ABSOLUTE CONTINUITY OF SYMMETRIC MARKOV PROCESSES

Z.-Q. Chen, P.J. Fitzsimmons, M. Takeda, J. Ying and T.-S. Zhang

Department of Mathematics University of Washington Seattle, WA 98195, USA.

                        Email: zchen@math.washington.edu

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

                          Email: pfitz@euclid.ucsd.edu

Mathematical Institute Tohoku University Sendai 980-8578, Japan

                         Email: takeda@math.tohoku.ac.jp

Institute of Mathematics Fudan University Shanghai 200433, China

                            Email: jying@fudan.edu.cn

Department of Mathematics University of Manchester Oxford Road, Manchester M13 9PL England, U.K.

                          Email: tzhang@maths.man.ac.uk

Abstract. We study Girsanov’s theorem in the context of symmetric Markov processes, extending earlier work of Fukushima-Takeda and Fitzsimmons on Girsanov transformations of “gradient type”. We investigate the most general Girsanov transformation leading to another symmetric Markov process. This investigation requires an extension of the forward-backward martingale method of Lyons-Zheng, to cover the case of processes with jumps.

(July 25, 2003)

1991 Mathematics Subject Classification. Primary 31C25, 60J45, secondary 60J57. Key words and phrases. absolute continuity, symmetric Markov process, Dirichlet form, forward-backward martingale de-

composition, Girsanov theorem, dual predictable projection, supermartingale multiplicative functional.

The research of the first-named author was supported, in part, by NSF Grant DMS-0071486, that of the last-named author by the Norwegian Research Council.

Typeset by AMS-TEX

1. Introduction and Preliminaries.

Our aim in this work is to study change-of-measure phenomena (“Girsanov” theorems) for general symmetric Markov processes. Our results extend both the earlier work of Fukushima and Takeda [9], and that of Fitzsimmons [5] (who was concerned only with symmetric diffusions). Our work also contains Theorem 2.7 of [12] as a special case.

Before setting down the precise context in which we shall be working, let us briefly describe our results. Let X = (X_{t}) be a symmetric (i.e., reversible) Markov process, with symmetry measure m, state space E, and distribution ℙ_{x} when started in state x ∈ E. Given a strictly positive element ρ of the Dirichlet space of X, let M^{ρ} be the martingale part in the Fukushima decomposition of ρ(X_{t}) − ρ(X_{0}), define a local martingale M by the formula M_{t} := ∫_{0}^{t}[ρ(X_{s−})]^{−1} dM_{s}^{ρ}, and let Z^{ρ} be the “stochastic exponential” of M; that is, Z^{ρ} is the unique solution of Z_{t}^{ρ} = 1 + ∫_{0}^{t} Z_{s−}^{ρ} dM_{s}. Then Z^{ρ} is a positive supermartingale, and so determines a new family (ℙ̂_{x})_{x∈E} of probability measures governing a second symmetric Markov process X̂ on E, with symmetry measure μ(dx) := ρ(x)^{2} m(dx).

Suppose, for example, that E = ℝ^{d} and that the (non-positive definite) infinitesimal generator L of X has the form

Lf(x) = L^{c}f(x) + ∫_{ℝ^{d}}[f(y) − f(x)] N(x, dy) − k(x)f(x),  x ∈ ℝ^{d},  (1.1)

where L^{c} is a second-order differential operator with L^{c}1 = 0, N is a kernel on ℝ^{d}, and k ≥ 0. Suppose also that D(L), the domain of L, is closed under the formation of products, and that ρ ∈ D(L) is strictly positive. Then the generator L̂ of the transformed process X̂ is given by

\begin{aligned}L̂f(x) & = ρ(x)^{−1} [L(ρf)(x) − f(x)Lρ(x)] \\ & = L^{c}f(x) + B_{ρ}f(x) + ∫_{ℝ^{d}}[f(y) − f(x)] N̂(x, dy), \end{aligned}  (1.2)

where B_{ρ}f := ρ^{−1}[L^{c}(ρf) − fL^{c}ρ − ρL^{c}f] is a first-order differential operator and N̂(x, dy) = [ρ(y)/ρ(x)]N(x, dy). It is an important but very challenging problem to characterize the domain D(L̂) of L̂. An equivalent, but more tractable, way to proceed is to characterize the bilinear Dirichlet form associated with L̂. Let E(f, g) := − ∫ Lf · g dm and Ê(f, g) := − ∫ L̂f · g dμ denote the Dirichlet forms corresponding to X and X̂. When expressed in terms of these bilinear forms, (1.1) and (1.2) become

E(f, g) = E^{c}(f, g) + ∫_{ℝ^{d}} ∫_{ℝ^{d}}[f(y) − f(x)] · [g(y) − g(x)] J(dx, dy) + ∫_{ℝ^{d}} f(x)g(x)k(x) m(dx),  (1.3)

and

Ê(f, g) = E_{ρ}^{c}(f, g) + ∫_{ℝ^{d}} ∫_{ℝ^{d}}[f(y) − f(x)] · [g(y) − g(x)] Ĵ(dx, dy),  (1.4)

where E^{c}(f, g) := − ∫ L^{c}f · g dm, E_{ρ}^{c}(f, g) := − ∫ [L^{c} + B_{ρ}]f · g dμ, J(dx, dy) := \frac{1}{2}m(dx)N(x, dy), and Ĵ(dx, dy) := \frac{1}{2}μ(dx)N̂(x, dy) = ρ(x)ρ(y)J(dx, dy). In the general context in which we shall be working, Dirichlet forms are more convenient objects than their associated infinitesimal generators, and formulas like (1.4) will be the main focus of our study.

The following is concrete instance in which (1.1) holds.

Example 1.1. Let E = ℝ^{d}, m(dx) = dx (Lebesgue measure), and let X be the Lévy process on ℝ^{d} that is the sum of Brownian motion on ℝ^{d} and an independent rotationally symmetric α-stable process on ℝ^{d}, for some 0 < α < 2 and d ≥ 3. Clearly X is an m-symmetric Hunt process. Its Dirichlet form (E, F) is given by

F = {f ∈ L^{2}(ℝ^{d}, dx) : ∇f ∈ L^{2}(ℝ^{d}, dx)}

and

E(f, g) = \frac{1}{2} ∫_{ℝ^{d}} ∇f(x) · ∇g(x) dx + ∫_{ℝ^{d}} ∫_{ℝ^{d}}[f(y) − f(x)] · [g(y) − g(x)]\frac{c(d, α)}{|x − y|^{d+α}} dx dy,  (1.5)

where c(d, α) > 0 is a constant depending only on d and α. In this case we have, in (1.1),

L^{c} = \frac{1}{2}Δ,  N(x, dy) = \frac{c(d, α)}{|x − y|^{d+α}} dy,  k(x) ≡ 0.  (1.6)

In the case at hand, (1.4) becomes

Ê(f, g) = \frac{1}{2} ∫_{ℝ^{d}} ∇f(x) · ∇g(x) ρ(x)^{2} dx + ∫_{ℝ^{d}} ∫_{ℝ^{d}}[f(y) − f(x)] · [g(y) − g(x)]\frac{c(d, α)ρ(x)ρ(y)}{|x − y|^{d+α}} dy. (1.7)
□

The type of change of measure considered above was studied in [9], for ρ the α-potential of a bounded strictly positive function on E. In section 2, we extend and complete the work of Fukushima and Takeda by obtaining a complete description of the Dirichlet form associated with X̂ for an arbitrary strictly positive ρ in the Dirichlet space of X. Our method is a modification of that found in the work of Chen and Zhang [3]. Of critical importance is Lemma 2.2 which extends the Lyons-Zheng forward-backward martingale decomposition to the context of symmetric Markov processes with jumps.

When X is a diffusion, the change of measure determined by Z^{ρ} is (modulo localization) the most general Girsanov transformation leading to another symmetric diffusion. This assertion is the principal result of [5]. The situation is more complex when X has jumps. In sections 3 and 4 we investigate the most general change of measure leading to a second symmetric Markov process X̂, and we take the first steps in describing the associated Dirichlet form. A formula like (1.4) holds even in this more general situation, although a zeroorder term might be present, and the density ρ(x)ρ(y) linking the measures Ĵ and J must be replaced by a more general symmetric function of (x, y). Our results here are somewhat less comprehensive than those of section 2 or of [5], the main unresolved difficulty being the description of a core for the Dirichlet space of X̂. Nevertheless, we find an explicit expression for the Dirichlet form of X̂ that is valid for a large class of functions in the Dirichlet space. It may be helpful for the reader to keep in mind the concrete Example 1.1 above when considering the general results of this paper.

In the remainder of this section we establish our setting and notation. Let E be a topological space that is homeomorphic to a co-analytic subset of a compact metric space (a “metric co-Souslin space”), with Borel σ-algebra B(E). Let m be a σ-finite measure on B(E) with supp[m] = E. We denote by B(E × E) the product σ-algebra on E × E. Let X = (Ω, M, M_{t}, θ_{t}, X_{t}, ℙ_{x}) be a Borel right Markov process with state space E, lifetime ζ, transition semigroup (P_{t})_{t>0} and resolvent (U^{q})_{q>0}. In more detail, the right-continuous process [0, +∞[∋ t ↦ X_{t} is defined on the sample space (Ω, M), with its minimal (augmented) admissible filtration {M_{t}}_{t≥0}, and under the law ℙ_{x} is a strong Markov process with initial condition X_{0} = x. The shift operators θ_{t}, t ≥ 0, satisfy X_{s} ∘ θ_{t} = X_{s+t} identically for s, t ≥ 0. Adjoined to the state space E is an isolated point Δ ∉ E; the process X retires to Δ at its “lifetime” ζ := inf{t : X_{t} = Δ}. Throughout this paper we assume that X is m-symmetric; more precisely (P_{t}) may be extended into a symmetric operator semigroup on L^{2}(m); i.e.,

(f, P_{t}g) = (P_{t}f, g), f, g ∈ L^{2}(m),

where (u, v) := ∫_{E} u(x)v(x) m(dx) is the natural inner product in L^{2}(m) := L^{2}(E; m). By the theory of Dirichlet forms, there exists a symmetric Dirichlet form (E, F) associated with X:

\begin{aligned}F & = \{u ∈ L^{2}(m) : sup_{t>0} \frac{1}{t}(u − P_{t}u, u) < ∞\}, \\ E(u, v) & = lim_{t↓0} \frac{1}{t}(u − P_{t}u, v), u, v ∈ F. \end{aligned}

For any q > 0, set

E_{q}(u, v) := E(u, v) + q(u, v), u, v ∈ F.

Then F becomes a Hilbert space with inner product E_{q} for any q > 0. We call the corresponding norm the E_{q}-norm. In view of the work in [6] and [19], the Dirichlet form (E, F) is quasi-regular. Thus, by Theorem 3.7 of [2], the process is quasi-homeomorphic to the Hunt process associated with a regular Dirichlet form on a locally compact separable metric space, so all of the results of [8] (established there for regular Dirichlet forms) apply to X and its Dirichlet form. For the reader uninterested in applications to infinite-dimensional settings, it is safe to assume from now on that X is the Hunt process on a locally compact metric state space (such as a Euclidean domain) associated with a regular Dirichlet form. See Chapter VI of [19] for more on the “transfer method,” by which the quasi-regular case is reduced to the regular case.

Let A = (A_{t}) be any increasing additive functional (AF), by which we mean that besides additivity and right continuity we also assume that 0 ≤ A_{t} < ∞ for t < ζ. We can (and do) take its perfected version. We use f ∗ A to denote the functional

(f ∗ A)_{t} := ∫_{0}^{t} f(X_{s})dA_{s},

if f is a Borel function on E, and

(F ∗ A)_{t} := ∫_{0}^{t} F (X_{s−}, X_{s})dA_{s},

if F is a Borel function on E × E. The bivariate Revuz measure ν_{A} of A (computed with respect to X and m) is defined for F ∈ pB(E × E) by

ν_{A}(F) :=↑ lim_{t↓0} \frac{1}{t}𝔼_{m}(F ∗ A)_{t}.

The usual Revuz measure μ_{A} of A is nothing but the second marginal measure of ν_{A}, namely, μ_{A}(dx) = ν_{A}(1 ⊗ dx). The mapping A ↦ μ_{A} establishes a one-to-one correspondence between the class of positive continuous additive functionals (PCAFs) of X and the class of smooth measures of (E, F), and is usually known as the Revuz correspondence.

A well-known consequence of symmetry is that for q.e. x ∈ E,

ℙ_{x}({ω ∈ Ω : X_{t−}(ω) exists in E for all t < ζ}) = 1.  (1.5)

Without any real loss of generality, we assume the exceptional set (of those x ∈ E for which (1.5) fails) to be empty. Adjoin the cemetery Δ to E as an isolated point to form E_{Δ}; the Borel σ-field on E_{Δ} is denoted B(E_{Δ}). The jump behavior of X is described by a pair (N, H), the Lévy system of X, in which N is a kernel from (E, B(E)) to (E_{Δ}, B(E_{Δ})) satisfying N(x, {x}) = 0 for any x ∈ E, and H a PCAF of X with bounded 1-potential, such that for any measurable function F ∈ pB(E × E), the dual predictable projection (or compensator) of the homogeneous random measure

η(ω, dt) := ∑_{s>0} F (X_{s−}(ω), X_{s}(ω))1_{{X_{s−}(ω)≠X_{s}(ω)}}ε_{s}(dt)

is NF ∗ H, where NF (x) := ∫_{E_{Δ}} N(x, dy)F (x, y). [Here ε_{s} is the unit point mass at s.] The special case (1.1) occurs when H_{t} ≡ t. Set J(dx, dy) := \frac{1}{2}1_{E}(y) μ_{H}(dx)N(x, dy), the jump measure for X. Note that J is invariant with respect to the mapping (x, y) → (y, x).

We will use ζ_{p} and ζ_{i} to denote, respectively, the predictable and totally inaccessible parts of the lifetime ζ. Let κ(dx) := N(x, {Δ}) μ_{H}(dx) be the Revuz measure of the compensator A^{κ} of the additive functional 1_{[[ζ_{i},∞} ; κ is the killing measure for X.

Each u ∈ F admits a quasi-continuous m-version ũ for which the process t ↦ ũ(X_{t}) is right-continuous on [0, ∞[ with left limits (equal to ũ(X_{t−}) for t ≠ ζ_{p}) on ]0, ∞[, ℙ_{x}-a.s. for q.e. x ∈ E. For such u ∈ F we have Fukushima’s decomposition [8; Theorem 5.2.2]

ũ(X_{t}) − ũ(X_{0}) = M_{t}^{u} + N_{t}^{u},

where M^{u} is a martingale additive functional (MAF) of finite energy and N^{u} is a continuous additive functional (CAF) of zero energy. Moreover M^{u} may be further decomposed into the sum of a continuous part and a purely discontinuous part

M^{u} = M^{u,c} + M^{u,d}.

Let E^{c}(u, u) := \frac{1}{2}μ_{⟨M^{u,c}⟩}(E). Then E admits a Beurling-Deny decomposition as in [8; Theorem 5.3.1]:

E(u, u) = E^{c}(u, u) + ∫_{E×E}(u(y) − u(x))^{2}J(dx, dy) + κ(u^{2}), u ∈ F,

where the “diffusion” piece E^{c} is strongly local in the sense that E(u, v) = 0 whenever u, v ∈ F and u is constant m-a.e. on a neighborhood of the support of the measure |v| · m. Here and in the sequel, we always take u ∈ F to be represented by its quasi-continuous version, and we usually drop the “tilde” from our notation.

Notations and Conventions: The notation ‘:=’ should be read ‘is defined to be’. For a class F of functions, we use bF (resp. pF (or F^{+})) to denote the set of bounded (resp. nonnegative) functions in F. We use both ‘nonnegative’ and ‘positive’ to mean ≥ 0, and ‘strictly positive’ to mean > 0. For a measure μ and a function f, μ(f) := ∫ fdμ. We sometimes write L^{p} or L^{p}(m) for L^{p}(E, m), and (·, ·) for the inner product in L^{2}(m). For f, g ∈ B(E), f ⊗ g(x, y) := f(x)g(y) and f̄(x, y) := f(y) − f(x) for x, y ∈ E. For a right continuous process H_{t} of finite variation on bounded intervals, we use H^{p} to denote its dual predictable projection and H to denote H − H^{p}, all computed with respect to (X, ℙ_{x}, x ∈ E). The jump M_{t} − M_{t−} will be abbreviated as ΔM_{t}. The first hitting time of a set G is denoted T_{G} := inf{t > 0 : X_{t} ∈ G}. A hitting time is an example of a terminal time, which is a stopping time T such that t + T^{∘}θ_{t} = T on {t < T }.

Acknowledgment: We gratefully acknowledge the comments of an anonymous referee, whose careful reading of the manuscript helped to improve the exposition of this paper

2. Girsanov Transform by Multiplicative Functional Related to M^{log ρ}.

In this section, we study Girsanov transforms of a type investigated earlier by Fukushima and Takeda [9]; our results extend and complete the work found there. Our method is a modification of that found in Chen and Zhang [3].

Throughout this section, ρ is a nonnegative element of F. We can (and do) assume that ρ is quasicontinuous, and we assume that ρ > 0 q.e. on E. (Otherwise, we would deal with the part process X killed upon leaving {x : ρ(x) > 0}.) We will use the convention that any function defined on E is extended to be zero at the cemetery point Δ; in particular ρ(Δ) = 0. By Fukushima’s decomposition,

ρ(X_{t}) − ρ(X_{0}) = M_{t}^{ρ} + N_{t}^{ρ}, ℙ_{x}-a.s. for q.e. x ∈ E,

where M^{ρ} is a square-integrable martingale AF and N^{ρ} is a CAF of zero energy. Note that s ↦ ρ(X_{s∧ζ_{i}−}) is strictly positive and left continuous on ]0, ζ_{p}[. Define a local martingale M on the random time interval [[0, ζ_{p} by

M_{t} = ∫_{0}^{t∧ζ_{i}} \frac{1}{ρ(X_{s−})}dM_{s}^{ρ}  0 ≤ t < ζ_{p}.  (2.1)

Note that for t < ζ_{p},

\begin{aligned}ΔM_{t} & = \frac{1}{ρ(X_{t∧ζ_{i}−})} (M_{t∧ζ_{i}}^{ρ} − M_{t∧ζ_{i}−}^{ρ}) = \frac{1}{ρ(X_{t∧ζ_{i}−})} (ρ(X_{t∧ζ_{i}}) − ρ(X_{t∧ζ_{i}−})) \\ & = \frac{ρ(X_{t∧ζ_{i}})}{ρ(X_{t∧ζ_{i}−})} − 1. \end{aligned}

The solution Z_{t}^{ρ} of the SDE

Z_{t}^{ρ} = 1 + ∫_{0}^{t} Z_{s−}^{ρ}dM_{s},  0 ≤ t < ζ_{p},  (2.2)

is a positive local martingale on the random time interval [[0, ζ_{p} , hence a positive supermartingale. Consequently, the formula

dℙ̂_{x} = Z_{t}^{ρ} dℙ_{x} on M_{t} ∩ {t < ζ_{p}} for x ∈ E,

uniquely determines a family of probability measures on (Ω, M_{∞}). It is known that under these new measures, X is a right Markov process on E; see [24; §62]. We will use (X̂, M, M_{t}, ℙ̂_{x}, x ∈ E) to denote the transformed process. Here X̂_{t}(ω) = X_{t}(ω) but we use X̂_{t} for emphasis when working with ℙ̂_{x}. Let P̂_{t} be the semigroup of X̂. That is,

P̂_{t}f(x) = 𝔼̂_{x}[f(X̂_{t})] = 𝔼_{x} [Z_{t}^{ρ}f(X_{t})] = 𝔼_{x} [Z_{t}^{ρ}f(X_{t}); t < ζ] .

[These transition operators need not preserve Borel measurability; this minor inconvenience can be dealt with as in Cor. 3.23 of [6].]

Before stating the next result, let us recall the definition of time-reversal operator r_{t} on the path space. Given a path ω ∈ {t < ζ}, define

r_{t}(ω)(s) := \{ \begin{matrix}ω((t − s)−) & for 0 ≤ s < t, \\ ω(0) & for s ≥ t. \end{matrix}

Here for r > 0, ω(r−) := lim_{s↑r} ω(s). It is known (see Lemma 4.1.2 of [8]) that the mapping r_{t} preserves the measure ℙ_{m} on M_{t} ∩ {t < ζ}.

Definition 2.1. (1) A continuous additive functional A_{t} is called even if A_{t} ∘ r_{t} = A_{t} for all t < ζ.

(2) An m-measurable function u : E → ℝ is locally in F (u ∈ F^{•}_{loc}) provided there is a nest {G_{n}} of finely open sets and a sequence {u_{n}} ⊂ F such that u = u_{n}, m-a.e. on G_{n} for each n.

We recall from [4; Thm. 2.1] that a CAF with paths locally of bounded variation (or merely of zero quadratic variation) is necessarily even.

Clearly each u ∈ F^{•}_{loc} has a quasi-continuous version, and for such u the continuous local martingale AF M^{u,c} is well defined by

M_{t}^{u,c} := M_{t}^{u_{n},c} for t ≤ T_{E_{Δ}\G_{n}}, n = 1, 2, ….  (2.3)

The following can be regarded as an extension to functions in F^{•}_{loc} of both Fukushima’s decomposition and the Lyons-Zheng forward-backward martingale decomposition.

Lemma 2.2. For u ∈ F^{•}_{loc} and t > 0, ℙ_{m}-a.s. on {t < ζ},

u(X_{t}) − u(X_{0}) = \frac{1}{2} (M_{t}^{u,c} − M_{t}^{u,c}^{∘}r_{t}) + lim_{ε↓0} ∑_{0<s≤t} (u(X_{s}) − u(X_{s−})) 1_{\{|u(X_{s})−u(X_{s−})|>ε\}}

The limit above exists in the sense of convergence in probability under ℙ_{x}, for m-a.e. x ∈ E.

Proof. Note that when u ∈ F, the martingale part M_{t}^{u} in Fukushima’s decomposition can be decomposed as

M_{t}^{u} = M_{t}^{u,c} + M_{t}^{u,j} + M_{t}^{u,k},

where M_{t}^{u,c} is the continuous part of martingale M^{u}, and

\begin{aligned}M_{t}^{u,j} & = lim_{ε↓0} \{ ∑_{0<s≤t}(u(X_{s}) − u(X_{s−}))1_{\{|u(X_{s})−u(X_{s−})|>ε\}}1_{\{s<ζ\}} \\ & − ∫_{0}^{t} (∫_{\{y∈E: |u(y)−u(X_{s})|>ε\}}(u(y) − u(X_{s}))N(X_{s}, dy)) dH_{s}\} , \\ M_{t}^{u,k} & = ∫_{0}^{t} u(X_{s})N(X_{s}, Δ)dH_{s} − u(X_{ζ−})1_{\{t≥ζ_{i}\}}, \\ & = ∫_{0}^{t} u(X_{s}) dA_{s}^{κ} − u(X_{ζ−})1_{\{t≥ζ_{i}\}}, \end{aligned}  (2.4)

are the jump and killing parts M^{u}, respectively. See [8; Theorem A.3.9.]. The limit in the expression for M^{u,j} is in the sense of convergence in the norm of the space of square-integrable martingales and convergence in probability under ℙ_{x} for m-a.e. x ∈ E (see [8]). So ℙ_{m}-a.s. on {t < ζ} it follows that

\begin{aligned}u(X_{t}) − u(X_{0}) & = \frac{1}{2}(M_{t}^{u} − M_{t}^{u}^{∘}r_{t}) \\ & = \frac{1}{2}(M_{t}^{u,c} − M_{t}^{u,c}^{∘}r_{t}) + lim_{ε↓0} ∑_{0<s≤t}(u(X_{s}) − u(X_{s−}))1_{\{|u(X_{s})−u(X_{s−})|>ε\}}. \end{aligned}

For u ∈ F^{•}_{loc}, let {G_{n}} be a nest of finely open sets and {u_{n}} a sequence of functions in F such that u = u_{n} q.e. on G_{n}. For each u_{n} ∈ F, ℙ_{m}-a.s. on {t < ζ} we have

\begin{aligned}u_{n}(X_{t}) − u_{n}(X_{0}) & = \frac{1}{2}(M_{t}^{u_{n},c} − M_{t}^{u_{n},c}^{∘}r_{t}) \\ & + lim_{ε↓0} ∑_{0<s≤t}(u_{n}(X_{s}) − u_{n}(X_{s−}))1_{\{|u_{n}(X_{s})−u_{n}(X_{s−})|>ε\}}. \end{aligned}

As [0, t] ∋ s ↦ X_{s} is right continuous with left limits in E on {t < ζ}, the lemma now follows from the above display, using (2.3) to pass to the limit as n → ∞. □

Remark 2.3. By [8; Theorem A.3.9], we can also replace the indicator 1_{{|u(X_{s})−u(X_{s−})|>ε}} by 1_{{|e^{u(Xs)}−e^{u(X_{s−})}|>ε}}, and the set {y ∈ E : |u(y) − u(X_{s})| > ε} by {y ∈ E : |e^{u(y)} − e^{u(X_{s})}| > ε}. Thus Lemma 2.2 can also take the following form: For u ∈ F^{•}_{loc},

u(X_{t}) − u(X_{0}) = \frac{1}{2} (M_{t}^{u,c} − M_{t}^{u,c}^{∘}r_{t})

+ lim_{ε↓0} ∑_{0<s≤t} (u(X_{s}) − u(X_{s−})) 1_{{|e^{u(Xs)}−e^{u(X_{s−})}|>ε}} ℙ_{m}-a.s. on {t < ζ}.

The convergence in the above expression is in the sense of convergence in probability under each ℙ_{x} for m-a.e. x ∈ E.

Lemma 2.4. P̂_{t} is symmetric on L^{2}(E, ρ^{2}m).

Proof. Let f, g ∈ bB^{+}(E). By time reversal, we have

\begin{matrix}(P̂_{t}f, g)_{ρ^{2}m} = & 𝔼_{m}[Z_{t}^{ρ}f(X_{t})g(X_{0})ρ^{2}(X_{0})] \\ = & 𝔼_{m}[Z_{t}^{ρ} ∘ r_{t} g(X_{t})ρ^{2}(X_{t})f(X_{0})]. \end{matrix}

To show

(P̂_{t}f, g)_{ρ^{2}m} = (f, P̂_{t}g)_{ρ^{2}m} = 𝔼_{m}[Z_{t}^{ρ}g(X_{t})ρ^{2}(X_{0})f(X_{0})],

it suffices to prove the following identity

Z_{t}^{ρ} ∘ r_{t} = Z_{t}^{ρ} \frac{ρ^{2}(X_{0})}{ρ^{2}(X_{t})}  ℙ_{m}-a.s. on \{t < ζ\}.  (2.5)

To this end, note that by the Doléans-Dade formula [13; Theorem 9.39], on {t < ζ},

Z_{t}^{ρ} = exp (M_{t} − \frac{1}{2}⟨M^{c}⟩_{t}) ∏_{0<s≤t} (1 + ΔM_{s}) e^{−ΔM_{s}}
= exp (M_{t} − \frac{1}{2}⟨M^{c}⟩_{t}) ∏_{0<s≤t} \frac{ρ(X_{s})}{ρ(X_{s−})} exp (1 − \frac{ρ(X_{s})}{ρ(X_{s−})}) .

(2.6)

It follows from (2.1) and (2.4) that on {t < ζ},

\begin{aligned}M_{t} & = M_{t}^{c} + lim_{ε↓0} \{ ∑_{0<s≤t} (\frac{ρ(X_{s})}{ρ(X_{s−})} − 1) 1_{\{|ρ(X_{s})−ρ(X_{s−})|>ε\}} \\ & − ∫_{0}^{t} (∫_{\{y∈E_{Δ}:|ρ(y)−ρ(X_{s})|>ε\}} (\frac{ρ(y)}{ρ(X_{s})} − 1) N(X_{s}, dy)) dH_{s}\} , \end{aligned}

where

M_{t}^{c} = ∫_{0}^{t} \frac{1}{ρ(X_{s−})}dM_{s}^{ρ,c}.

Since ρ > 0 q.e. on E, we see that log ρ ∈ F^{•}_{loc} (see [18; Cor. 6.2]), and therefore, by Lemma 2.2 and Remark 2.3, we have ℙ_{m}-a.s. on {t < ζ},

log ρ(X_{t}) − log ρ(X_{0}) = \frac{1}{2} (M_{t}^{c} − M_{t}^{c} ∘ r_{t})
+ lim_{ε↓0} ∑_{0<s≤t} (log ρ(X_{s}) − log ρ(X_{s−})) 1_{{|ρ(X_{s})−ρ(X_{s−})|>ε}}.  (2.7)

Since both ⟨M^{c}⟩_{t} and ∫_{0}^{t} ∫_{{y∈E_{Δ}:|ρ(y)−ρ(X_{s})|>ε}} (ρ(y)ρ(X_{s})^{−1} − 1) N(X_{s}, dy)dH_{s} are even CAFs of X, identity (2.5) follows from (2.6) and (2.7). □

The following result appears for symmetric diffusions as Lemma 4.4 in [5]; the proof given there is valid for general symmetric Borel right processes.

Theorem 2.5. If A = (A_{t}) is a PCAF of X with Revuz measure μ, then the Revuz measure of A as a PCAF of X̂ is ρ^{2}μ.

In what follows, if f ∈ B(E) then we write f ∈ L^{2}(ρ ⊗ ρ · J) to mean that \overline{f}(x, y) := f(y) − f(x) is square integrable with respect to ρ ⊗ ρ · J.

Theorem 2.6. Let (Ê, F̂) be the symmetric Dirichlet form on L^{2}(E, ρ^{2}m) associated with X̂. Then (a)

{f ∈ F : ∫ ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) < ∞} ∩ L^{2}(ρ ⊗ ρ · J) ∩ L^{2}(ρ^{2}m) ⊂ F̂

and for f in F ∩ L^{2}(ρ ⊗ ρ · J) ∩ L^{2}(ρ^{2}m) with ∫ ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) < ∞,

Ê(f, f) = \frac{1}{2} ∫_{E} ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + ∫_{E×E}(f(x) − f(y))^{2}ρ(x)ρ(y)J(dx, dy);  (2.8)

(b) 1 ∈ F̂ and Ê(1, 1) = 0, so the transformed process X̂ has infinite lifetime and is conservative in the ergodic theory sense (“recurrent” in the sense of [8; p. 48]).

Proof. Our proof is a modification of the proof of Theorem 3.6 in [3]. For the reader’s convenience, we spell out the details. Let F_{n}^{(1)} = {x : ρ(x) ≥ 1/n}. Then {F_{n}^{(1)}; n ≥ 1} is an E-nest. By the probabilistic characterization of E-nest, {F_{n}^{(1)}}_{n≥1} is an Ê-nest (for X̂) as well. So there is an Ê-nest {F_{n}; n ≥ 1} of compact sets and a sequence of g_{n} ∈ F̂ such that F_{n} ⊂ F_{n}^{(1)} and g_{n} = 1 on F_{n} for each n ≥ 1. Again by the probabilistic characterization of E-nest, {F_{n}}_{n≥1} is also an E-nest (for X). If we let ρ_{F_{n}} be the E_{1}-orthogonal projection of ρ onto F_{F_{n}} := {u ∈ F : u = 0 q.e. on F_{n}^{c}}, then ρ_{F_{n}} converges to ρ in (F, E_{1}). Let ρ_{n} = (0 ∨ ρ_{F_{n}}) ∧ ρ. Then ρ_{n} converges to ρ in (F, E_{1}) as well (cf. Theorem 1.4.2(v) in [8]). Taking a subsequence if necessary, we may assume that ρ_{n} converges to ρ, E-q.e. on E. For n ≥ 1, define h_{n} = ρ_{n}/ρ. Since ρ ≥ 1/n E-q.e. on F_{n} and ρ_{n} = 0 E-q.e. on F_{n}^{c}, we have h_{n} ∈ F, by the contraction property of (E, F). Note that 0 ≤ h_{n} ≤ 1 and h_{n} → 1 q.e. on E as n → ∞. By a calculation found in the proof of Lemma 6.3.3 of [8], it can be shown that

\frac{1}{2} ∫_{E} ρ(x)^{2}μ_{⟨h_{n}⟩}^{c}(dx) + ∫_{E×E}(h_{n}(x) − h_{n}(y))^{2}ρ(x)ρ(y)J(dx, dy) → 0  (2.9)

as n → ∞.

Let u be a bounded function in F with

\frac{1}{2} ∫_{E} ρ(x)^{2}μ_{⟨u⟩}^{c}(dx) + ∫_{E×E}(u(x) − u(y))^{2}ρ(x)ρ(y)J(dx, dy) + ∫ u(x)^{2}ρ(x)^{2}m(dx) < ∞.

Fix n ≥ 1 and define f := uh_{n}. Clearly f is a bounded function in F satisfying the above inequality with f in place of u. The process f(X_{t}) admits the following Lyons-Zheng forward-backward martingale decomposition

f(X_{t}) − f(X_{0}) = \frac{1}{2} (M_{t}^{f} − M_{t}^{f} ∘ r_{t})  ℙ_{m}-a.s. on \{t < ζ\},  (2.10)

where M_{t}^{f} is the martingale part in Fukushima’s decomposition of f(X_{t}) − f(X_{0}). Recall that dℙ̂_{x} = Z_{t}^{ρ}dℙ_{x} on M_{t} ∩ {t < ζ_{p}}. Hence

K_{t} := M_{t}^{f} − ∫_{0}^{t} \frac{1}{Z_{s−}^{ρ}} d⟨M^{f}, Z^{ρ}⟩_{s} = M_{t}^{f} − ⟨M^{f}, M⟩_{t}, t < ζ_{p},

is a local martingale AF under ℙ̂ = (ℙ̂_{x} : x ∈ E) and

[K]_{t}(ℙ̂) = [M^{f}]_{t}(ℙ)  ℙ̂_{m}-a.s. for t < ζ_{p}.  (2.11)

Here [K](ℙ̂) is the square bracket process for the martingale K under the family ℙ̂, and [M^{f}](ℙ) is the square bracket for martingale M^{f} under the family ℙ. We will use ⟨K⟩(ℙ̂) and ⟨M^{f}⟩(ℙ) to denote the dual predictable projections of [K](ℙ̂) and [M^{f}](ℙ) under the respective families ℙ̂ and ℙ. It follows from (2.11) that for t < ζ_{p},

⟨K⟩_{t}(ℙ̂) = ⟨M^{f}⟩_{t}(ℙ) + ∫_{0}^{t} \frac{1}{Z_{s−}^{ρ}} d⟨[M^{f}], Z^{ρ}⟩_{s}(ℙ)
= ⟨M^{f}⟩_{t}(ℙ) + ⟨[M^{f}], M⟩_{t}(ℙ);

see, for example, Chapter 12 of [13]. By the quasi-left continuity of the processes X and X̂, all of the “sharp bracket” processes involved are continuous, so we have for t ≥ 0,

\begin{aligned}⟨K⟩_{t}(ℙ̂) & = ⟨M^{f}⟩_{t}(ℙ) + ⟨[M^{f}], M⟩_{t}(ℙ) \\ & = ⟨M^{f}⟩_{t}(ℙ) + ( ∑_{0<s≤t} (f(X_{s}) − f(X_{s−}))^{2} (\frac{ρ(X_{s})}{ρ(X_{s−})} − 1))^{p} (ℙ) \\ & = ⟨M^{f}⟩_{t}(ℙ) + ∫_{0}^{t} ∫_{E_{Δ}} (f(X_{s}) − f(y))^{2} (\frac{ρ(y)}{ρ(X_{s})} − 1) N(X_{s}, dy) dH_{s}, \end{aligned}  (2.12)

with the convention that 0/0 = 1. Thus, by Theorem 2.5, the Revuz measure for the PCAF ⟨K⟩_{t}(ℙ̂) of X̂ is

\begin{matrix}ρ(x)^{2}μ_{⟨f⟩}(dx) + 2ρ(x)^{2} & ∫_{y∈E} (f(x) − f(y))^{2} (\frac{ρ(y)}{ρ(x)} − 1) J(dx, dy) − f(x)^{2}ρ(x)^{2}κ(dx) \\ = & ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + 2 ∫_{y∈E} (f(x) − f(y))^{2} ρ(x)ρ(y)J(dx, dy). \end{matrix}  (2.13)

Now the CAF ⟨M^{f}, M⟩ is even, so by (2.10)

f(X_{t}) − f(X_{0}) = \frac{1}{2}(K_{t} − K_{t} ∘ r_{t})  ℙ_{m}-a.s. on \{t < ζ\}.  (2.14)

Let ν = ρ^{2}m and

ℙ̂_{ν}(·) = ∫_{E} ℙ̂_{x}(·)ν(dx).

Applying Theorem 2.5 and noting that the time reversal operator r_{t} also leaves the measure ℙ̂_{ν} invariant on M_{t} ∩ {t < ζ}, we have by (2.13)-(2.14),

lim_{t→0} \frac{1}{t}𝔼̂_{ν} [(f(X̂_{t}) − f(X̂_{0}))^{2}; t < ζ] ≤ lim_{t→0} (\frac{1}{2t}𝔼̂_{ν} [(K_{t})^{2}; t < ζ] + \frac{1}{2t}𝔼̂_{ν} [(K_{t} ∘ r_{t})^{2}; t < ζ])

= lim_{t→0} \frac{1}{t}𝔼̂_{ν} [⟨K⟩_{t}(ℙ̂); t < ζ]

≤ ∫_{E} ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + 2 ∫_{E×E}(f(x) − f(y))^{2}ρ(x)ρ(y)J(dx, dy)

< ∞.

Recall that f = 0 m-a.e. on F_{n}^{c} and g_{n} ∈ F̂ with g_{n} = 1 m-a.e. on F_{n}. Thus f = fg_{n} and

lim_{t→0} \frac{1}{t} ∫_{E}(f(x) − P̂_{t}f(x))f(x)ν(dx)

= lim_{t→0} \frac{1}{t} (\frac{1}{2} 𝔼̂_{ν} [(f(X̂_{t}) − f(X̂_{0}))^{2}; t < ζ] + ∫_{E} f(x)^{2}(1 − P̂_{t}1)(x)ν(dx))

≤ lim sup_{t→0} \frac{1}{2t}𝔼̂_{ν} [(f(X̂_{t}) − f(X̂_{0}))^{2}; t < ζ] + lim sup_{t→0} \frac{1}{t} ∫_{E}(fg_{n})(x)^{2}(1 − P̂_{t}1)(x)ν(dx)

≤ lim sup_{t→0} \frac{1}{2t}𝔼̂_{ν} [(f(X̂_{t}) − f(X̂_{0}))^{2}; t < ζ] + ∥f∥_{∞}^{2} lim sup_{t→0} \frac{1}{t} ∫_{E} g_{n}(x)^{2}(1 − P̂_{t}1)(x)ν(dx)

< ∞.

Therefore f ∈ F̂, so f admits a Fukushima decomposition:

f(X̂_{t}) − f(X̂_{0}) = M̂_{t}^{f} + N̂_{t}^{f},  ℙ̂_{ν}-a.s.,

where M̂_{t}^{f} is a ℙ̂_{x}-square integrable martingale AF and N̂_{t}^{f} is a CAF of zero-energy; in particular, N̂_{t}^{f} is a process of zero quadratic variation. On the other hand, f(X_{t}) has a Fukushima decomposition under the family ℙ = (ℙ_{x} : x ∈ E):

f(X_{t}) − f(X_{0}) = M_{t}^{f} + N_{t}^{f}.

By Girsanov’s theorem the process K_{t} = M_{t}^{f} − ⟨M^{f}, M⟩_{t} is a local martingale under ℙ̂_{x} on [[0, ζ_{p} , so by uniqueness we have

M̂_{t}^{f} = K_{t}  for t < ζ_{p}.  (2.15)

To express Ê(f, f), we first calculate the killing measure κ̂ for the transformed process {X̂, ℙ̂_{x}, x ∈ E}. Now κ̂ is the Revuz measure of the PCAF (1_{{t≥ζ_{i}}})^{p}(ℙ̂), the dual predictable projection of the increasing AF t → 1_{{t≥ζ_{i}}} under ℙ̂. By the same reasoning as for (2.12), for q.e. x ∈ E,

(1_{{t≥ζ_{i}}})^{p}(ℙ̂) = (1_{{t≥ζ_{i}}})^{p}(ℙ) + ⟨1_{{t≥ζ_{i}}}, M⟩(ℙ)
= (1_{{t≥ζ_{i}}})^{p}(ℙ) − (1_{{t≥ζ_{i}}})^{p}(ℙ)
= 0.

Thus κ̂ = 0. Now by (2.13) and (2.15),

Ê(f, f) = lim_{t→0} \frac{1}{2t}𝔼̂_{ν} [(f(X̂_{t}) − f(X̂_{0}))^{2}] = lim_{t→0} \frac{1}{2t}𝔼̂_{ν} [(M̂_{t}^{f})^{2}] = lim_{t→0} \frac{1}{2t}𝔼̂_{ν} [⟨K⟩_{t}]
=\frac{1}{2} ∫_{E} ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + ∫_{E×E}(f(x) − f(y))^{2}ρ(x)ρ(y)J(dx, dy).

(2.16)

Applying the above argument to h_{n} (in place of f) we see that that h_{n} ∈ F̂ and, by (2.9), that Ê(h_{n}, h_{n}) → 0 as n → ∞. Since h_{n} → 1 q.e. on E, this implies that 1 ∈ F̂_{e} ∩L^{2}(E, ρ^{2}m) = F̂ and Ê(1, 1) = 0 (see Theorem 1.5.2 of [8]). Consequently, X̂ is recurrent by Theorem 1.6.3 of [8]. This proves part (b) of the Theorem.

So far we have proved that f = uh_{n} ∈ F̂ and that (2.16) holds for f. Note that

\begin{aligned}Ê(uh_{n}, uh_{n}) ≤ & 2∥u∥_{∞}^{2} Ê(h_{n}, h_{n}) + ∫_{E} ρ(x)^{2}μ_{⟨u⟩}^{c}(dx) \\ & + 2 ∫_{E×E}(u(x) − u(y))^{2}ρ(x)ρ(y)J(dx, dy) \end{aligned}

which is uniformly bounded. As |uh_{n}| ≤ |u|, uh_{n} → u, we see that u can be approximated in (F̂, Ê_{1}) by the Cesàro means of a subsequence of {uh_{n}}_{n≥1}. Hence u is in F̂. Repeating the computation for f shows that (2.16) holds for u as well. This proves part (a) of the Theorem. □

Remark 2.7. Suppose that ρ is in F^{•}_{loc} with ρ > 0 q.e. on E and that t ↦ ∑_{0<s≤t} |ρ(X_{s}) − ρ(X_{s−})| is locally ℙ_{x}-integrable for q.e. x ∈ E. This is the case if ρ is bounded, for example. If we define

M_{t}^{ρ} = M_{t}^{ρ,c} + ( ∑_{0<s≤t}(ρ(X_{s}) − ρ(X_{s−})) ,

where the superscript indicates compensated sum, and use (2.1)-(2.2) to define Z^{ρ}, then Theorem 2.6(i) remains valid (with the same proof). We will not use this fact in the sequel.

We now identify the domain of the Dirichlet space for X̂.

Theorem 2.8. Under the condition of Theorem 2.6, the domain F̂ of the Dirichlet form (Ê, F̂) for the Girsanov transformed process X̂ is the Ê_{1}-completion of

{f ∈ F : ∫ ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) < ∞} ∩ L^{2}(ρ ⊗ ρ · J) ∩ L^{2}(ρ^{2}m),

where Ê_{1} = Ê + (·, ·)_{L^{2}(E,ρ^{2}m)}.

We first prepare a lemma.

Lemma 2.9. Define N̂(x, dy) := \frac{ρ(y)}{ρ(x)} N(x, dy). Then (N_{X}, H) is a Lévy system for X̂. Consequently if Ĵ denotes the jump measure of X̂, then Ĵ = ρ ⊗ ρ · J.

Proof. Let K = (K_{t}) be a predictable process on Ω and f a non-negative Borel function on E ×E vanishing on diagonal. By [24; (62.13)] we find

ℙ̂_{x} [∑_{s≤t} K_{s}f(X̂_{s−}, X̂_{s})] = ℙ_{x} [∑_{s≤t} K_{s}f(X_{s−}, X_{s})Z_{s}^{ρ}; t < ζ]

= ℙ_{x} [∑_{s≤t} K_{s}Z_{s−}^{ρ}f(X_{s−}, X_{s}) \frac{ρ(X_{s})}{ρ(X_{s−})} ; t < ζ]

= ℙ_{x} [∫_{0}^{t} K_{s}Z_{s}^{ρ}ρ(X_{s})^{−1}N(ρf)(X_{s}) dH_{s}; t < ζ]

= ℙ̂_{x} [∫_{0}^{t} K_{s} N̂f(X̂_{s}) dH_{s}] .

The conclusion follows. □

Proof of Theorem 2.8. Define M^{ρ,o} = M^{ρ} + (ρ(X_{ζ−})1_{{·≥ζ_{i}}}) and M_{t}^{o} = ∫_{0}^{t}[ρ(X_{s−})]^{−1}dM_{s}^{ρ,o}. Clearly M^{ρ,o} is a ℙ_{x}-square integrable MAF of X and M^{o} is a locally ℙ_{x}-square integrable MAF of X. Define

M̂_{t} := −M_{t}^{o} + ⟨M^{c}⟩_{t}(ℙ) + ∑_{0<s≤t} \frac{(ρ(X̂_{s}) − ρ(X̂_{s−}))^{2}}{ρ(X̂_{s})ρ(X̂_{s−})} 1_{\{s<ζ\}}.  (2.17)

Then M̂ is a local MAF of X̂. To see this observe that, by Girsanov’s theorem,

M_{t}^{o} − ⟨M^{o}, M⟩_{t}(ℙ) = M_{t}^{o} − ⟨M^{c}⟩_{t}(ℙ) − ( ∑_{0<s≤·} (\frac{ρ(X_{s})}{ρ(X_{s−})} − 1)^{2} 1_{\{s<ζ\}})_{t}^{p} (ℙ)

is a local martingale AF of X̂. An application of Lemma 2.9 (to compute the compensator of the sum on the right side of (2.17)) now finishes the proof of the claim.

Let Ẑ_{t} be the solution to dẐ_{t} = Ẑ_{t−}dM̂_{t}. Denote by A^{κ} the PCAF of X associated with the killing measure κ. Recall that A^{κ} = (1_{{·≥ζ_{i}}})^{p}(ℙ). Using the Doléans-Dade formula, one sees that

\frac{dℙ_{x}}{dℙ̂_{x}} |_{M_{t}} = \frac{1}{Z_{t}^{ρ}} = Ẑ_{t} e^{−A_{t}^{κ}}.  (2.18)

(cf. the last paragraph in the proof of Theorem 3.7 in [3].)

Let F_{k} = \{x ∈ E : ρ(x) ≥ 1/k\}, which is an E-nest hence an Ê-nest as well. Define ρ_{k} = ρ − (ρ ∧ \frac{1}{k}) and h_{k} = ρ_{k}/ρ. Clearly 0 ≤ h_{k} ≤ 1, ρ_{k} ∈ F_{F_{k}}, and ρ_{k} → ρ in (F, E_{1}). Arguing as in the proof of Theorem 2.6, it can be shown that h_{k} ∈ L^{2}(ρ ⊗ ρ · J), and h_{k} ∈ F̂ with Ê(h_{k}, h_{k}) → 0 as k → ∞. Now for any u ∈ bF̂, we claim f := uh_{k} lies in bF_{F_{k}} and

∫ ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + ∫ (f(x) − f(y))^{2}ρ(x)ρ(y)J(dx, dy) < ∞.

First note that

∫ (f(x) − f(y))^{2} J(dx, dy)
≤ ∫_{F_{k}×F_{k}} \overline{f}(x, y)^{2} J(dx, dy) + ∥u∥_{∞}^{2} ∫_{(F_{k}×F_{k})^{c}} \overline{h}_{k}(x, y)^{2} J(dx, dy)
≤ k^{2} ∫_{F_{k}×F_{k}} \overline{f}(x, y)^{2}ρ(x)ρ(y) J(dx, dy) + ∥u∥_{∞}^{2} ∫_{(F_{k}×F_{k})^{c}} \overline{h}_{k}(x, y)^{2} J(dx, dy)
< ∞.

As f ∈ bF̂, by Fukushima’s decomposition,

f(X̂_{t}) − f(X̂_{0}) = M̂_{t}^{f} + N̂_{t}^{f}.

Define a family of measures ℚ = (ℚ_{x} : x ∈ E) through \frac{dℚ_{x}}{d_{ℙ_{x}}} |_{M_{t}} = Ẑ_{t}, where Ẑ is the Doléans-Dade exponential of M̂, and M̂ is as in (2.17). In this proof only we shall use X^{∗} to denote the coordinate process when referring to ℚ. Recall that X̂ is a right Markov process with symmetry measure ν(dx) := ρ(x)^{2}m(dx). It can be shown as in the proof of Lemma 2.4 that (X^{∗}, ℚ) is a right Markov process with symmetry measure ρ(x)^{−2}ν(dx) = m(dx). From Girsanov’s theorem

K := M̂^{f} − ⟨M̂^{f}, M̂⟩(ℙ̂)

is a ℚ-local martingale. Note that since [K]_{t}(ℚ) = [M̂^{f}]_{t}(ℙ̂),

⟨K⟩_{t}(ℚ) = ⟨M̂^{f}⟩_{t}(ℙ̂) + ⟨[M̂^{f}], M̂⟩_{t}(ℙ̂).  (2.19)

But

⟨[M̂^{f}], M̂⟩_{t}(ℙ̂) = ( ∑_{0<s≤t}(f(X̂_{s}) − f(X̂_{s−}))^{2} \frac{ρ(X̂_{s−}) − ρ(X̂_{s})}{ρ(X̂_{s})})^{p} (ℙ̂)

so its Revuz measure with respect to (X̂, ℙ̂) is, by Lemma 2.9,

∫_{y∈E}(f(x) − f(y))^{2}ρ(x)(ρ(x) − ρ(y))J(dx, dy).

Hence by applying Theorem 2.5 with X̂ and X^{∗} in the roles of X and X̂, and with ρ^{−1} in place of ρ, the Revuz measure of ⟨[M̂^{f}], M̂⟩(ℙ̂) viewed as a PCAF of X^{∗} is

\frac{1}{ρ(x)} ∫_{y∈E}(f(x) − f(y))^{2}(ρ(x) − ρ(y))J(dx, dy).

Likewise, by (2.19), the Revuz measure μ_{⟨K⟩}^{∗} of ⟨K⟩(ℚ) viewed as PCAF of X^{∗} is seen to be

μ_{⟨K⟩}^{∗}(dx) = \frac{1}{ρ(x)^{2}} μ_{⟨f⟩}^{c}(dx) + ∫_{y∈E}(f(x) − f(y))^{2}J(dx, dy).  (2.20)

As ρ ≥ 1/k on F_{k} and f = 0 on F_{k}^{c}, hence ∫ ρ(x)^{−2} μ_{⟨f⟩}^{c}(dx) < ∞ and therefore μ_{⟨K⟩}^{∗}(E) < ∞. But f(X_{t}^{∗}) − f(X_{0}^{∗}) = (K_{t} − K_{t} ∘ r_{t})/2, and we deduce, by reasoning similar to that used just below (2.14), that f is in the Dirichlet space of X^{∗}. In view of (2.18), we have f = uh_{k} ∈ bF_{F_{k}}. As both u and h_{k} are in bF̂, Lemma 2.9 yields

∫ (f(x) − f(y))^{2}ρ(x)ρ(y)J(dx, dy) < ∞.

By a calculation similar to that used in the proof of Lemma 2.9, the jump measure of X^{∗} is J. Likewise, by the argument appearing above between (2.15) and (2.16), one sees that the killing measure of X^{∗} is the zero measure. If we use μ_{⟨f⟩}^{∗} and μ_{⟨f⟩}^{∗,c} to denote the the energy measure of f and its strong local part, in the context of X^{∗}, then μ_{⟨f⟩}^{∗} = μ_{⟨K⟩}^{∗}. On the other hand,

μ_{⟨f⟩}^{∗}(dx) = μ_{⟨f⟩}^{∗,c}(dx) + ∫_{y∈E}(f(x) − f(y))^{2}J(dx, dy).

Hence from (2.20), we see that

μ_{⟨f⟩}^{∗,c}(dx) = \frac{1}{ρ(x)^{2}} μ_{⟨f⟩}^{c}(dx).

As the Feynman-Kac transformation by the multiplicative functional exp(−A_{t}^{κ}) does not change the strongly local part of the energy measure, we have

μ_{⟨f⟩}^{c}(dx) = \frac{1}{ρ(x)^{2}} μ_{⟨f⟩}^{c}(dx)

and so

∫ ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + ∫ (f(x) − f(y))^{2}ρ(x)ρ(y)J(dx, dy) < ∞.

Since |uh_{k}| ≤ |u|, uh_{k} → u and

Ê(uh_{k}, uh_{k}) ≤ 2∥u∥_{∞}^{2}Ê(h_{k}, h_{k}) + 2Ê(u, u),

which is uniformly bounded, we see that u can be approximated in (F̂, Ê_{1}) by the Cesàro mean of a subsequence of {uh_{k}}_{k≥1}. Hence u is in the Ê_{1}-closure of

{f ∈ F : ∫ ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) < ∞} ∩ L^{2}(ρ ⊗ ρ · J) ∩ L^{2}(ρ^{2}m).

This proves the Theorem. □

3. Supermartingale Multiplicative Functional.

In this section we prove a representation theorem for a general class of supermartingale multiplicative functionals (MFs) of X. This result, is a sharpening of results of Kunita [16; Theorem 3.1] and Sharpe [23; Theorem 7.1]; see also [17; §6] and [15; §4]. For a stopping time T, we will use I(T) to denote the stochastic interval [[0, T ∪ T_{i} , where T_{i} is the totally inaccessible part of T. By a slight abuse of notation, we shall often write “t ∈ I(T)” to mean “(t, ω) ∈ I(T),” where ω is the (suppressed, as usual) sample path.

Theorem 3.1. Let Z be a supermartingale MF of X such that Z_{0} ≡ 1. Then there is a local martingale AF M, a PCAF A, and a Borel function φ : E × E_{Δ} → [−1, +∞[ such that

ΔM_{t} := M_{t} − M_{t−} = φ(X_{t−}, X_{t}),  ∀t ∈ I(ζ), ℙ_{m}-a.s., and  (3.1)
Z_{t} = e^{M_{t}− \frac{1}{2} ⟨M^{c}⟩_{t}−A_{t}} ∏_{0<s≤t}[1 + φ(X_{s−}, X_{s})]e^{−φ(X_{s−},X_{s})},  ∀t ∈ I(ζ), ℙ_{m}-a.s.  (3.2)

The AF M and the PCAF A are determined by Z up to ℙ_{m}-evanescence. In particular, φ is uniquely determined by Z modulo null sets of the measure J^{∗}(B) := J(B ∩ (E × E)) + κ(π_{1}(B ∩ (E × {Δ})), where π_{1}(x, y) := x. Moreover,

∫_{0}^{t} N(1_{{|φ|≤1}}φ^{2} + 1_{{φ>1}}φ)(X_{s}) dH_{s} + ∫_{0}^{t} φ(X_{s}, Δ) dA_{s}^{κ} < +∞  (3.3)

for all t ∈ I(ζ), ℙ_{m}-a.s. Finally,

S := inf{t > 0 : Z_{t} = 0} = inf{t > 0 : φ(X_{t−}, X_{t}) = −1}  (3.4)

ℙ_{m}-a.s. on {S < ζ_{p}}.

For a semimartingale N, let Exp(N) denote the unique solution Y of

Y_{t} = 1 + ∫_{0}^{t} Y_{s−}dN_{s}.

Exp(N) is called the stochastic exponential (in the sense of Doléans-Dade) of N. Formula (3.2) amounts to the statement that Z = Exp(M − A) at least on I(ζ). Before turning to the proof of Theorem 3.1 we prepare the way with a

Lemma 3.2. Let B = (B_{t})_{t≥0} be an AF of X. Then there is a Borel function b : E × E_{Δ} → ℝ with b(x, x) = 0 for all x ∈ E such that

ΔB_{t} := B_{t} − B_{t−} = b(X_{t−}, X_{t}),  ∀t ∈]0, ζ_{p}[, ℙ_{m}-a.s.  (3.5)

If b′ is another such function, then J^{∗}(b′ ≠ b) = 0.

Proof. It follows from [10; (16.5)] that there is a Borel function b_{0} : E×E → ℝ such that ΔB_{t} = b_{0}(X_{t−}, X_{t}) for all t ∈]0, ζ[, ℙ_{m}-a.s. Fix ϵ > 0 and define T := inf{t > 0 : |ΔB_{t}| ≥ ϵ}. Then T is a thin terminal time; i.e., ℙ_{x}[T = 0] = 0 for all x ∈ E. Consequently, T_{p}, the predictable part of T, is a thin predictable terminal time; by [10; (16.21)], ℙ_{m}(T_{p} < ζ) = 0. Owing to [24; §73], it follows that {t ∈ [0, ζ[: |ΔB_{t}| > 0} ⊂ {t > 0 : X_{t−} ≠ X_{t}}. Thus, modifying b_{0} if necessary, we can arrange that b_{0}(x, x) = 0 for all x ∈ E. Next, notice that ΔB_{ζ_{i}} is measurable over the germ σ-field F_{[ζ_{i}−,ζ_{i}]}^{m}. Because X ≡ Δ on [[ζ_{i}, ∞ , this germ σ-field is generated (modulo ℙ_{m}-null sets) by the random variables of the form g(X)_{ζ_{i}−} as g varies over the bounded 1-excessive functions of X; see [24; (24.32)(ii)]. But in the present context, natural AFs are continuous (by Cor. 3.17 in [6]), from which it follows that g(X)_{ζ_{i}−} = g(X_{ζ_{i}−}), ℙ_{m}-a.s. Therefore there is a Borel function b_{Δ} : E → ℝ such that ΔB_{ζ_{i}} = b_{Δ}(X_{ζ_{i}−}), ℙ_{m}-a.s. on {ζ_{i} < ∞}; cf. [10; (16.4)]. Defining b(x, y) = b_{0}(x, y)1_{E×E}(x, y) + b_{Δ}(x)1_{{Δ}}(y), we obtain the representation (3.5). The proof of the uniqueness assertion is left as an exercise to the reader. □

Proof of Theorem 3.1. We begin with a discussion of the terminal time S defined in (3.4); for related work see [26], [15], and [11]. Clearly S is a thin terminal time. Define the sequence {S^{(n)} : n ≥ 1} of iterates of S by setting S^{(1)} := S and S^{(n+1)} := S^{(n)} + S^{∘}θ_{S^{(n)}} for n = 1, 2, . . .. (As a matter of convention, if S^{(n)}(ω) = +∞, then S^{(k)}(ω) = +∞ for all k > n.) Next, define

C_{t} := ∑_{n} 1_{{S^{(n)}≤t}}, t ≥ 0

and

S^{(∞)} :=↑ lim_{n} S^{(n)}.

Let [Δ] denote the sample path ω such that X_{t}(ω) = Δ for all t ≥ 0. With the convention Z_{t}([Δ]) ≡ 1, we have S([Δ]) = +∞. Thus, if S^{(n)}(ω) = ζ(ω), then S^{(k)}(ω) = +∞ for all k > n. Consequently, {S^{(n)} : n ≥ 1} announces S^{(∞)} on {S^{(∞)} < +∞}. That is, S^{(∞)} is a thin predictable terminal time, hence S^{(∞)} ≥ ζ_{p}, ℙ_{m}-a.s. by [10; (16.21)]. Since {C_{t} = ∞, t < ζ} = {S^{(∞)} ≤ t < ζ}, it follows that C is finite on I(ζ), ℙ_{m}-a.s. Let us now apply Lemma 3.2 to C, taking into account the fact that ΔC takes values in {0, 1} in the present situation. We find that there is a Borel set Λ ⊂ E × E_{Δ}, disjoint from the ‘diagonal’ {(x, x) : x ∈ E}, such that

C_{t} = ∑_{0<s≤t} 1_{Λ}(X_{s−}, X_{s})

provided t ∈ I(ζ), ℙ_{m}-a.s. In particular,

S = inf{t > 0 : (X_{t−}, X_{t}) ∈ Λ}  (3.6)

ℙ_{m}-a.s. on {S < ζ_{p}}. In fact, a little thought shows that the infimum in (3.6) is attained: (X_{S−}, X_{S}) ∈ Λ, ℙ_{m}-a.s. on {S < ζ_{p}}.

From [14; Théorème 1] and the quasi-left-continuity of the filtration (M_{t}) we know that Z_{−}^{−1} is locally bounded on I(S) in the sense that there is an increasing sequence (S_{n}) of stopping times with I(S) = ∪_{n=1}^{∞}[[0, S_{n} such that Z_{−}^{−1} ≤ n on [[0, S_{n} for all n, ℙ_{m}-a.s. Let Z = N − V be the Doob-Meyer decomposition of Z into local martingale and predictable increasing components N and V, respectively. Define M_{t}^{∘} := ∫_{0}^{t} Z_{s−}^{−1} dN_{s} and A_{t}^{∘} := ∫_{0}^{t} Z_{s−}^{−1} dV_{s}. Both of these integrals are well defined on I(S) because of the local boundedness of Z_{−}^{−1} just noted. It is not hard to check that M^{∘} is an AF of (X, S) (the process X killed at time S, defined to equal X before S and Δ at and after S) and a local martingale on I(S), and that A^{∘} is a predictable increasing AF of (X, S). Of course, A^{∘} is continuous except perhaps for a jump at S_{p} on {S_{p} = ζ_{p} < ∞}. (By the discussion in the preceding paragraph, S_{{S<ζ_{p}}} is totally inaccessible.)

We now extend M^{∘} and A^{∘} to AFs of X. Let us begin with A^{∘}. By an extension procedure detailed in Theorem (4.8) of [10], there is a diffuse homogeneous random measure α of X such that A_{t}^{∘} = α([0, t]) for all t < S, ℙ_{m}-a.s. We will show that α([0, t]) < +∞ for all t < ζ, ℙ_{m}-a.s., and then use the recipe A_{t} := α([0, t ∧ ζ]), t ≥ 0, as the desired extension of A^{∘}. Define R := inf{t : α([0, t]) = +∞}. Clearly R is a terminal time, and R ≥ S, so that R is thin and hence exact, in the sense that t + R^{∘}θ_{t} decreases to R as t decreases to 0. Now define R_{n} := inf{t : α([0, t]) ≥ n}, and notice that R_{n} ≤ R. The strong Markov property (applied at the stopping time R) shows that the event {R_{n} = R < ζ} differs from the event {R_{n} = R < ζ, S^{∘}θ_{R} > 0} by a ℙ_{m}-null set. Suppose that ω is a point of {R_{n} = R < ζ, S^{∘}θ_{R} > 0}. Then α(ω, [0, t]) < n for 0 ≤ t < R(ω) but α(ω, [0, t]) = +∞ for t > R(ω). In particular, if t > R(ω), then

+∞ = α(ω, [0, t]) = α(ω, [0, R(ω)]) + α(θ_{R}ω, (0, t − R(ω)]).

The furthest right term above is finite when t is sufficiently close to R(ω) since S(θ_{R}ω) > 0. Therefore α(ω, [0, R(ω)] = +∞. On the other hand, α(ω, [0, R(ω))) = α(ω, [0, R_{n}(ω))) ≤ n. But α(ω, {R(ω)}) = 0 because α is diffuse. It follows that ℙ_{m}(R_{n} = R < ζ) = 0, so that R′ := R_{{R<ζ}} is a thin predictable terminal time. Thus R′ ≥ ζ, which forces R ≥ ζ as well. This shows that t ↦ α([0, t]) defines a PCAF of X.

Turning to M^{∘}, notice that ΔM_{t}^{∘} = ΔZ_{t}/Z_{t−} = Z_{t}/Z_{t−} − 1 ≥ −1. Define B_{t} := ∑_{0<s≤t} ΔM_{t}^{∘}1_{{ΔM_{t}^{∘}>1}}, let B_{t}^{p} denote the dual predictable projection of B, and now define M_{t}^{∘,2} := B_{t} −B_{t}^{p} and M_{t}^{∘,1} := M_{t}^{∘} −M_{t}^{o,2}. Both M^{∘,1} and M^{∘,2} are local martingales on I(S) and AFs of (X, S). By the argument of the last paragraph, the CAF (of (X, S)) B^{p} extends to a CAF \overline{B} of X. Moreover, by (3.5) there is a Borel function φ : E ×E_{Δ} → [−1, +∞[ such that ΔM^{∘} = φ(X_{−}, X) on I(S). Then

M_{t}^{∗,2} := ∑_{0<s≤t} φ(X_{s−}, X_{s})1_{\{φ>1\}}(X_{s−}, X_{s}) − \overline{B}_{t}

defines an extension of M^{∘,2} to a local martingale AF of X. Next, given a locally square-integrable martingale AF N, consider the covariation process Φ^{∘}(N) := ⟨M^{∘,1}, N⟩, viewed as a CAF of (X, S). As before, this CAF admits a unique extension Φ(N) to a CAF of X. In addition, we have the Kunita-Watanabe estimate

⟨M^{∘,1}, N⟩_{t}^{2} ≤ ⟨M^{∘,1}⟩_{t} · ⟨N⟩_{t}, ∀t ∈ [0, S[,

and it is easy to check that [Φ(N)_{t}]^{2} ≤ D_{t} · ⟨N⟩_{t}, where D is the extension of the CAF ⟨M^{∘,1}⟩. A result of Kunita (Prop. 2.4 in [15]) now tells us that there is a local martingale AF M^{∗,1} such that Φ(N) ≡ ⟨M^{∗,1}, N⟩ for all N. Of course, M^{∗,1} ≡ M^{∘,1} on I(S). The local martingale AF M := M^{∗,1} + M^{∗,2} is the desired extension of M^{∘}. Notice that ΔM ≡ φ(X_{−}, X) on I(ζ). □

4. Absolute Continuity and Dirichlet Forms.

Let Y = (Ω, M, M_{t}, Y_{t}, ℚ_{x}, x ∈ E) be another symmetric Markov process with symmetry measure ν, which is realized on the same (canonical) path space Ω as (X, ℙ_{x}, x ∈ E). Here Y_{t}(ω) = X_{t}(ω) but we use Y for emphasis when referring to Y. As with the process X, we assume that Y is a Borel right process.

We note that Lemmas 2.5, 3.4 and 3.9 in [5] are valid in the setting of symmetric Borel right processes. The first result of this section is the analogue of [5; Thm. 3.2]

Theorem 4.1. Assume ℚ_{ν} ≪_{loc} ℙ_{m}, in that ℚ_{ν}|_{M_{t}∩{t<ζ}} is absolutely continuous with respect to ℙ_{m}|_{M_{t}∩{t<ζ}} for each t > 0. Then ν ≪ m and there is a (Y, ν)-inessential Borel set N ⊂ E which is X-finely closed, and a version ρ^{2} of the Radon-Nikodym derivative dν/dm such that 0 < ρ(x) < ∞ for all x ∈ E \ N and

(a) t → ρ(X_{t}) is right continuous on [0, T_{N}[ with left limits on ]0, T_{N} ∧ ζ[, P_{x}-a.s. for all x ∈ E \ N; in particular, ρ|_{E\N} is (X, T_{N})-finely continuous; (b) log ρ ∈ F^{•}_{loc}(X, T_{N}) and there exists a local martingale AF M_{t} satisfying

log ρ(X_{t}) − log ρ(X_{0}) = (M_{t} − M_{t} ∘ r_{t})/2  ℙ_{m}-a.s. on {t < T_{N} ∧ ζ}.  (4.1)

Proof. The existence of N follows from the proofs of Lemmas 3.4 and 3.9 in Fitzsimmons [5]. By Kunita [16; Theorem 5.1], there is a supermartingale multiplicative functional Z_{t} of X satisfying

\frac{dℚ_{ν}}{dℙ_{m}} |_{M_{t}∩\{t<ζ\}} = Z_{t}.  (4.2)

We set Z_{t} = 0 for t ≥ ζ. Since X is symmetric under the measure ℙ_{m}, we have

ρ^{2}(X_{0})Z_{t} = ρ^{2}(X_{t})Z_{t} ∘ r_{t} ℙ_{m}-a.s. on {t < ζ}.

This implies that

2 log ρ(X_{t}) − 2 log ρ(X_{0}) = log Z_{t} − log Z_{t} ∘ r_{t}  ℙ_{m}-a.s. on {t < T_{N} ∧ ζ}.  (4.3)

But, by Theorem 3.1,

log Z_{t} − log Z_{0} = M_{t} − \frac{1}{2}⟨M^{c}⟩_{t} − A_{t} + H_{t}, 0 ≤ t < ζ,

where

H_{t} := ∑_{0<s≤t} (log (1 + φ(X_{s−}, X_{s})) − φ(X_{s−}, X_{s})) ,

which is absolutely convergent in view of (3.3). Clearly H is quasi-left continuous since X is so; thus its dual predictable projection H^{p} is a CAF. Define

M_{t}^{∗} := \frac{1}{2} (M_{t} + H_{t} − H_{t}^{p}) , A_{t}^{∗} := \frac{1}{2}(H_{t}^{p} − \frac{1}{2}⟨M^{c}⟩_{t} − A_{t}).

Then M_{t}^{∗} is a local martingale AF, A_{t}^{∗} is a CAF of finite variation with

log Z_{t} − log Z_{0} = 2M_{t}^{∗} + 2A_{t}^{∗}.

Note that Z_{0} = ρ^{2}(X_{0}) and A_{t}^{∗} is even. So on {t < T_{N} ∧ ζ},

log Z_{t} − log Z_{t}^{∘}r_{t} + 2 log ρ(X_{t}) − 2 log ρ(X_{0}) = 2M_{t}^{∗} − 2M_{t}^{∗}^{∘}r_{t}.

Hence by (4.3),

log ρ(X_{t}) − log ρ(X_{0}) = (M_{t}^{∗} − M_{t}^{∗} ∘ r_{t})/2.

Theorem 4.2 below now implies that log ρ ∈ F^{•}_{loc}(X, T_{N}). Therefore ρ has a quasi-continuous version with respect to subprocess (X, T_{N}). □

Theorem 4.2. Let f be a quasi-continuous Borel function on E, and suppose that there is a local martingale AF M such that

f(X_{t}) − f(X_{0}) = (M_{t} − M_{t} ∘ r_{t})/2, ℙ_{m}-a.s. for each t ∈ [0, ζ[.

Then f ∈ F^{•}_{loc}.

Proof. We first assume that the jumps of M are bounded, and then we show how to reduce to this special case.

If there is a constant C such that |ΔM_{t}(ω)| ≤ C, then it is easy to check that [M] is locally integrable, so ⟨M⟩ exists and is a PCAF of X. Things being so, the argument in [5; Lemma 3.15] can be used to reach the desired conclusion.

In general, define T := inf{t > 0 : |ΔM_{t}| > 1}. Then T is a thin terminal time, and the subprocess (X, T) is m-symmetric with state space E. Theorem 4.1 in [27] provides a precise description of the Dirichlet form of (X, T), telling us, in particular, that the Dirichlet space F(X, T) of (X, T) is a subspace of F. Evidently, X and (X, T) have the same fine topologies (modulo X-exceptional sets). Clearly if N is X-exceptional, then it is (X, T)-exceptional. Conversely, if N is (X, T)-exceptional, then it is X-exceptional. To see this let {T_{n}} denote the sequence of iterates of T. By assumption, X does not encounter N during any of the open intervals ]T_{n}, T_{n+1}[. Thus X visits N at most countably often, ℙ_{m}-a.s. That is, N is m-semipolar, hence exceptional since X is symmetric. Using quasi-left continuity, one now checks that any increasing sequence {G_{n}} of finely open sets is an X-nest if and only if it is an (X, T)-nest.

Now for f under the assumptions of the Theorem, by modifying M at time T we can produce a local martingale AF of (X, T), call it M^{∗}, with jumps bounded by 1, such that

f(X_{t}) − f(X_{0}) = (M_{t}^{∗} − M_{t}^{∗} ∘ r_{t})/2, ℙ_{m}-a.s. for each t ∈ [0, T [.

By the first paragraph, f ∈ F^{•}(X, T)_{loc}. The preceding paragraph tells us that F^{•}(X, T)_{loc} ⊂ F^{•}_{loc} since F(X, T) ⊂ F. This completes the proof. □

For t > 0, we say that two sample paths ω and ω′ are pre-t-equivalent provided ω(s) = ω′(s) for all s ∈ [0, t[. Observe that if A = (A_{t}) is a finite CAF of X and if ω and ω′ are pre-t-equivalent, then

A_{s}(ω) = A_{s}(ω′) for all 0 ≤ s ≤ t.

It is easy to check that r_{t}θ_{s}ω is pre-t-equivalent to r_{t+s}ω and that θ_{t}r_{t+s}ω is pre-s-equivalent to r_{s}ω. This will be used repeatedly in the proof of next theorem. Define

Â_{t} = A_{t}^{∘}r_{t} on {t < ζ}.

Following Walsh [26], we have the following result.

Theorem 4.3. Â = (Â_{t} : 0 ≤ t < ζ) is a continuous additive functional of X.

Proof. First we need to show that Â_{t} is an additive functional. On {t + s < ζ},

\begin{matrix}Â_{t+s} & = A_{t+s}^{∘}r_{t+s} = (A_{t} + A_{s}^{∘}θ_{t})^{∘}r_{t+s} = A_{t}^{∘}r_{t+s} + A_{s}^{∘}θ_{t}^{∘}r_{t+s} \\ = & A_{t}^{∘}r_{t}^{∘}θ_{s} + A_{s}^{∘}r_{s} = Â_{s} + Â_{t}^{∘}θ_{s}. \end{matrix}

Note that on {t < ζ}, for 0 < u < t,

Â_{t} − Â_{t−u} = Â_{u}^{∘}θ_{t−u} = A_{u}^{∘}r_{u}^{∘}θ_{t−u} = A_{u}^{∘}r_{t}.

Hence

lim_{u↓0}(Â_{t−u} − Â_{t}) = − lim_{u↓0} A_{u}^{∘}r_{t} = 0.

This shows that Â is left continuous. Let us now prove the right continuity. Note that on {t + u < ζ},

Â_{t+u} − Â_{t} = Â_{u}^{∘}θ_{t} = A_{u}^{∘}r_{u}^{∘}θ_{t} = A_{u}^{∘}r_{t+u}

so it suffices to show that lim_{u↓0} A_{u}^{∘}r_{t+u} = 0. For any s > u > 0, since θ_{s−u}^{∘}r_{t+s}ω is pre-(t + u)-equivalent to r_{t+u}ω, we have

(A_{s} − A_{s−u})^{∘}r_{t+s} = A_{u}^{∘}θ_{s−u}^{∘}r_{t+s} = A_{u}^{∘}r_{t+u}.

Thus

lim_{u↓0} A_{u}^{∘}r_{t+u} = lim_{u↓0}(A_{s} − A_{s−u})^{∘}r_{t+s} = (A_{s} − A_{s−})^{∘}r_{t+s} = 0.

This proves the theorem. □

For simplicity from now on we will assume that ℚ_{x} ≪_{loc} ℙ_{x} for all x and that Z > 0 on [[0, ζ . It is easy to reduce to this case by killing X at a terminal time and removing a Y -exceptional set N from E.

Defining ℓ := log ρ, we have ℓ ∈ F^{•}_{loc} by Theorem 4.1. Recall that the density process Z in (4.2) is a non-negative supermartingale MF of X, which is strictly positive on [[0, ζ . Hence by Theorem 3.1, Z = Exp(M − A), where M is a local martingale MF and A is a PCAF of X. Let M^{c} and M^{d} be the continuous and purely discontinuous components of M, and let φ be the Borel function: E ×E_{Δ} → [−1, +∞[ with φ(x, x) = 0 for all x ∈ E such that

ΔM_{t}^{d} = ΔM_{t} = φ(X_{t−}, X_{t}) ℙ_{m}-a.s..

In particular, M^{d} is the compensated local martingale corresponding to ∑_{0<s≤·} φ(X_{s−}, X_{s}). We now deduce from the identity (4.1) that

ℓ(X_{t}) − ℓ(X_{0}) = \frac{1}{2}(M_{t}^{c} − M_{t}^{c} ∘ r_{t}) + \frac{1}{2} ∑_{0<s≤t} log (\frac{1 + φ(X_{s−}, X_{s})}{1 + φ(X_{s}, X_{s−})})  (4.4)

ℙ_{m}-a.s. on {t < ζ} for every t > 0. The infinite series in (4.4) is to be understood in the following sense:

∑_{0<s≤t} log (\frac{1 + φ(X_{s−}, X_{s})}{1 + φ(X_{s}, X_{s−})}) = ∑_{0<s≤t} (log (1 + φ(X_{s−}, X_{s})) − φ(X_{s−}, X_{s}))

− ∑_{0<s≤t} (log (1 + φ(X_{s}, X_{s−})) − φ(X_{s}, X_{s−})) ,

(4.5)

both sums on the right being absolutely convergent, ℙ_{m}-a.s. on {t < ζ}. By Theorem 4.3, (4.4) in fact holds for all t ∈ [0, ζ[, ℙ_{m}-a.s..

As ℓ ∈ F^{•}_{loc} by Theorem 4.1, it follows from Lemma 2.2 that

\begin{matrix}ℓ(X_{t})− & ℓ(X_{0}) = \frac{1}{2}(M_{t}^{ℓ,c} − M_{t}^{ℓ,c} ∘ r_{t}) \\ + lim_{ε↓0} ∑_{0<s≤t}(ℓ(X_{s}) − ℓ(X_{s−}))1_{\{|ℓ(X_{s})−ℓ(X_{s−})|>ε\}} \end{matrix}  (4.6)

for all t ∈ [0, ζ[, ℙ_{m}-a.s. Identities (4.4)-(4.6) yield ℙ_{m}-a.s.

M_{t}^{c} − M_{t}^{c} ∘ r_{t} = M_{t}^{ℓ,c} − M_{t}^{ℓ,c} ∘ r_{t} for all t ∈ [0, ζ[.

Using the fact that an even martingale CAF must vanish [5; (3.25)], we deduce from the above that

M^{c} = M^{ℓ,c} = ρ^{−1}(X_{−}) • M^{ρ,c},

where ρ^{−1}(X_{−}) • M^{ρ,c} is an Itô integral, and the last equality follows from [8; Thm. 5.6.2]. It follows from Lemma 3.2 and (4.5) that

ℓ(x) − ℓ(y) = \frac{1}{2} log (\frac{1 + φ(x, y)}{1 + φ(y, x)}) J^{∗}-a.e. on E × E,

and so

\frac{ρ(y)^{2}}{ρ(x)^{2}} = \frac{1 + φ(x, y)}{1 + φ(y, x)}.

Thus, the function γ defined by

γ(x, y) := ρ(x)^{2}[1 + φ(x, y)]  (4.7)

is symmetric, J^{∗}-a.e.

Recall that by Theorem 3.1 the density Z_{t} in (4.2) can be written as

Z = Exp(M − A) = Exp(M)Exp(−A),

where M is a local martingale MF and A is a PCAF of X. As ΔZ_{t} = Z_{t−}·ΔM_{t}, we have Z_{t} = Z_{t−}(1+ΔM_{t}) = Z_{t−}(1 + φ(X_{t−}, X_{t})). By an argument used in the proof of Lemma 2.9, one sees that (N^{Y} , H) is a Lévy system of Y, where N^{Y} (x, dy) := (1 + φ(x, y))N(x, dy). Hence the jump measure of Y is

J^{Y} (dx, dy) = ρ(x)^{2}(1 + φ(x, y))J(dx, dy) = γ(x, y)J(dx, dy)  (4.8)

(cf. [5; Lemma 4.4]).

To find the killing measure κ^{Y} of Y, note that ℚ = (ℚ_{x} : x ∈ E) and ℙ = (ℙ_{x} : x ∈ E) are related by first making a Girsanov transformation using Exp(M) and then a Feynman-Kac transformation using Exp(−A). Hence κ^{Y} is the Revuz measure for the PCAF

(1_{{·≥ζ_{i}}})^{p}(ℚ) = (1_{{·≥ζ_{i}}})^{p}(ℙ) + ⟨1_{{·≥ζ_{i}}}, M⟩(ℙ) + A
= (1_{{·≥ζ_{i}}})^{p}(ℙ) + (ΔM_{ζ_{i}}1_{{·≥ζ_{i}}})^{p}(ℙ) + A
= (1_{{·≥ζ_{i}}})^{p}(ℙ) + (φ(X_{ζ_{i}−}, Δ)1_{{·≥ζ_{i}}})^{p}(ℙ) + A

So

κ^{Y} (dx) = ρ(x)^{2}(1 + φ(x, Δ))κ(dx) + ρ(x)^{2}μ_{A}(dx).  (4.9)

The above discussion proves half of the theorem to follow. Let (E^{Y} , F^{Y}) be the Dirichlet form for process (Y, ℚ). From its probabilistic characterization, {F_{k}}_{k≥1} is an E^{Y} -nest if and only if it is an E-nest. So we can choose E^{Y} -nest {F_{k}}_{k≥1} of compact sets (see, e.g., VI.3 of [19]) such that 1/k ≤ ρ ≤ k q.e. on F_{k}.

Theorem 4.4. Let (E^{Y} , F^{Y}) and {F_{k}}_{k≥1} be as above. Then

⋃_{k=1}^{∞} F_{F_{k}} ∩ L^{2}(γ · J) ∩ L^{2}(κ^{Y}) ⊂ F^{Y}  (4.10)

and for f in the set on the left side of (4.10),

E^{Y} (f, f) = \frac{1}{2} ∫ ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + ∫ (f(x) − f(y))^{2}J^{Y} (dx, dy) + ∫ f(x)^{2}κ^{Y} (dx),  (4.11)

with J^{Y} and κ^{Y} given by (4.8) and (4.9). Proof. To calculate the continuous part of the energy measure, we proceed as in Section 2, using the method of forward-backward martingale decompositions together with martingale theory. Define ℚ̃ = (ℚ̃_{x} : x ∈ E) by dℚ̃_{x}/dℙ_{x}|_{M_{t}∩{t<ζ_{p}}} = Exp(M_{t}). Then (X, ℚ̃) is ν-symmetric. For f ∈ bF_{F_{k}} ∩ L^{2}(κ^{Y}) ∩ L^{2}(γ · J), by the Lyons-Zheng forward-backward martingale decomposition,

f(X_{t}) − f(X_{0}) = \frac{1}{2} (M_{t}^{f} − M_{t}^{f} ∘ r_{t}) ℙ_{m}-a.s. on \{t < ζ\},

where M_{t}^{f} is the martingale part in Fukushima’s decomposition of f(X_{t}) − f(X_{0}). Hence

K_{t} := M_{t}^{f} − ⟨M^{f}, M⟩_{t}, t < ζ_{p},

is a local martingale AF under ℚ̃ and

[K]_{t}(ℚ̃) = [M^{f}]_{t}(ℙ)  ℙ̂_{x}-a.s. for t < ζ_{p}.

Hence for t < ζ_{p},

⟨K⟩_{t}(ℚ̃) = ⟨M^{f}⟩_{t}(ℙ) + ⟨M^{f}, M⟩_{t}(ℙ)
= ⟨M^{f}⟩_{t}(ℙ) + ∫_{0}^{t} ∫_{E_{Δ}}(f(X_{s}) − f(y))^{2}φ(X_{s}, y)N(X_{s}, dy)dH_{s}.

So the Revuz measure of ⟨K⟩ with respect to (X, ℚ̃) is

ρ(x)^{2}μ_{⟨f⟩}(dx) + 2ρ(x)^{2} ∫_{y∈E}(f(x) − f(y))^{2}φ(x, y)J(dx, dy) + ρ(x)^{2}φ(x, Δ)f(x)^{2}κ(dx)
= ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + 2 ∫_{y∈E}(f(x) − f(y))^{2}J^{Y} (dx, dy) + ρ(x)^{2}(1 + φ(x, Δ))f(x)^{2}κ(dx).

Let (Ê, F̂) (resp. (E^{Y} , F^{Y})) be the Dirichlet form for the process (X, ℚ̃) (resp. (Y, ℚ)). As (Y, ℚ) is obtained from (X, ℚ̃) through Feynman-Kac transform by Exp(−A), F^{Y} = F̂ ∩ L^{2}(ϕ^{2}μ_{A}) and F^{Y} (f, f) = Ê(f, f) + ∫ f(x)^{2}ϕ(x)^{2}μ_{A}(dx). Because f = 0 q.e. on F_{k}^{c} for some k ≥ 1, an argument similar to that used in the proof of Theorem 2.6 (between (2.14) and (2.15)) shows that f ∈ F̂. Applying Feynman-Kac, one has f ∈ F^{Y} with

E^{Y} (f, f) = \frac{1}{2} ∫ ρ(x)^{2}μ_{⟨f⟩}^{c}(dx) + ∫ (f(x) − f(y))^{2}J^{Y} (dx, dy) + ∫ f(x)^{2}κ^{Y} (dx),

where J^{Y} and κ^{Y} are given by (4.8)-(4.9). □

In the remainder of this section, we will focus on the special case in which the supermartingale Z_{t} in (4.2) is of pure jump type. That is, in the expression (3.2) for Z, we assume that M^{c} = 0, A = 0, φ is symmetric on E × E with φ > −1 on E × E and φ(x, Δ) = 0, so Z is strictly positive on [[0, ζ_{p} and t ↦ Z_{t} is continuous at ζ_{i}. As a consequence of Theorem 3.1, the integrability condition (3.3) holds.

Corollary 4.5. In the setting of Theorem 4.4, suppose there are real constants c_{1} and c_{2} such that −1 < c_{1} ≤ φ(x, y) ≤ c_{2} for all x, y ∈ E. Then F^{Y} = F and (4.11) holds for all f ∈ F.

Proof. By Theorem 4.4, F ⊂ F^{Y} and (4.11) holds for all f ∈ F. Moreover, as in the discussion at (2.18), we have dℙ_{x}|_{M_{t}}/dℙ̂_{x}|_{M_{t}} = Ẑ_{t}, where Ẑ is the exponential local martingale MF of Y determined by the purely discontinuous local martingale M̂ with M̂_{t} − M̂_{t−} = φ̂(Y_{t−}, Y_{t}), and φ̂ := −φ/(1 + φ). In short, X can be recovered from Y by a pure-jump Girsanov transformation of the same type that led from X to Y. A second application of Theorem 4.4, in which the roles of X and Y are reversed, shows that F^{Y} ⊂ F.

□

We now suppose that

∫_{0}^{t} N(|φ|)(X_{s}) dH_{s} < ∞,  ∀t ∈ [0, ζ[, ℙ_{m}-a.s.  (4.12)

Thus

M_{t} = ∑_{s≤t} φ(X_{s−}, X_{s}) − (Nφ ∗ H)_{t},

the infinite series converging absolutely for each t ∈ [0, ζ[, ℙ_{m}-a.s. Consequently, the local martingale AF M has paths of locally finite variation. In particular, Z can be expressed as

Z_{t} = ∏_{s≤t}(1 + φ(X_{s−}, X_{s})) exp(−(Nφ ∗ H)_{t}),  ∀t ∈ [0, ζ[,  (4.13)

the infinite product being absolutely convergent.

Conversely, if ϕ : E ×E →]−1, ∞[ is symmetric, ϕ(x, Δ) := 0 and (4.12) holds, then Z_{t} in (4.13) defines a positive local martingale MF. Under the family of measures ℚ = (ℚ_{x} : x ∈ E) defined by dℚ_{x}/dℙ_{x}|_{F_{t}} = Z_{t}, the process Y := (X, ℚ) is an m-symmetric Markov process whose law is locally absolutely continuous with respect to that of X. So we can just start with such a ϕ and construct the symmetric process Y in this way. We will now identify the Dirichlet space of Y.

Let φ^{+} := φ ∨ 0 and φ^{−} := (−φ) ∨ 0. Then 0 ≤ φ^{−} < 1, φ = φ^{+} − φ^{−} and 1 + φ = (1 + φ^{+})(1 − φ^{−}) on E × E. Define, for t ∈ [0, ζ[,

\begin{aligned}Z_{t}^{+} & := ∏_{s≤t}(1 + φ^{+}(X_{s−}, X_{s})) exp (∫_{0}^{t} Nφ^{−}(X_{s})dH_{s}); \\ Z_{t}^{−} & := ∏_{s≤t}(1 − φ^{−}(X_{s−}, X_{s})) exp (− ∫_{0}^{t} Nφ^{+}(X_{s})dH_{s}) . \end{aligned}  (4.14)

Clearly Z^{+} is increasing and Z^{−} is decreasing. Both Z^{+} and Z^{−} are MFs that are finite and strictly positive on [[0, ζ and Z = Z^{+} · Z^{−}. Let W = (W_{t}, ℙ_{x}^{W}) be the subprocess of (X, Z^{−}) (“X killed via the MF Z^{−}”). It is easy to see that W coincides with the subprocess of (Y, 1/Z^{+}).

The AF \frac{1}{2} ∑_{s≤t} φ(X_{s−}, X_{s}) is of bounded variation on compact subintervals of [0, ζ[. We write ν := φ · J for its bivariate Revuz measure, and define μ := 2ν(1 ⊗ ·). As before,

L^{2}(ν) := {u ∈ L^{2}(m) : u has a quasi-continuous version ũ

such that ∫_{E×E} \overline{u}(x, y)^{2} ν(dx, dy) < ∞\}. (4.15)

Of course, each element of F has a quasi-continuous version. Hence we can define

\begin{aligned}F^{ν} & := F ∩ L^{2}(ν) = \{u ∈ F : ∫_{E×E} \overline{u}(x, y)^{2} ν(dx, dy) < ∞\}; \\ E^{ν}(u, u) & := E(u, u) + ∫_{E×E} \overline{u}(x, y)^{2} ν(dx, dy), u ∈ F^{ν}. \end{aligned}  (4.16)

By Fatou’s lemma, (E^{ν}, F^{ν}) is a symmetric closable quadratic form on L^{2}(m) if φ ≥ c_{0} > −1.

Theorem 4.6. (i) Let φ : E × E →] − 1, ∞[ be a symmetric Borel function such that (4.12) holds. If φ(x, y) ≥ 0 for all x, y ∈ E, then

F ∩ L^{2}(μ) ⊂ F^{Y} ⊂ F^{ν},

where F ∩ L^{2}(μ) is dense in F^{Y} with respect to the E_{1}^{Y} -norm, and for u ∈ F^{Y},

E^{Y} (u, u) = E(u, u) + ∫_{E×E}(u(y) − u(x))^{2}ν(dx, dy).  (4.17)

(ii) If, in addition, φ is J-integrable, then F^{Y} = F^{ν}.

Proof. (i) Using the notation in (4.14), we now have (in view of the non-negativity of φ)

Z_{t}^{+} = ∏_{s≤t}(1 + φ(X_{s−}, X_{s})), Z_{t}^{−} = exp (− ∫_{0}^{t} Nφ(X_{s})dH_{s}) ,

and μ is the Revuz measure of Nφ ∗ H. Hence the Dirichlet form (E^{W} , F^{W}) associated with W and m is given by

\begin{aligned}F^{W} & = F ∩ L^{2}(μ); \\ E^{W} (u, u) & = E(u, u) + μ(u^{2}), u ∈ F^{W} . \end{aligned}  (4.18)

But W is also the subprocess (Y, 1/Z^{+}) of Y, so by (4.8) the bivariate Revuz measure of 1/Z^{+} computed with respect to Y and m is φ(1 + φ)^{−1} · J^{Y}, which is nothing but ν. It follows from [27; Theorem II.3.10] that

\begin{aligned}F^{W} & = F^{Y} ∩ L^{2}(ν); \\ E^{W} (u, u) & = E^{Y} (u, u) + ν(u ⊗ u), u ∈ F^{W} . \end{aligned}  (4.19)

Combining (4.18) and (4.19), and noting that μ is smooth with respect to both X and Y, we find that F ∩ L^{2}(μ) is contained in F^{Y} and is dense with respect to the E_{1}^{Y} -norm, and for u ∈ F ∩ L^{2}(μ), the formula (4.17) holds.

Assume that u ∈ F^{Y}. We may choose a sequence {u_{n}} ⊂ F ∩ L^{2}(μ) such that u_{n} ⟶ u in E_{1}^{Y} -norm. Then {u_{n}} is an E_{1}^{Y} -Cauchy sequence and by the result above it is also an E_{1}-Cauchy sequence. Therefore u ∈ F and u_{n} ⟶ u in E_{1}-norm and quasi-everywhere (at least along a suitable subsequence). Invoking Fatou’s lemma we have

E(u, u) + ν(ū^{2}) ≤ lim_{n}(E(u_{n}, u_{n}) + ν(ū_{n}^{2})) = E^{Y} (u, u) < ∞.

It follows that F^{Y} ⊂ F^{ν}. As ∪_{k=1}^{∞}F_{F_{k}} ∩ L^{2}(μ) is E + (·, , ·)_{L^{2}(μ)}-dense in F ∩ L^{2}(μ), by Theorem 4.4, (4.17) holds for u ∈ F^{Y}. [Here the nest {F_{k}} is as in the statement of Theorem 4.4.]

(ii) We now assume that φ is J-integrable. For any u ∈ F^{ν}, set u_{n} := (u ∧ n) ∧ (−n). Then the Jintegrability of φ guarantees u_{n} ∈ F ∩ L^{2}(μ) and u_{n} ⟶ u in E_{1}-norm and q.e.. Since |ū_{n}| ≤ |ū|, we may appeal the dominated convergence theorem and get u_{n} ⟶ u in E_{1}^{ν}-norm. This implies that {u_{n}} ⊂ F^{Y} is an E_{1}^{Y} -Cauchy sequence. Therefore u ∈ F^{Y}. □

In the more general case where (4.12) holds but φ is not necessarily positive, we have a weaker result by a similar approach. Let ν^{+} := φ^{+} · J, ν^{−} := φ^{−} · J, and let μ^{+} = ν^{+}(1 ⊗ ·), μ^{−} = ν^{−}(1 ⊗ ·) be the second marginal measures of ν^{+} and ν^{−} respectively. By [27; Theorem I.4.6] the bivariate Revuz measure of Z^{−}, computed with respect to X and m, is

\begin{aligned}ν_{Z^{−}}(dx, dy) & = φ^{−}(x, y)J(dx, dy) + Nφ^{+}(x)μ_{H}(dx)δ_{\{x\}}(dy) \\ & = ν^{−}(dx, dy) + μ^{+}(dx)δ_{\{x\}}(dy) \end{aligned}

and the bivariate Revuz measure of 1/Z^{+} computed with respect to (Y, m) is

ν_{1/Z^{+}}^{Y}(dx, dy) = \frac{φ^{+}}{1 + φ^{+}} J^{Y} (dx, dy) + Nφ^{−}(x)μ_{H}(dx)δ_{\{x\}}(dy)
= φ^{+}(1 − φ^{−})J(dx, dy) + μ^{−}(dx)δ_{{x}}(dy)
= ν^{+}(dx, dy) + μ^{−}(dx)δ_{{x}}(dy).

It is now clear that

ν_{Z^{−}}(1 ⊗ ·) = ν_{1/Z^{+}}^{Y}(1 ⊗ ·) = |μ|.

Hence we have by [27; Theorem II.3.10]

\begin{aligned}F^{W} & = F ∩ L^{2}(|μ|) \\ E^{W} (u, u) & = E(u, u) + ν_{Z^{−}}(u ⊗ u), u ∈ F^{W} , \end{aligned}  (4.20)

and also

\begin{aligned}F^{W} & = F^{Y} ∩ L^{2}(|μ|) \\ E^{W} (u, u) & = E^{Y} (u, u) + ν_{1/Z^{+}}^{Y}(u ⊗ u), u ∈ F^{W} . \end{aligned}  (4.21)

Combining (4.20) and (4.21) we have the following theorem.

Theorem 4.7. Let φ : E × E →] − 1, ∞[ be a symmetric Borel function such that (4.12) holds. Then F ∩ L^{2}(|μ|) is densely contained in F^{Y} and for u ∈ F ∩ L^{2}(|μ|) we have

E^{Y} (u, u) = E(u, u) + \frac{1}{2} ∫_{0}^{t}(u(y) − u(x))^{2}ν(dx, dy).  (4.22)

Moreover if φ^{+} = 0, then F ⊂ F^{Y} ∩ L^{2}(φ^{−} · J) and (4.22) holds for all u ∈ F.

Proof. It only remains to show the last assertion. In this case, as φ^{+} = 0,

\frac{1}{Z_{t}} = ∏_{s≤t} (1 + \frac{φ^{−}(Y_{s−}, Y_{s})}{1 − φ^{−}(Y_{s−}, Y_{s})}) exp (− ∫_{0}^{t} N^{Y} \frac{φ^{−}}{1 − φ^{−}} (Y_{s}) dH_{s}) .  (4.23)

But φ^{−}/(1 − φ^{−}) = φ^{−}/(1 + φ) ≥ 0, so the last assertion follows from Theorem 4.5 with the roles of X and Y interchanged. □

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