The Dirichlet Form of a Gradient-type Drift Transformation of a Symmetric Diffusion

by

P. J. Fitzsimmons

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

June 30, 2006

ABSTRACT

In the context of a symmetric diffusion process X admitting a carré du champs operator, we give a precise description of the Dirichlet form of the process obtained by subjecting X to a drift transformation of gradient type. This description relies on boundary-type conditions restricting an associated reflecting Dirichlet form.

Key words and phrases. Diffusion, Dirichlet form, Girsanov theorem, drift perturbation, Markovian extension, uniqueness.

1990 AMS Subject classification. Primary 60J60; secondary 31C25, 60J35

Running Head: Gradient-type Drift Transformations 1. Introduction Let X = (X_{t}, P^{x}) and Y = (Y_{t}, Q^{x}) be symmetric (i.e., reversible) Markov diffusion processes with respective symmetry measures m and μ on a common state space E. Let ζ denote the lifetime of either process. It is shown in [6; §3] that if

(1.1)  Q^{μ}|_{F_{t}∩{t<ζ}} ≪ P^{m}|_{F_{t}∩{t<ζ}},  ∀t ≥ 0,

then the associated Radon-Nikodym density process L_{t} := dQ^{μ}|_{F_{t}∩{t<ζ}}/dP^{m}|_{F_{t}∩{t<ζ}} has the form

(1.2)  L_{t} = exp(M_{t}^{ℓ} − \frac{1}{2}⟨M^{ℓ}⟩_{t})1_{\{t<T_{N} \}},

where N is an X-finely closed subset of E, ℓ: = \frac{1}{2} log(dμ/dm) is locally in the Dirichlet space of (X, T_{N}) (the process X killed at the hitting time T_{N} of N), and M^{ℓ} is the local martingale continuous additive functional of (X, T_{N}) associated with ℓ by Fukushima’s decomposition.

Conversely, given an X-finely closed set N_{0} and a function ℓ locally in the Dirichlet space of (X, T_{N_{0}}), we can set N = N_{0} ∪ {ℓ̃ ∉ R}, and then use the right side of (1.2) to define a process L = (L_{t}). (Here and in what follows, the tilde indicates that a quasi-continuous m-version of the function has been selected.) It is standard that L so defined is a supermartingale multiplicative functional of X and a local martingale on [0, T_{N}[. The formulae Q^{x}|_{F_{t}∩{t<ζ}}: = L_{t} · P^{x}|_{F_{t}∩{t<ζ}}, x ∈ E \ N, then determine the law of a diffusion Y with state space E \ N and symmetry measure μ := ψ^{2} · m, where ψ := exp(ℓ). Evidently, Q^{μ} ≪_{loc} P^{m} in the sense of (1.1).

The Dirichlet forms (E^{X}, D^{X}) and (E^{Y} , D^{Y}) of X and Y are easily related, at least formally. To this end recall that with each pair (u, v) of elements of D^{X} one can associate a signed measure μ_{⟨u,v⟩}^{X} (the mutual energy measure of u and v) such that the mapping (u, v) ↦ μ_{⟨u,v⟩}^{X} is symmetric and bilinear, and

(1.3)  E^{X}(u, v) = \frac{1}{2}μ_{⟨u,v⟩}^{X}(E),  ∀u, v ∈ D^{X}.

As is customary, we write μ_{⟨u⟩} for μ_{⟨u,u⟩}. Now let (G_{n}) be an increasing sequence of X-finely open subsets of E \ N such that (i) E \ N differs from ∪_{n}G_{n} by an X-polar set, (ii) ℓ̃|_{G_{n}} is bounded for all n ∈ N, and (iii) μ_{⟨ℓ⟩}(G_{n}) < ∞ for all n. (The properties of ℓ noted earlier ensure that μ_{⟨ℓ⟩} can be defined on Borel subsets of E \ N by a localization argument. These same properties imply the existence of a sequence (G_{n}) with the listed properties.) It follows from [6; (4.9)] that the vector space

(1.4)  C := ∪_{n}{u ∈ D^{X} ∩ L^{∞}(m) : ũ = 0 quasi-everywhere on E \ G_{n} }

is dense in D^{Y} and that

(1.5)  E^{Y} (u, v) = \frac{1}{2} ∫_{E\N} ψ̃^{2} dμ_{⟨u,v⟩}^{X},  ∀u, v ∈ C.

In fact, μ_{⟨u,v⟩}^{Y} = ψ̃^{2}μ_{⟨u,v⟩}^{X} for u, v ∈ C.

It is desirable to specify a “core” for D^{Y} in simpler terms; e.g., without reference to a “nest” such as (G_{n}). The work of Eberle [3] shows that this is indeed possible under some restrictions on ℓ, at least when X admits a carré du champs operator. Our goal in this paper is to extend and amplify the results of Eberle, in a general context. A special case of our main result can be stated as follows. Let N and ℓ be as in the first paragraph of this introduction, and put ψ = exp(ℓ) as before. Recall that the symmetry measure of Y is μ = ψ^{2} · m. Fix 0 < p < ∞, define

(1.6)  C_{p}^{ψ} := {u ∈ D^{X} ∩ L^{∞}(m) : u ∈ L^{2}(ψ^{2} · m), ψ̃ ∈ L^{2}(μ_{⟨u⟩}^{X}), ũ ∈ L^{p}(μ_{⟨ψ⟩}^{X})},

and let E^{ψ}(u, v) be defined on C_{p}^{ψ} by the right side of (1.5). We shall prove that the symmetric bilinear form (E^{ψ}, C_{p}^{ψ}) is closable, and that its closure is precisely (E^{Y} , D^{Y}).

The condition ũ ∈ L^{p}(μ_{⟨ψ⟩}^{X}) in (1.6) may be thought of as an approximate boundary condition. Suppressing this condition we obtain C_{0}^{ψ} := {u ∈ D^{X} ∩ L^{∞}(m) : u ∈ L^{2}(ψ^{2} · m), ψ̃ ∈ L^{2}(μ_{⟨u⟩}^{X})}. The right side of (1.5) still serves to define E^{ψ} on C_{0}^{ψ}; the form so defined is closable and its closure (E_{0}^{ψ}, D_{0}^{ψ}) is an extension of (E^{Y} , D^{Y}). In particular, D^{Y} ⊂ D_{0}^{ψ}, and this containment can be strict as we see by taking X to be the 3-dimensional Bessel process on E =]0, ∞[ and Y the absorbed Brownian motion on E. See (3.9) for details.

The work of [6] has been extended to general symmetric Markov processes in [2], and it seems likely that results similar to those found in this paper can be obtained in that context.

In section 2 we describe in detail the setting of the paper and we record some preliminary results. Section 3 contains the proof of the main result outlined above.

2. Preliminaries Let (E, B(E)) be a Lusin metrizable topological space; i.e., E is homeomorphic to a Borel

subset of some compact metric space and B(E) is the class of Borel sets in E. Let m be a σ-finite measure on E and let (E, D) be a quasi-regular [13; IV.3] Dirichlet form on L^{2}(m). We assume that (E, D) is strongly local in the sense that

(2.1) F, G ∈ C_{c}^{∞}(R), F constant on the support of G ⟹ E(F_{0}^{∘}u, G_{0}^{∘}u) = 0, ∀u ∈ D,

where F_{0} := F − F (0), G_{0} := G − G(0). Under these conditions there is a right Markov process X = (Ω, F, F_{t}, X_{t}, θ_{t}, P^{x}) that is symmetric with respect to m and properly associated with (E, D). Consequently, if (P_{t})_{t≥0} denotes the transition semigroup of X, then

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

where (u, v)_{m} := ∫_{E} uv dm is the natural inner product in L^{2}(m). Viewed as a semigroup operating in L^{2}(m), (P_{t}) is strongly continuous, and P_{t}f is B(E)-measurable for each bounded B(E)- measurable function f: E → R. Conditions (2.1) and (2.2) imply that X is a diffusion in the following strong sense:

(2.3)(i) The P^{m}-completion (F_{t})_{t≥0} of the natural filtration σ{X_{s}; 0 ≤ s ≤ t}, t ≥ 0, is quasi-left-continuous and the lifetime of X, denoted ζ, is an (F_{t}) predictable stopping time;

(2.3)(ii) t ↦ X_{t} is continuous on [0, ζ [, P^{m}-a.s.

Consequently, every (F_{t})-stopping time is predictable, and every (F_{t})-martingale has continuous paths (P^{m}-a.s.). See [14; §47] and [13; IV.3].

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 that hold the value Δ ∉ E after time ζ(ω). As usual, any function f defined on E is automatically extended to the cemetery state Δ by the convention f(Δ) = 0.

The Dirichlet form (E, D) is related to the transition semigroup (P_{t}) by

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

E(u, v) = lim_{t→0} \frac{1}{t}(u, v − P_{t}v)_{m}, u, v ∈ D.

Endowed with the inner product E_{1}(u, v) := E(u, v) + (u, v)_{m}, D is a Hilbert space.

Each element u ∈ D admits an m-modification ũ (a quasi-continuous version) such that t ↦ ũ(X_{t}) is continuous on [0, ∞ [, P^{m}-a.s. We then have Fukushima’s decomposition [8; Thm. 5.2.2]:

(2.4)  ũ(X_{t}) − ũ(X_{0}) = M_{t}^{u} + N_{t}^{u}, ∀t ≥ 0, P^{x}-a.s. for q.e. x ∈ E,

where M^{u} and N^{u} are continuous additive functionals (CAFs) of X, M^{u} is a martingale such that sup_{t>0} t^{−1}P^{m}[(M_{t}^{u})^{2}] < ∞, and lim_{t→0} t^{−1}P^{m}[(N_{t}^{u})^{2}] = 0. This decomposition is unique. In the sequel we shall refer to M^{u} as the martingale part of u.

Let τ(G) denote the first exit time from G ⊂ E; that is, τ(G) := inf{t > 0 : X_{t} ∉ G}. A function u: E → R is locally in D (notation: u ∈ D_{loc} ) provided there is an increasing sequence {G_{n}} of finely open sets with lim_{n} τ(G_{n}) = ζ, P^{m}-a.s., and a sequence {u_{n}} of elements of D such that u = u_{n} a.e.-m on G_{n} for each n ∈ N. Each u ∈ D_{loc} admits a quasi-continuous modification ũ such that (2.4) holds for t ∈ [0, ζ [, where now M^{u} is a local martingale CAF and N^{u} is a CAF locally of zero energy. See [8; §5.5]. For convenience, in the sequel we will always take u ∈ D_{loc} to be represented by its quasi-continuous version, and will drop the ˜ from the notation.

Given u ∈ D_{loc}, the local martingale CAF M^{u} admits a quadratic variation process ⟨M^{u}⟩; i.e., ⟨M^{u}⟩ is predictable and (M^{u})^{2} − ⟨M^{u}⟩ is a local martingale on [0, ζ[. Since X is a diffusion, ⟨M^{u}⟩ is a CAF. The Revuz measure of ⟨M^{u}⟩ (the so-called energy measure of u) is the smooth measure μ_{⟨u⟩} on E determined by

(2.5)  μ_{⟨u⟩}(f) =↑ lim_{t→0} t^{−1}P^{m} ∫_{0}^{t} f(X_{s}) d⟨M^{u}⟩_{s},  f ∈ pB(E).

If u ∈ D then μ_{⟨u⟩} has finite total mass; indeed,

E(u, u) = \frac{1}{2}μ_{⟨u⟩}(E), u ∈ D.

Given two elements u and v of D_{loc}, the quadratic covariation ⟨M^{u}, M^{v}⟩ is a (signed) CAF of X and since \frac{1}{4}[⟨M^{u+v}⟩ − ⟨M^{u−v}⟩], we have

(2.6)  μ_{⟨u,v⟩} = \frac{1}{4}[μ_{⟨u+v⟩} − μ_{⟨u−v⟩}].

(This formula can be taken literally if both u and v lie in D. In general there is a sequence {G_{n}} of finely open sets with P^{m}[lim_{n} τ(G_{n}) < ζ] = 0, and sequences {u_{n}}, {v_{n}} from D with u = u_{n} and v = v_{n} m-a.e. on G_{n} for each n; things being so, the two sides of (2.6) determine the same finite signed measure on each G_{n}.) We then have

(2.7)  E(u, v) = \frac{1}{2}μ_{⟨u,v⟩}(E),  ∀u, v ∈ D.

It can be shown [6; §5] that (u, v) ↦ μ_{⟨u,v⟩} is a continuous mapping of D × D into the space of finite signed measures on E endowed with the total variation norm. (Convergence in D is with respect to the norm (E_{1})^{1/2}.) See [8; Ch. 1] for further details.

If F : R → R is locally absolutely continuous with F′ ∈ L_{loc}^{∞}(R) and if u ∈ D_{loc}, then F^{∘}u ∈ D_{loc} and by Itô’s formula ⟨M^{F ∘u}⟩_{t} = ∫_{0}^{t}[F′^{∘}u]^{2}(X_{s}) d⟨M^{u}⟩_{s}. From this and (2.5) it follows that

(2.8)  μ_{⟨F ∘u⟩} = [F′^{∘}u]^{2}μ_{⟨u⟩}.

Polarizing the special case F (t) = t^{2} of (2.8), we see that if u, v ∈ D_{loc} (in which case uv ∈ D_{loc}), then μ_{⟨uv⟩} = u^{2}μ_{⟨v⟩} + 2uvμ_{⟨u,v⟩} + v^{2}μ_{⟨u⟩}. Also, the positivity of μ_{⟨u+tv⟩} = μ_{⟨u⟩} + 2tμ_{⟨u,v⟩} + t^{2}μ_{⟨v⟩} for all t ∈ R implies that |μ_{⟨u,v⟩}(f)| ≤ μ_{⟨u⟩}(f)^{1/2}μ_{⟨v⟩}(f)^{1/2} for any f ∈ L^{1}(μ_{⟨u⟩} + μ_{⟨v⟩}). These remarks yield the useful estimate:

(2.9)  μ_{⟨uv⟩} ≤ 2[u^{2}μ_{⟨v⟩} + v^{2}μ_{⟨u⟩}],  u, v ∈ D_{loc}.

We shall use M to denote the class of martingale CAFs of finite energy. More precisely, define the mutual energy e(M, M′) of two martingale CAFs M and M′ by

e(M, M′) := lim_{t↓0}(2t)^{−1}P^{m}[M_{t}M′_{t}] = lim_{t↓0}(2t)^{−1}P^{m}⟨M, M′⟩_{t},

whenever the limit exists, and write e(M) = e(M, M). Then

M := {M :M is a CAF of X, P^{x}[M_{t}^{2}] < ∞ and
P^{x}[M_{t}] = 0 for quasi-every x ∈ E, e(M) < ∞}.

When equipped with the inner product (M, M′) ↦ e(M, M′), M becomes a Hilbert space; the associated norm is e(·)^{1/2}.

It can be shown that the Dirichlet form (E_{ref}, D_{ref}) discussed below coincides with the active reflected Dirichlet from associated with (E, D) as in [1]. Since we shall only need the fact that the active reflected Dirichlet space is an extension of (E_{ref}, D_{ref}) (and this assertion is obvious from the definitions), we shall not prove this claim. The reader is referred to [1] and [10] for more on these matters.

(2.10) Proposition. The form defined by

D_{ref} := {u ∈ D_{loc} ∩ L^{2}(m) : μ_{⟨u⟩}(E) < ∞},
E_{ref}(u, v) := \frac{1}{2}μ_{⟨u,v⟩}(E), u, v ∈ D_{ref},

is a strongly local Dirichlet form on L^{2}(m). Moreover, bD is an algebra ideal in bD_{ref}.

Proof. Let us first check that (E_{ref}, D_{ref}) is a Dirichlet form in L^{2}(m). For this it suffices to check that (E_{ref}, D_{ref}) is closed. Let {u_{n}} ⊂ D_{ref} be an (E_{ref})_{1}-Cauchy sequence with u_{n} → u in L^{2}(m). Let M^{n} denote the (local) martingale part in the Fukushima decomposition of u_{n}. The quadratic variation of M^{n} satisfies

P^{m}[⟨M^{n}⟩_{t}] ≤ 2tE_{ref}(u_{n}, u_{n}),

from which it follows that M^{n} ∈ M. Moreover, e(M^{n} −M^{k}) = E_{ref}(u_{n} −u_{k}, u_{n} −u_{k}) so that (M^{n}) is a Cauchy sequence in M. Thus, there exists M ∈ M such that e(M^{n} − M) → 0 as n → ∞. According to [8; Thm. 5.2.1], this means that sup_{0≤s≤t} |M_{s}^{n} − M_{s}| → 0 as n → ∞, in L^{2}(P^{m}), for all t > 0. Now the Lyons-Zheng decomposition [12, 11], as developed in [5], asserts that

(2.11)  u_{n}(X_{t}) − u_{n}(X_{0}) = [M_{t}^{n} − M_{t}^{n}^{∘}r_{t}]/2,  P^{m}-a.s. on {t < ζ}, ∀t > 0.

Here, for t > 0, r_{t} : {t < ζ} → Ω denotes the time-reversal operator on [0, t]:

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

[The m-symmetry of X is equivalent to the assertion that for each t > 0, r_{t} is a measure-preserving transformation of the measure space ({t < ζ}, F_{t} ∩ {t < ζ}, P^{m}[·; t < ζ)].] Passing to the limit in (2.11) (along a subsequence if necessary), using the measure-preserving property of r_{t}, we find that for fixed t > 0,

(2.12)  u(X_{t}) − u(X_{0}) = [M_{t} − M_{t}^{∘}r_{t}]/2,  on {t < ζ}, P^{m}-a.s.

Using [6; (4.15)] we can deduce from (2.12) that u ∈ D_{loc} and that M is the martingale part in the Fukushima decomposition of u. This yields μ_{⟨u⟩}(E) = 2e(M) < ∞ so that u ∈ D_{ref}. Thus, (E_{ref}, D_{ref}) is closed.

To check the asserted ideal property, we need to show that gu ∈ D whenever g ∈ bD and u ∈ bD_{ref}. But for t > 0 we have

(2.13)  \begin{aligned}t^{−1}(gu, gu − P_{t}(gu))_{m} = (2t)^{−1} & P^{m}[(gu(X_{t}) − gu(X_{0}))^{2}; t < ζ] \\ & + t^{−1}P^{m}[(gu(X_{0}))^{2}; ζ ≤ t]. \end{aligned}

Since u is bounded, the second term on the right side of (2.13) is dominated by a multiple of t^{−1}P^{m}[(g(X_{0}))^{2}; ζ ≤ t], which remains bounded as t → 0 because g ∈ D. To treat the first term on the right in (2.13), note that gu ∈ D_{loc}. The Lyons-Zheng decomposition now yields the estimate

(2t)^{−1}P^{m}[(gu(X_{t}) − gu(X_{0}))^{2}; t < ζ] ≤ Cμ_{⟨gu⟩}(E) ≤ 2C [∥u∥_{∞}^{2}E(g, g) + ∥g∥_{∞}^{2}E_{ref}(u, u)] < ∞.

Thus, sup_{t>0} t^{−1}(gu, gu − P_{t}(gu))_{m} < ∞. Since gu ∈ L^{2}(m), we have shown that gu ∈ D.

3. Drift Transformations Our first task is to discuss the Girsanov transformation of X induced by ψ ∈ pD_{loc}^{X}. Fix

ψ ∈ pD_{loc}^{X} and let N_{0} be a properly exceptional set for (X, m) ([8; p. 134]) such that the Fukushima decomposition

(3.1) ψ(X_{t}) − ψ(X_{0}) = M_{t}^{ψ} + N_{t}^{ψ}, ∀t ≥ 0, P^{x}-a.s., x ∈ E \ N_{0},

holds. We can (and do) assume that ψ is finely continuous and finite-valued on E \ N_{0}. Define N_{1} := {x ∈ E \ N_{0} : ψ(x) = 0} and N = N_{ψ} := N_{0} ∪ N_{1}. Then N is X-finely closed, and ℓ := log ψ is an element of (D_{E\N}^{X})_{loc}, so we have the (local) Fukushima decomposition

(3.2) ℓ(X_{t}) − ℓ(X_{0}) = M_{t}^{ℓ} + N_{t}^{ℓ}, ∀0 ≤ t < T_{N}, P^{x}-a.s., x ∈ E \ N,

where T_{N} := inf{t > 0 : X_{t} ∈ N} is the hitting time of N, M^{ℓ} is a local martingale CAF of (X, T_{N}) and N^{ℓ} is a CAF of (X, T_{N}) locally of zero energy. Here (X, T_{N}) denotes the process X killed at time T_{N} and (E_{E\N}^{X}, D_{E\N}^{X}) is the associated Dirichlet space. The stochastic exponential L_{t} := exp(M_{t}^{ℓ} − \frac{1}{2}⟨M^{ℓ}⟩_{t})1_{t<T_{N}} of M^{ℓ} is a (positive) supermartingale multiplicative functional of (X, T_{N}) and a local martingale on [0, T_{N}[. By standard theory (e.g. [14; § 62]) there is a family (Q^{x})_{x∈E} = (Q_{ψ}^{x})_{x∈E\N} of probability laws on (Ω, F, F_{t}) such that

(3.3) Q^{x}(F; t < ζ) = P^{x}[F · L_{t}; t < ζ], F ∈ bF_{t}, t ≥ 0, x ∈ E \ N.

For emphasis we use (Y_{t})_{t≥0} to denote the coordinate process under the Q^{x}.

Let μ denote the measure ψ^{2} · m. The process Y: = (Y_{t}, Q^{x}) is a μ-symmetric diffusion with state space E_{ψ} := E \ N_{ψ}; see, for example, [5; § 4]. We refer to Y as the Girsanov transformation of X based on ψ. Let (E^{Y} , D^{Y}) denote the Dirichlet form of Y. This is a strongly local Dirichlet form on L^{2}(μ) = L^{2}(μ, E_{ψ}).

The following summary is drawn from [6; §4]. (3.4) Proposition. Fix ψ ∈ pD_{loc}^{X}and write N for N_{ψ}.

(a) Let G be an X-finely open subset of E \ N on which ψ is bounded away from zero and infinity. Furthermore, assume μ_{⟨ψ⟩}(G) < ∞. Then D_{G}^{Y} = D_{G}^{X}, and if u, v ∈ D_{G}^{Y} then E^{Y} (u, v) = \frac{1}{2} ∫_{G} ψ^{2} dμ_{⟨u⟩}.

(b) Let {G_{n}} ⊂ E be an increasing sequence of X-finely open subsets of E \ N such that (i) (E \ N) \ ∪_{n}G_{n} is (X, m)-polar, (ii) μ_{⟨ψ⟩}(G_{n}) < ∞ for all n ∈ N, and (iii) there are constants 0 < C_{n} < ∞ such that C_{n}^{−1} ≤ ψ^{2} ≤ C_{n} on G_{n}, ∀n. Then ∪_{n}D_{G_{n}}^{X} is E_{1}^{Y} -dense in D^{Y}, and

(3.5)  E^{Y} (u, u) = \frac{1}{2} ∫_{E} ψ^{2} μ_{⟨u⟩},  ∀u ∈ ∪_{n}D_{G_{n}}^{X}.
(c) D_{loc}^{Y} = (D_{E_{ψ}}^{X})_{loc}, and μ_{⟨u⟩}^{Y} = ψ^{2}μ_{⟨u⟩} for all u ∈ D_{loc}^{Y}.
(d) Any (X, T_{N})-nest is a Y -nest.

Of course, to apply (3.4)(b) we require the existence of a sequence (G_{n}) as specified there. This matter is addressed in the following (3.6) Lemma. If ψ ∈ pD_{loc}^{X}, then there is a sequence (G_{n}) of X-finely open sets satisfying the conditions listed in (3.4)(b).

Proof. Since ψ ∈ D_{loc}^{Y}, the measure μ_{⟨ψ⟩} is smooth; indeed it is the smooth measure corresponding to the CAF ⟨M^{ψ}⟩. By a standard construction [8; Lemma 5.1.7], there is a bounded 1-excessive function f ∈ D such that {f = 0} is X-exceptional and μ_{⟨ψ⟩}(f) < ∞. The set H_{n} := {f > 1/n} evidently satisfies μ_{⟨ψ⟩}(H_{n}) < ∞, and E \ ∪_{n}H_{n} = {f = 0}. Consequently, the sequence (G_{n}) defined by G_{n} := H_{n} ∩ {n^{−1} < ψ < n} has the required properties.

We now introduce the vector space

C_{0}^{ψ} := {u ∈ D^{X} ∩ L^{∞}(m) : u ∈ L^{2}(μ), ψ ∈ L^{2}(μ_{⟨u⟩})}

on which is defined the symmetric form

(3.7)  E^{ψ}(u, v) := \frac{1}{2} ∫_{E_{ψ}} ψ^{2} dμ_{⟨u,v⟩}.

In view of (2.8), normalized contractions operate on (E^{ψ}, C_{0}^{ψ}), and the form (E^{ψ}, C_{0}^{ψ}) is strongly local.

(3.8) Proposition. Fix ψ ∈ pD_{loc}^{X} and let the form (E^{ψ}, C_{0}^{ψ}) be as defined above. Let Y denote the Girsanov transformation of X based on ψ, with associated Dirichlet form (E^{Y} , D^{Y}).

(a) The form (E^{ψ}, C_{0}^{ψ}) is closable in L^{2}(μ, E_{ψ}), and its closure (E_{0}^{ψ}, D_{0}^{ψ}) is a strongly local Dirichlet form. Moreover, D_{0}^{ψ} ⊂ D_{loc}^{Y} and E_{0}^{ψ} = E^{ψ} on D_{0}^{ψ} × D_{0}^{ψ}.

(b) D^{Y} ⊂ D_{0}^{ψ}, E^{Y} = E^{ψ} on D^{Y} × D^{Y}, and bD^{Y} is an algebra ideal in bD_{0}^{ψ}.

Proof. (a) By (3.4)(c) and (2.11) applied to Y, the form E^{ψ} on the domain

D_{ref}^{Y} := {u ∈ D_{loc}^{Y} ∩ L^{2}(μ) : ψ ∈ L^{2}(μ_{⟨u⟩})}

is closed. Its restriction (E^{ψ}, C_{0}^{ψ}) is therefore closable. Because of [8; Thm. 3.1.1], the closure (E_{0}^{ψ}, D_{0}^{ψ}) of (E^{ψ}, C_{0}^{ψ}) is a Dirichlet form. Also, since (E^{ψ}, D_{ref}^{Y}) is closed, D_{0}^{ψ} ⊂ D_{ref}^{Y} ⊂ D_{loc}^{Y}. For the same reason, E_{0}^{ψ} = E^{ψ} on D_{0}^{ψ} × D_{0}^{ψ}. Finally, E_{0}^{ψ} is strongly local by virtue of (3.7) and the polarization of (2.8).

(b) The inclusion D^{Y} ⊂ D_{0}^{ψ} is an obvious consequence of (3.4)(b). The second assertion follows immediately from (2.10) since D_{0}^{ψ} ⊂ D_{ref}^{Y} by part (a).

The significance of (3.8)(b) is this: According to [15; Thm. 20.1], there is a quasi-regular representation (Ẽ, D̃) of (E^{ψ}, D_{0}^{ψ}) with state space Ẽ such that E_{ψ} is quasi-open in Ẽ, and such that if Ỹ = (Ỹ_{t}, Q̃^{x}) is the symmetric process associated with (Ẽ, D̃) then Ỹ killed at the first exit time from E_{ψ} coincides (in law) with Y. Consequently, D^{Y} may be identified with the set of elements of D̃ that vanish on Ẽ \ E_{ψ}. Our goal is to give a more concrete description of D^{Y} through the use of what might be called “pseudo boundary conditions”.

The reader will find in [3] conditions under which D^{Y} = D_{0}^{ψ}. Roughly speaking, the class of drifts admitted for consideration in [3] is restricted to those for which localization is needed only at “metric infinity”. Here is an example to show that the inclusion D^{Y} ⊂ D_{0}^{ψ} may be strict.

(3.9) Example. Let X be a three-dimensional Bessel process; that is, X is the radial part of a three dimensional Brownian motion. We take the state space of X to be ]0, +∞[. The associated Dirichlet form is

E^{X}(u, v) = \frac{1}{2} ∫_{0}^{∞} u′(x)v′(x) x^{2} dx

acting on the class D^{X} of all locally absolutely continuous functions u: ]0, ∞[→ R such that both u and u′ are square integrable with respect to the symmetry measure m(dx) = x^{2} dx. Now take ψ(x) = 1/x ∈ D_{loc}^{X}. In this case the process Y is just the absorbed Brownian motion on ]0, ∞[, so D^{Y} is the class of all locally absolutely continuous functions u: ]0, ∞[→ R such that (i) lim_{x↓0} u(x) = 0 and (ii) both u and u′ are square integrable with respect to the symmetry measure μ(dx) := dx. Meanwhile, D_{0}^{ψ} consists of the class obtained when the boundary condition (i) above is dropped. The inclusion D^{Y} ⊂ D_{0}^{ψ} is strict—if u: [0, ∞[→ R is any smooth function of compact support and if u is constant on some interval of the form [0, b[, then the restriction of u to ]0, ∞[ is an element of D_{0}^{ψ} \ D^{Y}.

We now introduce certain vector subspaces of C_{0}^{ψ}. Let Σ denote the class of function σ : [0, ∞[→ [0, ∞[ such that (i) σ is increasing, (ii) σ(t) > 0 for t > 0, (iii) σ(s + t) ≤ C_{σ}[σ(s) + σ(t)] for all s, t ≥ 0 and some constant 0 < C_{σ} < ∞. Now define, for ψ ∈ pD_{loc}^{X} and σ ∈ Σ,

C_{σ}^{ψ} := {u ∈ C_{0}^{ψ} : σ(|u|) ∈ L^{1}(μ_{⟨ψ⟩})}.

Properties (i) and (iii) imply that C_{σ}^{ψ} is a vector space closed under composition with normalized contractions. It is easy to see that

C_{0+}^{ψ} := ∪_{σ∈Σ}C_{σ}^{ψ} = {u ∈ C_{0}^{ψ} : μ_{⟨ψ⟩}(|u| > t) < ∞, ∀t > 0},

so that

C_{σ}^{ψ} ⊂ C_{0+}^{ψ} ⊂ C_{0}^{ψ}, ∀σ ∈ Σ.

Of course, C_{0+}^{ψ} is also a contraction-stable vector space.

Here is the main result of the paper.

(3.10) Theorem. Fix σ ∈ Σ ∪ {0+}. The symmetric form (E^{ψ}, C_{σ}^{ψ}) is closable, and its closure (E_{σ}^{ψ}, D_{σ}^{ψ}) coincides with (E^{Y} , D^{Y}).

Proof. Fix σ ∈ Σ ∪ {0+}. Let us begin by observing that the closability of (E^{ψ}, C_{σ}^{ψ}) follows as in the proof of part (a) of (3.8) since C_{σ}^{ψ} ⊂ D_{ref}^{Y}. Also, D^{Y} ⊂ D_{σ}^{ψ} by virtue of (3.4)(b). Thus, to complete the proof we need only show that C_{σ}^{ψ} ⊂ D^{Y}, and for this it suffices to restrict attention to σ ∈ Σ. Define, for t ≥ 0,

g(t) := ∫_{0}^{t}[σ(s) ∧ 1] ds.

Notice that ∥g′∥_{∞} ≤ 1 and that g(t)/t ≤ 1 ∧ \sqrt{σ(t)} for all t ≥ 0.

Now fix u ∈ pC_{σ}^{ψ}. Let (f_{n}) ⊂ C_{c}^{∞}(R) be chosen such that (i) f_{n} = 1 on [−(n − 1), n − 1], (ii) f_{n} = 0 off [−n, n], and (iii) ∥f′_{n}∥_{∞} ≤ 2. Define u_{n} := (u − n^{−1})^{+} and

w_{n} := u · g^{∘}u_{n} · f_{n}^{∘} log ψ, n = 1, 2, ….

Clearly w_{n} ∈ D^{X} ∩ L^{2}(μ) and w_{n} vanishes off the set

G_{n} := {u > 1/n} ∩ {| log ψ| < n}

which differs from a finely open set by at most an (X, m)-polar subset of E_{ψ}. Moreover,

μ_{⟨ψ⟩}(G_{n}) ≤ μ_{⟨ψ⟩}(u > 1/n) = μ_{⟨ψ⟩}(σ^{∘}u ≥ σ(1/n)) ≤ μ_{⟨ψ⟩}(σ^{∘}u)/σ(1/n) < ∞.

It now follows from (3.4)(a) that w_{n} ∈ D^{Y}. By (2.8), since g(t) ≤ t,

μ_{⟨w_{n}⟩} ≤ 4[ u^{2}(g^{∘}u_{n})^{2}(f′_{n}^{∘} log ψ)^{2}ψ^{−2}μ_{⟨ψ⟩}
+ u^{2}(f_{n}^{∘} log ψ)^{2}(g′^{∘}u_{n})^{2}μ_{⟨u_{n}⟩} + (g^{∘}u_{n})^{2}(f_{n}^{∘} log ψ)^{2}μ_{⟨u⟩}]
≤ 4 [u^{2}(g^{∘}u)^{2}ψ^{−2}μ_{⟨ψ⟩} + ∥u∥_{∞}^{2}μ_{⟨u_{n}⟩} + ∥u∥_{∞}^{2}μ_{⟨u⟩}]
≤ 4 [u^{2}(g^{∘}u)^{2}ψ^{−2}μ_{⟨ψ⟩} + 2∥u∥_{∞}^{2}μ_{⟨u⟩}] .

Recalling that g(t)^{2} ≤ t^{2}σ(t), we therefore have

E^{ψ}(w_{n}) ≤ 4∥u∥_{∞}^{4}μ_{⟨ψ⟩}(σ^{∘}u) + 16∥u∥_{∞}^{2}E^{ψ}(u) < ∞, ∀n ∈ N.

Since μ(w_{n}^{2}) ≤ ∥u∥_{∞}^{2}μ(u^{2}) < ∞, the Banach-Saks theorem implies that there is a subsequence of (w_{n}) whose Cesàro means converge strongly in D^{Y} to some w ∈ D^{Y}. This limit must coincide with the (μ-a.e.) pointwise limit u · g^{∘}u = lim_{n} w_{n}. Therefore u · g^{∘}u is an element of D^{Y}. Now u, being an element of C_{σ}^{ψ}, lies in D_{ref}^{Y}. As such, u admits a decomposition u = u_{0} + h where u_{0} ∈ bpD_{e}^{Y} (D_{e}^{Y} is the extended Dirichlet space associated with Y) and h is is bounded, positive and harmonic with respect to Y. More precisely, h can be represented as

(3.11)  h(x) = Q^{x}(Ψ),  (Y, μ)-q.e. x ∈ E_{ψ},

where Ψ is a positive bounded random variable vanishing off {ζ < ∞}, and Ψ^{∘}θ_{t} = Ψ for all t ∈ [0, ζ[, Q^{μ}-a.s. See [1]. By [4; (5.7)] applied to Y, lim_{t↑ζ} u_{0}(Y_{t}) = 0, Q^{μ}-a.s. since u_{0} ∈ D_{e}^{Y}. For the same reason, lim_{t↑ζ} w(Y_{t}) = 0, Q^{μ}-a.s. on {ζ < ∞}. This in turn implies that lim_{t↑ζ} u(Y_{t}) = 0, Q^{μ}-a.s. on {ζ < ∞}, because w = u · g^{∘}u and g(t) > 0 for t > 0. Thus, lim_{t↑ζ} h(Y_{t}) = 0, Q^{μ}-a.s. Now h is bounded, so choosing a sequence (T_{n}) of Y -stopping times announcing ζ, we have, using (3.11) and the strong Markov property,

(3.12)

h(x) = Q^{x}(Ψ) = Q^{x}(Ψ; T_{n} < ζ)
= Q^{x}(Ψ^{∘}θ_{T_{n}}; T_{n} < ζ) = Q^{x}(h(Y_{T_{n}}); T_{n} < ζ) → 0, n → ∞, Y -q.e. x ∈ E_{ψ}.

Thus, h = 0 and u = u_{0}, μ-a.e. Consequently, u = u_{0} ∈ D_{e}^{Y} ∩ L^{2}(μ) = D^{Y}.

In the following corollary of Theorem (3.10) we present three simple sufficient conditions under which D^{Y} = D_{0}^{ψ}.

(3.13) Corollary. Fix ψ ∈ pD_{loc}. Then each of the following conditions implies that D^{Y} = D_{0}^{ψ} : (a) μ_{⟨ψ⟩}(E) < ∞.

(b) There exist p ≥ 1 and a finite constant C_{p} > 0 such that [∫_{E} |u|^{p} dμ_{⟨ψ⟩}]^{1/p} ≤ C_{p} ·\sqrt{E(u, u)} for all u ∈ D^{X}.

(c) μ_{⟨ψ⟩}U ≤ C · m for some finite constant C > 0.

Proof. The sufficiency of condition (a) follows from Theorem (3.10) with σ(t) ≡ 1, and that of (b) with σ(t) = t^{p}. The sufficiency of (c) follows from (b), because [7; (1.19)] tells us that if (c) holds then ∫_{E} u^{2} dμ_{⟨ψ⟩} ≤ C^{2}E(u, u) for all u ∈ D^{X}.

(3.14) Remark. Let Cap be the 1-capacity associated with (E, D). If p ≥ 2, then condition (b) of Corollary (3.13) holds if and only if there is a finite constant C̃_{p} > 0 such that

(3.15)  μ_{⟨ψ⟩}(K) ≤ C̃_{p}Cap(K)^{p/2},  ∀ compact K ⊂ E.

See Theorem 3.1 of [9].

Our final result is an extension of the principal work (Theorem 1.5) of [3]. The special case b ≡ 1 recovers (3.13)(a).

(3.16) Theorem. Suppose there exists b ∈ pD_{loc} such that μ_{⟨b⟩} ≤ m. If ψ ∈ pD_{loc} satisfies μ_{⟨ψ⟩}[b ≤ n] < ∞ for each n ∈ N, then D^{Y} = D_{0}^{ψ}.

Proof. The proof is similar to that of Theorem (3.10), though much simpler; as there it suffices to show that C_{0}^{ψ} ⊂ D^{Y}. Define ρ_{n} := (n − b)^{+} ∧ 1 for n ∈ N. Then ρ_{n} ∈ bpD_{loc}, and if u ∈ pC_{0}^{ψ} then as in the proof of (3.1) we have w_{n} := uρ_{n} ∈ D^{Y} and sup_{n}[E^{ψ}(w_{n}) + μ(w_{n}^{2})] < ∞. The (μ-a.e.) pointwise limit u = lim_{n} w_{n} is therefore an element of D^{Y}. This proves that C_{0}^{ψ} ⊂ D^{Y}.

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