)m by use of [37, Lemma 2.2]. Consider M, M̂ ∈ M_{loc} and C_{t} := ∫_{0}^{t} c(X_{s}) ds and suppose that Γ(M)^{1/2} ∈ L^{p}(m), Γ(M̂)^{1/2} ∈ L^{q}(m) and c ∈ L^{r/2}(m) for p, q, r ≥ d. Then μ_{⟨M⟩}, μ_{⟨M̂⟩} and |c|m are in the Hardy class and δ_{0} can be taken to be arbitrarily small. Note that C_{t}^{+} = ∫_{0}^{t} c^{+}(X_{s}) ds, where c^{+} := max{−c, 0} and C^{+} is the PCAF associated with the positive part ν^{+} of the signed smooth measure ν := cm.

Example 5.2 (Defective log-Sobolev inequality, cf. [21, 43]). Assume that m is a Borel probability measure on E and that the quasi-regular strongly local Dirichlet form (E, D) on L^{2}(m) satisfies the defective log-Sobolev inequality: there exist constants λ_{1}, λ_{2} > 0 such that

∫_{X} |u(x)|^{2} log(|u(x)|/∥u∥_{2}) m(dx) ≤ λ_{1}E(u, u) + λ_{2}∥u∥_{2}^{2}, ∀u ∈ D.

We assume (5.6) and take M, M̂ ∈ M_{loc} and C to be as in the preceding paragraph. Suppose that there exist p_{1}, p_{2}, q_{1}, q_{2} ∈ [1, ∞[, c_{1} ∈]0, ∞[ such that e^{c_{1}p_{1}Γ(M)}, e^{c_{1}p_{2}Γ(M̂)}, e^{q_{1}c^{+}}, e^{q_{2}c^{−}} ∈ L^{1}(m) and

\sqrt{\frac{2λ_{1}}{c_{1}p_{1}}} + \sqrt{\frac{2λ_{1}}{c_{1}p_{2}}} + \frac{2λ_{1}}{q_{1}} < 1. (5.7)

Then μ_{⟨M⟩}, μ_{⟨M̂⟩} and |c|m are of the Hardy class with δ_{0} = \sqrt{\frac{2λ_{1}}{c_{1}p_{1}}} + \sqrt{\frac{2λ_{1}}{c_{1}p_{2}}} + \frac{2λ_{1}}{q_{1}} by virtue of Young’s inequality: st ≤ s log s − s + e^{t}, s ≥ 0, t ∈ ℝ, and the defective log-Sobolev inequality. Hence we recover the main result of [43] under the milder assumption (5.7), because δ_{0} ≤ 2λ_{1}(\frac{1}{c_{1}p_{1}} + \frac{1}{c_{1}p_{2}} + \frac{1}{q_{1}} ) + \frac{1}{2} and the latter quantity was assumed in [43] to be less than 1.

Example 5.3 (Feynman-Kac formula for CAF locally of zero energy, cf. [7]). Let (E, D) be a quasi-regular strongly local Dirichlet form on L^{2}(m) as in section 1. Let M = M̂ = −\frac{1}{2}M^{[ρ]} = −M^{[ρ/2]} for some ρ ∈ D_{loc} and let C = 0. Suppose that μ_{⟨ρ⟩} ∈ S_{H} with δ(μ_{⟨ρ⟩}) < 1/2. Then assumption (1.2) is satisfied: δ_{0} = 2\sqrt{2δ(−M^{[ρ/2]})} = 2\sqrt{2^{−1}δ(μ_{⟨ρ⟩})} < 2\sqrt{4^{−1}} = 1. Since N^{[ρ]} is the even part of −M^{[ρ]}, we have Z_{t} = exp[N_{t}^{[ρ]}], P^{m}-a.s. on {t < ζ}, hence Q_{t}f(x) = E^{x}[exp(N_{t}^{[ρ]})f(X_{t})], m-a.e. x ∈ E for f ∈ L^{2}(m). When ρ ∈ D_{e}, (Q, D) has the following expression:

Q(u, v) = E(u, v) + E(uv, ρ) for u, v ∈ bD. (5.8)

Here D_{e} is the extended Dirichlet space of (E, D); see p. 36 in [17].

Example 5.4 (Hardy inequality in upper half plane). Let ℝ_{+}^{d} := {x = (x̄, x_{d}) ∈ ℝ^{d} | x̄ ∈ ℝ^{d−1}, x_{d} > 0} be the upper half plane in ℝ^{d} (d ≥ 2). We consider the classical Dirichlet form (E, D) := (\frac{1}{2}D, H_{0}^{1}(ℝ_{+}^{d})) on L^{2}(ℝ_{+}^{d}) associated with the Brownian motion (Ω, B_{t}^{ℝ_{+}^{d}}, P^{x}) on ℝ_{+}^{d} with absorbing boundary ℝ_{0}^{d} := {x = (x̄, x_{d}) ∈ ℝ^{d} | x̄ ∈ ℝ^{d−1}, x_{d} = 0}.

Then the following Hardy inequality holds (see [36, Section 2.1.6], [10, Thm. 6]):

∫_{ℝ_{+}^{d}} \frac{u(x)^{2}}{|x_{d}|^{2}} dx ≤ 4 ∫_{ℝ_{+}^{d}} |∇u(x)|^{2} dx for all u ∈ H_{0}^{1}(ℝ_{+}^{d}). (5.9)

Fix ε > 0 and let b(x) := (4\sqrt{2}(2+ε)x_{d})^{−1}e_{1}, b̂(x) := −(4\sqrt{2}(2+ε)x_{d})^{−1}e_{d}, and c(x) := (2(4+ ε))^{−1}x_{d}^{−2}, where e_{1} := (1, 0, ⋯, 0), e_{d} := (0, ⋯, 0, 1). We consider M_{t} := ∫_{0}^{t∧τ^{+}} b(B_{s}) dB_{s}, M̂_{t} := ∫_{0}^{t∧τ^{+}} b̂(B_{s}) dB_{s} and C_{t} := ∫_{0}^{t∧τ^{+}} c(B_{s}) ds, where τ^{+} := τ(ℝ_{+}^{d}) is the first exit time of d-dimensional Brownian motion (Ω, B_{t}, P^{x}) from ℝ_{+}^{d}. Then our assumptions are satisfied with δ_{0} = 1/(2 + ε) + 2/(4 + ε) < 1. Note |b|, |b̂|, c ∉ L^{p}(ℝ_{+}^{d}) for any p > 0.

Example 5.5 (cf. [18],[24],[6], [40], [35], (4.2) in [14]). When d ≥ 3, the following Hardy inequality holds: for any r > 0

∫_{B_{r}(0)} \frac{u(x)^{2}}{|x|^{2}} dx ≤ \frac{4}{(d − 2)^{2}} ∫_{B_{r}(0)} |∇u(x)|^{2} dx for u ∈ H_{0}^{1}(B_{r}(0)). (5.10)

Hence

∫_{ℝ^{d}} \frac{u(x)^{2}}{|x|^{2}} dx ≤ \frac{4}{(d − 2)^{2}} ∫_{ℝ^{d}} |∇u(x)|^{2} dx for u ∈ H^{1}(ℝ^{d}). (5.11)

Take φ ∈ C^{∞}([0, ∞[) with 0 < φ ≤ 1, φ ≡ 1 on [0, 1/2], and φ(s) ≤ 1/s^{d} for s ∈ [1, ∞[. Let b(x) = b̂(x) := \frac{d−2}{8\sqrt{2}+ε} \frac{φ(|x|)}{|x|^{2}}x and c(x) := (\frac{d−2}{2\sqrt{2}+ε})^{2} \frac{φ(|x|)^{2}}{|x|^{2}} for ε > 0. We consider M_{t} := ∫_{0}^{t} b(B_{s}) dB_{s}, M̂_{t} := ∫_{0}^{t} b̂(B_{s}) dB_{s} and C_{t} := ∫_{0}^{t} c(B_{s}) ds. Then |b|^{2}dx, |b̂|^{2}dx and |c|dx are of the Hardy class with respect to the Dirichlet form associated with d-dimensional Brownian motion X := (Ω, B_{t}, P^{x}) and δ_{0} = 4\sqrt{2}/(8\sqrt{2} + ε) + 4/(2\sqrt{2} + ε)^{2} < 1. Note that |b|, |b̂|, |c|^{1/2} ∉ L^{p}(ℝ^{d}) for p ≥ d, but |b|, |b̂|, |c|^{1/2} ∈ L^{p}(ℝ^{d}) for d/(d + 1) < p < d and |b|^{2}, |b̂|^{2}, |c| are not of the Kato class when the support of φ is bounded, by Theorem 1.3(viii) in [1].

Example 5.6 (Low dimensional case). Fix α ≥ 1 and r ≥ 1 with r^{α+1} ≥ α. Let ψ(x) := 1/|x|^{α}, x ∈ ℝ^{d}. Then |∇ψ| ≤ 1 and \frac{1}{2}Δψ ≤ α(α + 2 − d)/2ψ on G := \{y ∈ ℝ^{d} | |y| > r\}. We assume d = 1 if α = 1, or d = 1, 2 if α > 1. Then α(α + 2 − d)/2 > 1/2. By (4.2) in [14] we have the following Hardy inequality:

∫_{G} u(x)^{2}|x|^{2α} dx ≤ \frac{4}{α^{2}(α + 1 − d)^{2}} ∫_{G} |∇u(x)|^{2} dx for u ∈ H_{0}^{1}(G). (5.12)

On the other hand, the smooth measure |x|^{2α}dx is not of the Kato class with respect to the Dirichlet form corresponding to the Brownian motion (Ω, B_{t}^{G}, P^{x}) on G with absorbing boundary ∂B_{r}(0). Indeed, consider

sup_{x∈G} ∫_{\{|x−y|< \frac{r}{2} ,r≤d(x,G^{c})∧d(y,G^{c})\}} |y|^{2α} dy = ∞, (5.13)

where d(x, G^{c}) := inf_{z∈G^{c}} |x − z|. Let p_{t}^{G}(x, y) on (0, ∞) × G × G be the heat kernel for (B_{t}^{G}), defined by

p_{t}^{G}(x, y) := p_{t}(x, y) − E^{x}[p_{t−τ(G)}(B_{τ(G)}, y)I_{{τ(G)<t}}], (5.14)

where X = (Ω, B_{t}, P^{x}) is the d-dimensional Brownian motion, p_{t}(x, y) := \frac{1}{(2πt)^{d/2}} exp(−\frac{|x−y|^{2}}{2t}), x, y ∈ ℝ^{d}, is the heat kernel of X, and τ(G) is the first exit time of X from G. Then by [8, Proof of Thm. 2.4], for 0 < s < 1/d, |x − y| < r ≤ d(x, G^{c}) ∧ d(y, G^{c}),

p_{s}^{G}(x, y) ≥ \frac{1}{(2πs)^{d/2}} (e^{− \frac{|x−y|^{2}}{2s}} − e^{− \frac{r^{2}}{2s}}). (5.15)

Hence, for t/2 < s < t < 1/d, |x − y| < r/2 and r ≤ d(x, G^{c}) ∧ d(y, G^{c}),

p_{s}^{G}(x, y) ≥ \frac{1}{(4πt)^{d/2}} (e^{− \frac{r^{2}}{4t}} − e^{− \frac{r^{2}}{2t}}) =: C_{t} > 0. (5.16)

Therefore, for t < 1/d

\frac{t}{2C_{t}} sup_{x∈G} ∫_{\{|x−y|< \frac{r}{2} ,r≤d(x,G^{c})∧d(y,G^{c})\}} |y|^{2α} dy

≤ sup_{x∈G} ∫_{\{|x−y|< \frac{r}{2} ,r≤d(x,G^{c})∧d(y,G^{c})\}}(∫_{t/2}^{t} p_{s}^{G}(x, y)ds)|y|^{2α} dy

≤ sup_{x∈G} E^{x}[∫_{0}^{t} |B_{s}^{G}|^{2α} ds],

(5.17)

which implies that |x|^{2α}dx is not of the Kato class (or even of the Dynkin class). However, if b(x) := b̂(x) := α(α + 1 − d)x|x|^{α−1}/(8\sqrt{2} + ε) and c(x) := α^{2}(α + 1 − d)^{2}|x|^{2α}/(8 + ε), then our assumptions are satisfied with δ_{0} = 4\sqrt{2}/(8\sqrt{2} + ε) + 4/(8 + ε) < 1.

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P.J. Fitzsimmons — Department of Mathematics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0112, USA

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

K. Kuwae — Department of Mathematical Sciences, Yokohama City University, Yokohama 236-0027, JAPAN

   E-mail address: kuwae@yokohama-cu.ac.jp