Zeitschrift für Analysis und ihre Anwendungen

Journal for Analysis and its Applications Volume 18 (1999), No. 4, 1039–1064

Representation Formulas for Non-Symmetric Dirichlet Forms

S. Mataloni

Abstract. As well-known, for X a given locally compact separable Hausdorff space, m a positive Radon measure on X with supp [m] = X and C_{0}(X) the space of all continuous functions with compact support on X the Beurling and Deny formula states that any regular Dirichlet form (Ẽ, D(Ẽ)) on L^{2}(X, m) can be expressed as

Ẽ(u, v) = Ẽ^{c}(u, v) + ∫_{X} uvk(dx) + ∫∫_{X×X−d}(u(x) − u(y))((v(x) − v(y))j̃(dx, dy)

for all u, v ∈ D(Ẽ) ∩ C_{0}(X) where the symmetric Dirichlet form Ẽ^{c}, the symmetric measure j̃(dx, dy) and the measure k(dx) are uniquely determined by Ẽ. It is our aim to prove this formula in the non-symmetric case. For this we consider certain families of non-symmetric Dirichlet forms of diffusion type and show that these forms admit an integral representation involving a measure that enjoys some important functional properties as well as in the symmetric case. Keywords: Non-symmetric Dirichlet forms, Beurling-Deny formula, diffusion forms, energy

measures, differentiation formulas, differential operators

AMS subject classification: 28 A 99, 31 C 25, 35 J 70

0. Introduction

It is well known that the Dirichlet forms are suitable tools to describe the variational principles of irregular bodies. In order to formulate such principles, regular Dirichlet forms have been studied. For these forms a rich representation theory is available, based on the fundamental formula of Beurling and Deny (see [1, 2]) and extended by Silverstein [9, 10], Fukushima [4] and Le Jean [6].

Let X be a given locally compact separable Hausdorff space and let m be a positive Radon measure on X such that supp [m] = X. Let C_{0}(X) be the space of all continuous

S. Mataloni: Univ. Roma ”Tor Vergata”, Dept. Math., Via della Ricerca Scientifica, 00133 Roma, Italy; e-mail: Mataloni@mat.uniroma2.it Partially supported by the Italian National Project MURST 40% ”Problemi non lineari...” coord. Prof. Fasano. The author thanks also Prof. N. A. Tchou and Prof. U. Mosco for useful suggestions and comments.

ISSN 0232-2064 / $ 2.50 \begin{matrix}c \\ ◯ \end{matrix} Heldermann Verlag Berlin

functions with compact support on X. The Beurling and Deny formula states that any regular Dirichlet form (Ẽ, D(Ẽ)) on L^{2}(X, m) can be expressed as

Ẽ(u, v) = Ẽ^{c}(u, v) + ∫_{X} uvk(dx) + ∫∫_{X×X−d}(u(x) − u(y))((v(x) − v(y))j̃(dx, dy)

for all u, v ∈ D(Ẽ) ∩ C_{0}(X), the symmetric Dirichlet form Ẽ^{c}, the symmetric measure j̃(dx, dy) and the measure k(dx) occuring in the formula being uniquely determined by Ẽ. These terms are respectively called the diffusion part, the jumping measure and the killing measure of Ẽ.

A particular class of regular Dirichlet forms has been recently developed by many authors – it is the family of diffusion forms. A regular Dirichlet form is called a diffusion if both its killing and jumping measures vanish. In this case the form Ẽ ≡ Ẽ^{c} admits the integral representation

Ẽ^{c}(u, v) = ∫_{X} μ̃(u, v)(dx) ∀ u, v ∈ D(Ẽ^{c})

where μ̃(u, v) is a Radon measure-valued non-negative definite symmetric bilinear form on D(Ẽ). Such a measure is called local energy measure of E, or briefly energy measure of E. The measure μ has indeed a “local character” in X, that is, the restriction of the measure μ to any open subset A of X only depends on the restriction to A of the functions in its argument. Moreover, μ satisfies the following properties:

Leibniz property: For every u, v, w ∈ D(Ẽ) ∩ C_{0}(X),
μ̃(uv, w) = vμ̃(u, w) + uμ̃(v, w)

in the sense of measure.

Schwarz inequality: If φ ∈ L^{2}(X, μ̃(u, u)) and ψ ∈ L^{2}(X, μ̃(v, v)), then φψ is integrable with respect to the total variation of |μ̃(u, v)| and

∫_{X} |φψ| |μ̃(u, v)| (dx) ≤ (∫_{X} φ^{2}μ̃(u, u) (dx))^{\frac{1}{2}} (∫_{X} ψ^{2}μ̃(v, v) (dx))^{\frac{1}{2}}

for every u, v ∈ D(Ẽ) ∩ C_{0}(X).

Chain rule: Let u ∈ D(Ẽ) ∩ C_{0}(X) and let f ∈ C^{1}(ℝ), with bounded derivative and f(0) = 0. Then f(u) belongs to D(Ẽ) ∩ C_{0}(X) and for every v ∈ D(Ẽ) ∩ C_{0}(X)

μ̃(f(u), v) = f′(u)μ̃(u, v)

in the sense of measure.

Truncation rule: For every u, v ∈ D(Ẽ) ∩ C_{0}(X) we have

μ̃(u^{+}, v) = χ_{{u>0}}μ̃(u, v)

where χ_{A} denotes the characteristic function of the set A.

The aim of this paper is to show that these results hold without any symmetry assumption on the form Ẽ.

Actually, in Section 2 we prove that any regular non-symmetric Dirichlet form (E, D(E)) on L^{2}(X, m) can be expressed for u, v ∈ D(E) ∩ C_{0}(X) as

\begin{aligned}E(u, v) & = E^{c}(u, v) + ∫_{X} uvk(dx) \\ & + ∫∫_{X×X−d} [u(x)(v(x) − v(y)) + v(y)(u(y) − u(x))]j(dx, dy), \end{aligned}

the non-symmetric non-negative definite form E^{c}, the measures j(dx, dy) and k(dx) occuring in the formula being uniquely determined by E. We point out that the measure k is the Killing measure associated with the symmetric part Ẽ of E. The form E^{c} and the measure j(dx, dy) are not symmetric, that is E^{c}(u, v) ≠ E^{c}(v, u) and j(dx, dy) ≠ j(dy, dx).

In Section 3 we prove that E^{c} admits the integral representation

E^{c}(u, v) = ∫_{X} μ(u, v)(dx) ∀ u, v ∈ D(E^{c})

where μ(u, v) is a Radon measures-valued non-negative definite non-symmetric bilinear form on D(E) and we show that such a measure enjoys the same properties as in the symmetric case. This fact allows us to extend many results on diffusion forms to the non-symmetric case, for example in [7] we extend to non-symmetric Dirichlet forms, the results obtained by Dal Maso and A. Garroni in [3] for second order non-symmetric elliptic operators.

We conclude the paper with an example in Section 4.

1. Preliminaries on non-symmetric Dirichlet forms

We shall consider the Hilbert space H = L^{2}(X, m), where X is a given locally compact separable Hausdorff space and m a positive Radon measure on X such that supp [m] = X, i.e. m is a non-negative Borel measure on X which is finite on compact sets and strictly positive on each non-empty open set. By (u, v) = ∫_{X} uv dm we denote the inner product of H, and by || · || the related norm. Let D(E) be a linear subspace of H and let E : D(E) × D(E) → ℝ be a bilinear map.

Definition 1.1. A pair (E, D(E)) is called a closed form (on H) if D(E) is a dense linear subspace of H and E : D(E) × D(E) → ℝ is a non-negative definite bilinear form which is symmetric and D(E) is complete with respect to the intrinsic norm

||u||_{D(E)} = E_{1}(u, u)^{\frac{1}{2}} = (E(u, u) + (u, u))^{\frac{1}{2}} .

To give an analogous definition for non symmetric forms, one has to define the – symmetric part of E as Ẽ(u, v) = \frac{1}{2}(E(u, v) + E(v, u))

– antisymmetric part of E as Ĕ(u, v) = \frac{1}{2}(E(u, v) − E(v, u)).

Clearly, (E(u, v), D(E)) = (Ẽ(u, v) + Ĕ(u, v), D(E)).

Then we have the following

Definition 1.2. A pair (E, D(E)) is called a coercive closed form (on H) if D(E) is a dense linear subspace of H and E : D(E) × D(E) → ℝ is a bilinear form such that the following conditions hold:

i) Its symmetric part (Ẽ, D(E)) is a closed form on H.

ii) (E, D(E)) satisfies the following “weak sector condition ”: there exists a constant K > 0 (called continuity constant) such that

|E_{1}(u, v)| ≤ KE_{1}(u, u)^{\frac{1}{2}} E_{1}(v, v)^{\frac{1}{2}} ∀ u, v ∈ D(E),

i.e. (E_{1}, D(E)) is continuous with respect to the intrinsic norm on D(E).

Let us recall now the main definitions and properties of coercive closed forms which are necessary in the following sections (see [5] for furthers details), while we refer to [4, 8] for properties of symmetric closed form.

Definition 1.3. A coercive closed form (E, D(E)) (on H) satisfies the strong sector condition if there exists a constant K ∈ (0, ∞) such that

|E(u, v)| ≤ KE(u, u)^{\frac{1}{2}} E(v, v)^{\frac{1}{2}}  ∀ u, v ∈ D(E).  (1.1)

Definition 1.4. A family {G_{β}}_{β>0} of linear operators on H with D(G_{β}) = H for all β ∈ (0, ∞) is called a strongly continuous contraction resolvent on H if

i) βG_{β} is a contraction on H for all β > 0. ii) G_{α} − G_{β} = (β − α)G_{α}G_{β} for all α, β > 0. iii) lim_{β→∞} βG_{β}u = u for all u ∈ H.

For any given coercive closed form (E, D(E)), always there exists a strongly continuous contraction resolvent {G_{β}}_{β>0} associated with (E, D(E)). It is defined as follows: for all u ∈ H, G_{β}u is the unique element in H such that

E_{β}(G_{β}u, v) := E(G_{β}u, v) + β(G_{β}u, v) = (u, v) ∀ v ∈ D(E).

Moreover, the range R(G_{β}) of G_{β} is contained in D(E). Let us also observe that resolvent is not a self-adjoint operator actually, naming coresolvent the adjoint operator of G_{β}, we have that, for any given coercive closed form (E, D(E)), always there exists a strongly continuous contraction coresolvent {Ĝ_{β}}_{β>0} associated with (E, D(E)), and it results that, for all u ∈ H, Ĝ_{β}u is the unique element in H such that

\begin{matrix}E_{β}(v, Ĝ_{β}u) : & = E(v, Ĝ_{β}u) + β(v, Ĝ_{β}u) = (u, v) \\ E_{β}(v, Ĝ_{β}u) & = E_{β}(G_{β}u, v) \end{matrix} \} ∀ v ∈ D(E).

Finally, let us recall that, if G_{β} is the resolvent associated with the coercive closed form (E, D(E)), then we can define, for β > 0,

E^{(β)}(u, v) := β(u − βG_{β}u, v)  ∀ u, v ∈ H.  (1.2)

Thus

E^{(β)}(u, v) = E(βG_{β}u, v) ∀ u ∈ H, v ∈ D(E)

and

lim_{β→∞} E^{(β)}(u, v) = E(u, v)  ∀ u, v ∈ D(E).  (1.3)

For further details see [5].

We conclude this section giving the definition of non-symmetric Dirichlet form. As usually, we put for all u, v : X → ℝ

u ∨ v := sup (u, v), u ∧ v := inf (u, v), u^{+} := u ∨ 0, u^{−} := −(u ∧ 0).

Definition 1.5. A coercive closed form (E, D(E)) is called a non-symmetric Dirichlet form (on H) if, for all u ∈ D(E), it results

u^{+} ∧ 1 ∈ D(E)  and  \begin{matrix}E(u − u^{+} ∧ 1, u + u^{+} ∧ 1) ≥ 0 \\ E(u + u^{+} ∧ 1, u − u^{+} ∧ 1) ≥ 0 \end{matrix} \} .  (1.4)

Proposition 1.6. A coercive closed form (E, D(E)) on L^{2}(X, m) is a non-symmet-ric Dirichlet form if and only if the Markovian property

\begin{matrix}for each ε > 0 ∃ a function φ_{ε} : ℝ → [−ε, 1 + ε] such that \\ φ_{ε}(t) = t (t ∈ [0, 1]) and 0 ≤ φ_{ε}(t_{2}) − φ_{ε}(t_{1}) ≤ t_{2} − t_{1} if t_{1} ≤ t_{2} \end{matrix} \}  (1.5)

and, moreover,

\begin{matrix}φ_{ε} ∘ u ∈ D(E) \\ lim inf_{ε→0} E(φ_{ε} ∘ u, u − φ_{ε} ∘ u) ≥ 0 \\ lim inf_{ε→0} E(u − φ_{ε} ∘ u, φ_{ε} ∘ u) ≥ 0 \end{matrix} \} ∀ u ∈ D(E)

holds.

Proof. See [5: Proposition I.4.7]

Theorem 1.7. A coercive closed form (E, D(E)) is a non-symmetric Dirichlet form (on H) if and only if the resolvent {βG_{β}} associated with the form is Markovian (i.e. {βG_{β}} is bounded and 0 ≤ βG_{β}u ≤ 1 m-a.e., whenever u ∈ L^{2}(X, m), 0 ≤ u ≤ 1 m-a.e.).

Proof. See [5: Theorem 1.4.4]

In the following we study some particular non-symmetric Dirichlet forms, named regular non-symmetric Dirichlet forms. From now on we denote by C_{0}(X) the space of all continuous functions with compact support in X.

Definition 1.8. A pair (E, D(E)) is called a regular form if it possesses a core, that is a subset F of D(E) ∩ C_{0}(X), which is dense in C_{0}(X) with the uniform norm and in D(E) with the intrinsic norm.

Proposition 1.9. Let (E, D(E)) be a regular non-symmetric Dirichlet form on H = L^{2}(X, m). Let K be a compact set and let U be an open set of X with K ⊂ U. Then there exists a function Φ_{K} such that

Φ_{K} ∈ D(E) ∩ C_{0}(X)
Φ_{K} = 1 on K
supp [Φ_{K}] ⊂ U.

Proof. Since X is a locally compact Hausdorff space, by the Uryshon lemma, there exists a function f ∈ C_{0}(X) such that f = 1 on K and supp [f] ⊂ U. Let us consider 2f ∈ C_{0}(X). By regularity of E, D(E) ∩ C_{0}(X) is a dense subspace of C_{0}(X). Then there exists a sequence {ψ_{n}} ∈ D(E) ∩ C_{0}(X) such that

|ψ_{n}(x) − 2f(x)| < ||ψ_{n} − 2f||_{∞} < \frac{1}{n} ∀ n ∈ ℕ, x ∈ X.

For any fixed n̄, sufficiently large, it results that ψn̄ − \frac{1}{n̄} ≥ 1 on K and ψ := ψn̄ − \frac{1}{n̄} ≤ 0 on X − U. By (1.4) it follows that Φ_{K} := ψ^{+} ∧ 1 is the required function

2. Beurling-Deny formula in the non-symmetric case

We are concerned now with a general representation theorem on regular non-symmetric Dirichlet forms, firstly presented by Beurling and Deny in the symmetric case in their famous paper [2]. Let us start with two fundamental lemmas.

Lemma 2.1. The following statements are true:

(i) Let S be a positive linear operator on L^{2}(X, m) (i.e. Su ≥ 0 m-a.e. whenever u ∈ L^{2}(X, m), u ≥ 0 m-a.e.). Then there exists uniquely a positive Radon measure σ on the product space X × X such that, for any pair of Borel functions u, v ∈ L^{2}(X, m),

(Su, v) = ∫∫_{X×X} u(x)v(y)σ(dx, dy).  (2.1)

(ii) Let Ŝ the adjoint operator of a positive linear operator S on L^{2}(X, m) and let σ̂ be the unique positive Radon measure on the product space X × X associated with Ŝ by (i). Then for any pair of Borel functions u, v ∈ L^{2}(X, m) one has

(Su, v) = ∫∫_{X×X} u(y)v(x)σ̂(dx, dy).  (2.2)

(iii) Let B(X) be the family of all Borel subset of X. Then

σ(E × F) = σ̂(F × E)  ∀ E, F ∈ B(X).  (2.3)

(iv) If, in addition, S is Markovian (i.e. S is bounded and 0 ≤ Su ≤ 1 m-a.e. whenever u ∈ L^{2}(X, m), 0 ≤ u ≤ 1 m-a.e.), one has

σ(X × E) ≤ m(E)  ∀E ∈ B(X)  (2.4)

and, if Ŝ is Markovian, one has

σ(E × X) ≤ m(E)  ∀ E ∈ B(X).  (2.5)

Proof. Firstly we observe that, if S is a symmetric operator, the proof of the lemma is given in [4: Lemma 1.4.1] and it results that σ is a positive symmetric Radon measure.

Let us start to prove that (2.4) follows by (2.1). Let assume, at first, that m(X) < ∞ and let be χ_{E} the characteristic function of the set E ∈ B(X). Then

σ(X × E) = ∫∫_{X×E} σ(dx, dy)
= ∫∫_{X×X} χ_{X}(x)χ_{E}(y)σ(dx, dy)
= (Sχ_{X}, χ_{E})
≤ ∫_{X} χ_{E}(x) dm
= m(E)

since S is Markovian. Let now be m(X) = ∞. By the assumptions on X and m, m is σ-finite, actually let {X_{n}} be a sequence of compact sets such that X_{1} ⊂ X_{2} ⊂ ⋯ ⊂ X_{n} ⊂ ⋯ and lim_{n→∞} X_{n} = ∪_{n=1}^{∞}X_{n} = X. As before one easily check that

σ(X_{n} × E) ≤ ∫_{X_{n}} χ_{E}(x) dm.

Letting the limit as n → ∞, we obtain the thesis. Analogously, if Ŝ is Markovian, (2.2) implies σ̂(X × E) ≤ m(E) then, by (2.3), (2.5) follows.

To prove statement (i), let us denote by C_{0}(X × X) the family of all continuous functions with compact support on X × X. Let us denote by C̃_{0}(X × X) the set of functions f ∈ C_{0}(X × X) such that

f(x, y) = ∑_{i=1}^{l} u_{i}(x)v_{i}(y) ∀ u_{i}, v_{i} ∈ C_{0}(X), (x, y) ∈ X × X

and consider the functional

I(f) = ∑_{i=1}^{l}(Su_{i}, v_{i}) ∀ u_{i}, v_{i} ∈ C_{0}(X).

By some analogous arguments to those used in the proof of [4: Lemma 1.4.1], we can show that if f(x, y) ≥ 0, then I(f) ≥ 0. Since C̃_{0}(X × X) is dense in C_{0}(X × X), we can extend I to a positive linear functional on C_{0}(X × X). Then there exists a unique positive Radon measure σ such that

I(f) = ∫∫_{X×X} f(x, y)σ(dx, dy).

Choosing f(x, y) = u(x) v(y) and I(f) = (Su, v), we obtain

(Su, v) = ∫∫_{X×X} u(x)v(y)σ(dx, dy).

Statement (ii) follows by statement (i). Indeed, by (i) there exists uniquely a positive Radon measure σ̂ on the product space X ×X such that, for any pair of Borel functions u, v ∈ L^{2}(X, m), one has

(Ŝu, v) = ∫∫_{X×X} u(x)v(y)σ̂(dx, dy) = (v, Ŝu).

Then, interchanging the role of u and v, we obtain

(Su, v) = (u, Ŝv) = ∫∫_{X×X} u(y)v(x)σ̂(dx, dy).

It remains to prove statement (iii). Indeed, let E, F ∈ B(X). Then one has

σ(E × F) = ∫∫_{X×X} χ_{E}(x)χ_{F} (y)σ(dx, dy)
= (Sχ_{E}, χ_{F})
= (χ_{E}, Ŝχ_{F})
= ∫∫_{X×X} χ_{E}(y)χ_{F} (x)σ̂(dx, dy)
= σ̂(F × E)

and the proof is complete

Remark 2.2. Let us remark that, by Lemma 2.1/(iv), the measures σ(· × X) and σ(X×·) are absolutely continuous with respect to m and the Radon-Nikodym derivatives satisfy the relations

0 ≤ \frac{σ(dx × X)}{dm} ≤ 1 and 0 ≤ \frac{σ(X × dx)}{dm} ≤ 1  m-a.e.  (2.6)

Lemma 2.3. Let {βG_{β}}_{β>0} or {βĜ_{β}}_{β>0} be a Markovian resolvent or coresolvent in L^{2}(X, m), respectively (see Theorem 1.7). Then there exist some positive Radon measures σ_{β} or σ̂_{β} for every β > 0 such that the approximating form E^{(β)} associated with G_{β} or Ĝ_{β} for every β > 0 (see 1.6) has the representation form

E^{(β)}(u, v) = β∫_{X} uv(1 − s_{β}(x))dm + β∫∫_{X×X} u(x)(v(x) − v(y))σ_{β}(dx, dy)  (2.7)

where

s_{β}(x) = \frac{σ_{β}(dx × X)}{dm}  (2.8)

or

E^{(β)}(v, u) = β∫_{X} uv(1 − ŝ_{β}(x))dm + β∫∫_{X×X} u(x)(v(x) − v(y))σ̂_{β}(dx, dy)  (2.9)

where

ŝ_{β}(x) = \frac{σ_{β}(X × dx)}{dm},  (2.10)

respectively.

Proof. Firstly we prove that βG_{β} is a positive linear operator on L^{2}(X, m). For every u ∈ C_{0}(X), u ≥ 0, let us put v = \frac{u}{||u||_{∞}}. Since 0 ≤ v ≤ 1 and βG_{β} is Markovian, then 0 ≤ βG_{β}v ≤ 1. Hence 0 ≤ βG_{β}u ≤ ||u||_{∞}. Let u ∈ L^{2}(X, m), u ≥ 0. Then there exists a sequence {u_{n}} ∈ C_{0}(X), u_{n} ≥ 0, such that u_{n} → u in L^{2}(X, m). By the continuity of βG_{β} we have βG_{β}u_{n} → βG_{β}u with βG_{β}u_{n} ≥ 0, thus βG_{β}u ≥ 0. Now, by Lemma 2.1 one has

(βG_{β}u, v) = ∫∫_{X×X} u(x)v(y)σ_{β}(dx, dy).

Since, by (1.2), it results that

E^{(β)}(u, v) = β(u − βG_{β}u, v) = β(u, v) − β(βG_{β}u, v),

then

E^{(β)}(u, v) = β∫_{X} u(x)v(x) dm − β∫∫_{X×X} u(x)v(y)σ_{β}(dx, dy)
= β∫_{X} u(x)v(x) dm + β∫∫_{X×X} u(x)[v(x) − v(y)]σ_{β}(dx, dy)
− β∫∫_{X×X} u(x)v(x)σ_{β}(dx, dy).

Let us observe now that, by the Fubini Theorem,

∫∫_{X×X} u(x)v(x)σ_{β}(dx, dy) = ∫_{X} u(x)v(x)σ_{β}(dx × X)

hence, by (2.8),

\begin{aligned}E^{(β)}(u, v) & = β∫_{X} uv(dm − σ_{β}(dx × X)) + β∫∫_{X×X} u(x)[v(x) − v(y)]σ_{β}(dx, dy) \\ & = β∫_{X} uv(1 − s_{β}(x))dm + β∫∫_{X×X} u(x)[v(x) − v(y)]σ_{β}(dx, dy) \end{aligned}

and (2.7) is proved.

To prove (2.9) it is enough to start with

E^{(β)}(v, u) = β(u − βĜ_{β}u, v)

and to make the same arguments used to prove (2.7)

Remark 2.4. Interchanging the role of u and v and with a change of variables x and y in (2.9) and (2.7), we obtain two further representation forms of E^{(β)}:

E^{(β)}(u, v) = β∫_{X} uv(1 − ŝ_{β}(x)) dm + β∫∫_{X×X} v(y)(u(y) − u(x))σ_{β}(dx, dy) (2.11)
E^{(β)}(v, u) = β∫_{X} uv(1 − s_{β}(x)) dm + β∫∫_{X×X} v(y)(u(y) − u(x))σ̂_{β}(dx, dy). (2.12)

Remark 2.5. Let (E, D(E)) be a non-symmetric Dirichlet form. Let {βG_{β}}_{β>0} be the Markovian resolvent and let {βĜ_{β}}_{β>0} be the Markovian coresolvent associated with E. Moreover, let {βG̃_{β}}_{β>0} be the Markovian symmetric resolvent associated with

the symmetric part Ẽ of E. By simple considerations one has that G̃_{β} = \frac{Ĝ_{β}+G_{β}}{2} , hence denoting by σ_{β} and σ̂_{β} the measures associated with G_{β} and Ĝ_{β}, respectively, by Lemma 2.3, and by σ̃_{β} the measure associated with G̃_{β} by [4: Lemma 1.4.1], it results

σ̃_{β}(dx, dy) = \frac{σ_{β}(dx, dy) + σ̂_{β}(dx, dy)}{2}.  (2.13)

Moreover, putting

s̃_{β}(x) = \frac{σ̃_{β}(dx × X)}{dm}  (2.14)

by (2.8), (2.10) and (2.13) easily

s̃_{β}(x) = \frac{s_{β}(x) + ŝ_{β}(x)}{2}  (2.15)

follows. These considerations, (2.7), (2.9), (2.11), (2.12) yield

\begin{aligned}Ẽ^{(β)}(u, v) & = \frac{1}{2}(E^{(β)}(u, v) + E^{(β)}(v, u)) \\ & = β∫_{X} uv(1 − s̃_{β}(x)) dm + \frac{β}{2} ∫∫_{X×X}(u(x) − u(y))(v(x) − v(y))σ̃_{β}(dx, dy) \end{aligned}

according with [4: Formula (1.4.8)] for the symmetric case.

Now we are in a position to prove the Beurling-Deny formula for regular nonsymmetric Dirichlet forms.

Theorem 2.6. Any regular non-symmetric Dirichlet form on L^{2}(X, m) can be expressed as

\begin{aligned}E(u, v) & = E^{c}(u, v) + ∫_{X} uvk(dx) \\ & + ∫∫_{X×X−d} [u(x)(v(x) − v(y)) + v(y)(u(y) − u(x))]j(dx, dy) \end{aligned}  (2.16)

for all u, v ∈ D(E) ∩ C_{0}(X). Here E^{c} is a non-symmetric non-negative definite form with domain D(E^{c}) = D(E)∩C_{0}(X), j is a positive Radon measure on the product space X × X off the diagonal d, and k is a positive Radon measure on X. Such elements E^{c}, j and k are uniquely determined by E.

In order to prove this result we give the proof of two further theorems.

Theorem 2.7. Any regular non-symmetric Dirichlet form on L^{2}(X, m) can be expressed as

E(u, v) = G^{c}(u, v) + ∫_{X} uvg(dx) + 2∫∫_{X×X−d} u(x)(v(x) − v(y))j(dx, dy)  (2.17)

for all u, v ∈ D(E) ∩ C_{0}(X). Here G^{c} is a bilinear form with domain D(G^{c}) = D(E) ∩ C_{0}(X), such that G^{c}(u, v) = 0 for every u, v ∈ D(G^{c}) with disjoint supports and

G^{c}(u, v) = 0 ∀ u ∈ D(G^{c}), v ∈ Θ(u)

where Θ(u) = {w ∈ D(G^{c}) : w is constant on a neighbourhood of supp [u]}, j is a positive Radon measure on the product space X × X off the diagonal d and g is a positive Radon measure on X. Such elements G^{c}, j and g are uniquely determined by E.

Proof. Let us suppose that (2.17) holds. The proof of the uniqueness of j, g and G^{c} is the same of the symmetric case (see [4: Theorem 2.2.1]). In particular, it results that there exists uniquely j such that

E(u, v) = −2∫∫_{X×X−d} u(x)v(y)j(dx, dy)  (2.18)

for any u, v ∈ D(E) ∩ C_{0}(X) with disjoint supports, and there exists uniquely g such that

∫_{X} u(x)g(dx) = E(u, v) − 2∫∫_{X×X−d} u(x)(v(x) − v(y))j(dx, dy)

for any u, v ∈ D(E)∩C_{0}(X), with v identically equal to 1 on a neighbourhood of supp [u]. Let us prove now the existence of the measure j. By (2.7) and by (1.3), one has

E^{(β)}(u, v) = −β∫∫_{X×X} u(x)v(y)σ_{β}(dx, dy) → E(u, v)

as β → ∞ if u, v ∈ D(E) ∩ C_{0}(X) and supp [u] ∩ supp [v] = ∅. Let K_{1}, K_{2} be arbitrary compacts subset of X such that K_{1} ∩ K_{2} = ∅ and let u, v be a pair of functions such that

0 ≤u ∈ D(E) ∩ C_{0}(X), u = 1 on K_{1} and u = 0 on K_{2}
0 ≤v ∈ D(E) ∩ C_{0}(X), v = 0 on K_{1} and v = 1 on K_{2}.

It results that

βσ_{β}(K_{1} × K_{2}) = ∫∫_{K_{1}×K_{2}} βσ_{β}(dx, dy)
≤ ∫∫_{X×X} βu(x)v(y)σ_{β}(dx, dy)
= −E^{(β)}(u, v)
→ −E(u, v)
< ∞

as β → ∞ by (1.3). Then the family of measures {βσ_{β}}_{β>0} is uniformly bounded on each compact subset of X × X − d, hence a sequence β_{n}σ_{β_{n}} converges as β_{n} → ∞ vaguely on X × X − d to a positive Radon measure \overline{j}. Actually, if we define j = \frac{\overline{j}}{2}, we obtain that such a measure j satisfies (2.18), so we get that there exists a unique Radon measure j on X × X − d such that

βσ_{β} → 2j  vaguely on X × X − d as β → ∞.  (2.19)

Let us prove now the existence of the measure g. Let us fix a metric ρ on X which is compatible with the given topology and choose a sequence of relatively compact open sets {G_{l}} increasing to X and a sequence of numbers δ_{l} → 0 such that

Γ_{l} = {(x, y) ∈ G_{l} × G_{l} : ρ(x, y) ≥ δ_{l}}

is a continuous set with respect to the measure j for every l, that is lim_{l→∞} j(Γ_{l}) = j(∪_{l=1}^{∞}Γ_{l}). By (2.7), for any u ∈ D(E) ∩ C_{0}(X) such that supp [u] ⊂ G_{l}, one has

\begin{aligned}E^{(β_{n})}(u, u) & = β_{n}∫_{G_{l}} u^{2}(1 − s_{β_{n}}^{l}(x)) dm \\ & + β_{n}∫∫_{G_{l}×G_{l}} u(x)(u(x) − u(y))σ_{β_{n}}(dx, dy) \end{aligned}  (2.20)

where s_{β_{n}}^{l}(x) = \frac{σ_{βn} (dx×G_{l})}{dm} and \{β_{n}\} is such that β_{n}σ_{β_{n}} → 2j vaguely. Moreover, by (2.6) and (2.20)

0 ≤ β_{n}∫_{G_{l}} u^{2}(1 − s_{β_{n}}^{l}(x))dm
= E^{(β_{n})}(u, u) − β_{n}∫∫_{G_{l}×G_{l}−d} u(x)^{2}σ_{β_{n}}(dx, dy)
+ β_{n}∫∫_{G_{l}×G_{l}−d} u(x)u(y)σ_{β_{n}}(dx, dy)
≤ E^{(β_{n})}(u, u) + β_{n}∫∫_{G_{l}×G_{l}−d} u(x)u(y)σ_{β_{n}}(dx, dy).

Then, by (2.19),

0 ≤ lim sup_{n→∞} β_{n}∫_{G_{l}} u^{2}(1 − s_{β_{n}}^{l}(x))dm
≤ E(u, u) + 2∫∫_{G_{l}×G_{l}−d} u(x)u(y)j(dx, dy)
< ∞.

This implies, in particular, that the family of measures {β_{n}(1 − s_{β_{n}}^{l}(x))dm}_{β>0} is uniformly bounded on each compact subset of G_{l}. Hence an increasing sequence {β_{n}}, β_{n} → ∞ and some positive Radon measures g_{l} on G_{l} exist in such a way that

β_{n}(1 − s_{β_{n}}^{l}(x))dm → g_{l}  vaguely on G_{l} as β_{n} → ∞ ∀ l.  (2.21)

In view of (2.19) and (2.21) we have that, for every l such that supp [u] ⊂ G_{l},

\begin{aligned}E(u, v) & = lim_{β_{n}→∞} β_{n}∫∫_{G_{l}×G_{l},ρ<δ_{l}} u(x)(v(x) − v(y))σ_{β_{n}}(dx, dy) \\ & + 2∫∫_{Γ_{l}} u(x)(v(x) − v(y))j(dx, dy) + ∫_{G_{l}} u(x)v(x)g_{l}(dx). \end{aligned}  (2.22)

Let us extend the measures g_{l} to X, by setting g_{l}(E) = g_{l}(E ∩ G_{l}). Then by (2.21), {g_{l}} is non-increasing on each compact set. Let us denote the vague limit by g, i.e.

g_{l} → g vaguely on X as l → ∞.

Letting l → ∞ in (2.22), we get the expression (2.17) with

G^{c}(u, v) = lim_{l→∞} lim_{β_{n}→∞} β_{n}∫∫_{G_{l}×G_{l},ρ<δ_{l}} u(x)(v(x) − v(y))σ_{β_{n}}(dx, dy).  (2.23)

By (2.23) it is easy to prove that G^{c} has the asked properties

Theorem 2.8. Any regular non-symmetric Dirichlet form on L^{2}(X, m) can be expressed as

E(u, v) = F^{c}(u, v) + ∫_{X} uvh(dx) + 2∫∫_{X×X−d} v(y)(u(y) − u(x))j(dx, dy)  (2.24)

for all u, v ∈ D(E) ∩ C_{0}(X). Here F^{c} is a bilinear form with domain D(F^{c}) = D(E) ∩ C_{0}(X), such that F^{c}(u, v) = 0 for every u, v ∈ D(F^{c}) with disjoint supports and

F^{c}(u, v) = 0 ∀ v ∈ D(F^{c}), u ∈ Θ(v)

where Θ(v) = {w ∈ D(G^{c}) : w is constant on a neighbourhood of supp [v]}, j is a positive Radon measure on the product space X × X off the diagonal d and h is a positive Radon measure on X. Such elements F^{c}, j and h are uniquely determined by E.

Proof. The proof is the same of the previous one using (2.11) in place of (2.7). It is clear that

lim_{β_{n}→∞} lim_{l→∞} β_{n}(1 − ŝ_{β_{n}}^{l}(x))dm = h  vaguely on X  (2.25)

F^{c}(u, v) = lim_{l→∞} lim_{β_{n}→∞} β_{n}∫∫_{G_{l}×G_{l},ρ<δ_{l}} v(y)(u(y) − u(x))σ_{β_{n}}(dx, dy)  (2.26)

and j is the same of Theorem 2.7 since is obtained as in (2.19)

We are now in a position to prove Theorem 2.6.

Proof of Theorem 2.6. Expression (2.16) easily follows by Theorems 2.7 and 2.8 putting

E^{c}(u, v) = \frac{F^{c}(u, v) + G^{c}(u, v)}{2}

= lim_{l→∞} lim_{β_{n}→∞} \frac{β_{n}}{2}∫∫_{G_{l}×G_{l},ρ<δ_{l}}

× [u(x)(v(x) − v(y)) + v(y)(u(y) − u(x))]σ_{β_{n}}(dx, dy)

and

k(dx) = \frac{h(dx) + g(dx)}{2}.

E^{c} is a non-symmetric Dirichlet form. Indeed,

E^{c}(u, u) = lim_{l→∞} lim_{β_{n}→∞} \frac{β_{n}}{2}∫∫_{G_{l}×G_{l},ρ<δ_{l}}(u(x) − u(y))^{2}σ_{β_{n}}(dx, dy) ≥ 0

and the statement is proved

Remark 2.9. Let us denote by ĵ the positive Radon measure on the product space X × X − d such that j(dx, dy) = ĵ(dy, dx). We can also obtain ĵ as the vague limit of the sequence {βσ̂_{β}}_{β>0} that appear in (2.9). Moreover, using representation (2.9) in the proof of Theorem 2.6, we obtain

\begin{aligned}E(v, u) & = E^{c}(v, u) + ∫_{X} uvk(dx) \\ & + ∫∫_{X×X−d} [u(x)(v(x) − v(y)) + v(y)(u(y) − u(x))]ĵ(dx, dy) \end{aligned}

with

\begin{aligned}E^{c}(v, u) & = lim_{l→∞} lim_{β_{n}→∞} \frac{β_{n}}{2}∫∫_{G_{l}×G_{l},ρ<δ_{l}} \\ & × [u(x)(v(x) − v(y)) + v(y)(u(y) − u(x))]σ̂_{β_{n}}(dx, dy). \end{aligned}

Moreover, if (E, D(E)) is a symmetric Dirichlet form, (2.16) becames the well known Beurling-Deny formula.

3. Non-symmetric diffusion forms

By using the terminology of the symmetric case, we give the following

Definition 3.1. Let us call non-symmetric diffusion form a regular non-symmetric Dirichlet form when the positive Radon measures j and k in (2.16) vanish.

By Definition 3.1 and Theorem 2.6 it results that E ≡ E^{c}. Moreover, we observe that the positive Radon measure k vanishes if and only if the positive Radon measures g and h of (2.17) and (2.24) vanish; then E ≡ E^{c} ≡ G^{c} ≡ F^{c}.

Proposition 3.2. A non-symmetric diffusion form (E, D(E)) is strongly local, i.e.

\begin{aligned}E(u, v) & = 0  ∀ u ∈ D(E), v ∈ Θ(u) \\ E(u, v) & = 0  ∀ v ∈ D(E), u ∈ Θ(v) \end{aligned}

where Θ(w) = {f ∈ D(E) : f is constant on a neighbourhood of supp [w]}.

Proof. The thesis easily follows by the identities E ≡ E^{c} ≡ G^{c} ≡ F^{c} and by their integral representation forms (2.23) and (2.26)

In this section we are interested to consider non-symmetric diffusion forms. Our aim is to prove that, as well as in the symmetric case, these forms admit an integral representation.

From now on, let us denote

A_{l} := {(x, y) ∈ G_{l} × G_{l} : ρ(x, y) < δ_{l}}.

Let (E, D(E)) be a non-symmetric diffusion form. Then E ≡ E^{c} ≡ G^{c} ≡ F^{c}. By (2.23) and (2.26),

E^{c}(u, v) = lim_{l→∞} lim_{β→∞} [\frac{β}{2}∫∫_{A_{l}} u(x)(v(x) − v(y))σ_{β}(dx, dy)

+ \frac{β}{2}∫∫_{A_{l}} u(y)(v(y) − v(x))σ̂_{β}(dx, dy)]

= lim_{l→∞} lim_{β→∞} [\frac{β}{2}∫∫_{A_{l}} v(y)(u(y) − u(x))σ_{β}(dx, dy)

+ \frac{β}{2}∫∫_{A_{l}} v(x)(u(x) − u(y))σ̂_{β}(dx, dy)] .

Then using (2.13), the symmetric part Ẽ^{c} of E^{c} has the expression

Ẽ^{c}(u, v) = lim_{l→∞} lim_{β→∞} \frac{β}{2} ∫∫_{A_{l}}(u(x) − u(y))(v(x) − v(y))σ̃_{β}(dx, dy).

Moreover, the antisymmetric part Ĕ^{c} of E^{c} has the expression

Ĕ^{c}(u, v) = E^{c}(u, v) − Ẽ^{c}(u, v)

= lim_{l→∞} lim_{β→∞} \frac{β}{4} [ ∫∫_{A_{l}} u(x)(v(x) − v(y))σ_{β}(dx, dy)

+ ∫∫_{A_{l}} u(y)(v(y) − v(x))σ̂_{β}(dx, dy)

− ∫∫_{A_{l}} u(x)(v(x) − v(y))σ̂_{β}(dx, dy)

− ∫∫_{A_{l}} u(y)(v(y) − v(x))σ_{β}(dx, dy)]

= lim_{l→∞} lim_{β→∞} \frac{β}{4} [ ∫∫_{A_{l}}(u(x) + u(y))(v(x) − v(y))σ_{β}(dx, dy)

− ∫∫_{A_{l}}(u(x) + u(y))(v(x) − v(y))σ̂_{β}(dx, dy)]

= lim_{l→∞} lim_{β→∞} \frac{β}{2}∫∫_{A_{l}}(u(x) + u(y))(v(x) − v(y))(\frac{σ_{β} − σ̂_{β}}{2} )(dx, dy).

Let us define now

σ̆_{β}(dx, dy) = \frac{σ_{β} − σ̂_{β}}{2} (dx, dy).

Then it results that

Ĕ^{c}(u, v) = lim_{l→∞} lim_{β→∞} \frac{β}{2} ∫∫_{A_{l}}(u(x) + u(y))(v(x) − v(y))σ̆_{β}(dx, dy).  (3.1)

Remark 3.3. Let E(u, v) be a non-symmetric diffusion form. Then by the identity F^{c} ≡ G^{c} and by their integral representation forms, it follows that

lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(x)v(x)σ_{β}(dx, dy) = lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(y)v(y)σ_{β}(dx, dy).

Then, obviously, since σ̃_{β}(dx, dy) = σ̃_{β}(dy, dx) one has

lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(x)v(x)σ̃_{β}(dx, dy) = lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(y)v(y)σ̃_{β}(dx, dy)

and, since

\begin{matrix}σ_{β}(dx, dy) & = σ̃_{β}(dx, dy) + σ̆_{β}(dx, dy) \\ σ̆_{β}(dx, dy) & = −σ̆_{β}(dy, dx) \end{matrix} \}

it results that

lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(x)v(x)σ̆_{β}(dx, dy) = lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(y)v(y)σ̆_{β}(dx, dy)

= − lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(x)v(x)σ̆_{β}(dx, dy).

Then

lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(x)v(x)σ̆_{β}(dx, dy) = lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} u(y)v(y)σ̆_{β}(dx, dy)
= 0.

With these considerations there follows easily that the symmetric and antisymmetric part of a non-symmetric diffusion form are strongly local forms. Moreover, it is easy to check that Ĕ^{c}(u, v) = −Ĕ^{c}(v, u) hence (3.1) is a “good” representation.

Theorem 3.4. Let E^{c}(u, v) be a non-symmetric diffusion form and let Ĕ^{c}(u, v) be its antisymmetric part. There exists a signed Radon measure μ̆(u, v) such that

Ĕ^{c}(u, v) = ∫_{X} μ̆(u, v)(dx)  ∀ u, v ∈ D(E^{c}).  (3.2)

Proof. Let us consider, for any fixed pair of functions u, v ∈ D(E^{c}) ∩ C_{0}(X), the linear functional

⟨Ψ(u, v), φ⟩ := lim_{l→∞} lim_{β→∞} \frac{β}{2}∫∫_{A_{l}} φ(x)(u(x) + u(y))(v(x) − v(y))σ̆_{β}(dx, dy)  (3.3)

on C_{0}(X). If φ ≡ 1 on supp [u] ∩ supp [v], the functional Ψ(u, v) is well defined by (3.1). Moreover, by simple calculation using Cauchy criterion, the functional Ψ(u, v) is well defined for all φ ∈ C_{0}(X) and is bounded in C_{0}(X)^{∗} (the dual space of C_{0}(X)), hence by the Riesz representation theorem, for any fixed u, v ∈ D(E^{c}) ∩ C_{0}(X) there exists a unique signed bounded Radon measure μ̆(u, v) such that

⟨Ψ(u, v), φ⟩ = ∫_{X} φ(x)μ̆(u, v)(dx)  ∀ φ ∈ C_{0}(X).  (3.4)

Choosing now φ ≡ 1 on supp [u] ∩ supp [v], by (3.1) Ĕ^{c}(u, v) = ∫_{X} μ̆(u, v)(dx)

At this point let us define μ(u, v) := μ̃(u, v) + μ̆(u, v), where μ̃(u, v) is the energy measure associated with the symmetric form Ẽ^{c} (see [6: Proposition 1.4.1]). Hence we have proved the following

Theorem 3.5. Let E^{c}(u, v) be a non-symmetric diffusion form. Then there exists a signed bounded Radon measure μ(u, v) such that

E^{c}(u, v) = ∫_{X} μ(u, v)(dx).  (3.5)

Such a Radon measure μ(u, v) enjoys some important functional properties that will be describe below. The following proposition shows one of the most important properties of μ: its local character.

Proposition 3.6. Let E(u, v) be a non-symmetric diffusion form and μ(u, v) the measure associated with E by (3.5). Let u_{1}, u_{2}, v_{1}, v_{2} ∈ D(E) ∩ C_{0}(X) such that u_{1} = u_{2} and v_{1} = v_{2} m-a.e. on A open subset of X. Then it results

χ_{A}(x)μ(u_{1}, v_{1}) = χ_{A}(x)μ(u_{2}, v_{2}) on X.

Proof. The proof in the symmetric case is given in [6: Proposition 1.5.2] and in [4: Lemma 5.4.6]. Firstly let us observe that, by (3.3) and by the antisymmetry of the measure σ̆_{β}(u, v), if v = 0 on A or u = 0 on A, then

∫_{X} χ_{A}(x)μ̆(u, v) = 0.

Then, since

\begin{matrix}|∫_{X} χ_{A}(x)μ̆(u_{1}, v_{1}) − χ_{A}(x)μ̆(u_{2}, v_{2})| \\ ≤ |∫_{X} χ_{A}(x)μ̆(u_{1} − u_{2}, v_{1})| + |∫_{X} χ_{A}(x)μ̆(u_{2}, v_{1} − v_{2})| \end{matrix}

the thesis easily follows

The previous proposition states that for any open subset A of X, the restriction μ(u, v) to A depends only on the restriction of u and v to A. This allows us to define the domain of the form restricted to A, denoted by D_{0}(E, A) as the closure of D(E) ∩ C_{0}(A) in D(E).

Lemma 3.7. Let Ω be a relatively compact subset of X and let A be an open subset of Ω. Let (E, D_{0}(E, Ω)) be a regular non-symmetric Dirichlet form on L^{2}(X, m). Then there exists a sequence {φ_{n}} ∈ D(E) ∩ C_{0}(Ω) such that φ_{n} ≥ 0 for any n ∈ ℕ and φ_{n} ↗ χ_{A} in Ω.

Proof. Let us fix a metric ρ on X which is compatible with the given topology and let us consider

A_{n} := \{y ∈ A : ρ(y, ∂Ω) ≥ \frac{1}{n}\}

where ∂Ω denotes the boundary of Ω. For any n ∈ ℕ, A_{n} is closed and A_{n} ⊂ Ω, so A_{n} is compact. By Proposition 1.9 there exists a sequence Φ_{n} ∈ D(E) ∩ C_{0}(Ω) such that, for any n ∈ ℕ, Φ_{n} = 1 on A_{n} and supp [Φ_{n}] ⊂ A. Let us define φ_{n} := φ_{n−1} ∨ Φ_{n} and φ_{0} := Φ_{0}. Then φ_{n} ∈ D(E) ∩ C_{0}(Ω), φ_{n} = 1 on A_{n}, φ_{n} ≥ φ_{n−1} and φ_{n} ↗ χ_{A}. Indeed, for all x ∈ A, ρ(x, ∂Ω) ≥ ρ(x, ∂A) = δ > 0, hence x ∈ A_{n} for each n > δ^{−1} and φ_{n}(x) = 1 → 1. Moreover, for all x ∉ A, φ_{n}(x) = 0 → 0, then φ_{n}(x) → χ_{A}(x)

Proposition 3.8. Let E(u, v) be a non-symmetric diffusion form and μ(u, v) the measure associated with E by (3.5). For every u, v ∈ D(E) ∩ C_{0}(X), it results that

μ̆(u, v) = −μ̆(v, u)

in the sense of measure, that is

∫_{E} μ̆(u, v)(dx) = − ∫_{E} μ̆(v, u)(dx)

for every Borel set E of X.

Proof. Step I. Let Ω be an open set such that supp [u] ∩ supp [v] ⊂ \overline{Ω}. By Lemma 3.7, for any open subset A of Ω there exists a sequence {φ_{n}} ∈ D(E) ∩ C_{0}(Ω) such that φ_{n} → χ_{A}. By (3.2) and by the Lebesgue convergence theorem, we have

0 = lim_{n→∞} lim_{l→∞} lim_{β→∞} ∫∫_{A_{l}} φ_{n}(x)u(x)v(x)σ̆_{β}(dx, dy)

= lim_{l→∞} lim_{β→∞} ∫∫_{A_{l}} χ_{A}(x)u(x)v(x)σ̆_{β}(dx, dy)

= lim_{l→∞} lim_{β→∞} ∫∫_{{A×G_{l},ρ(x,y)<δ_{l}}} u(x)v(x)σ̆_{β}(dx, dy)

= lim_{l→∞} lim_{β→∞} ∫∫_{{A×G_{l},ρ(x,y)<δ_{l}}} u(y)v(y)σ̆_{β}(dy, dx).

Step II. We show now that for the sequence {φ_{n}} as before, we have

lim_{n→∞} ∫_{X} φ_{n}(x)μ̆(uφ_{n}, vφ_{n}) − φ_{n}(x)μ̆(u, v) = 0.

Indeed, for any n ∈ ℕ

\begin{matrix}|∫_{X} φ_{n}(x)[μ̆(uφ_{n}, vφ_{n}) − μ̆(u, v)]| \\ ≤ |∫_{X} φ_{n}(x)μ̆(uφ_{n} − u, vφ_{n})| + |∫_{X} φ_{n}(x)μ̆(u, vφ_{n} − v)| \end{matrix}

and, by the Lebesgue convergence theorem,

lim_{n→∞} ∫_{X} φ_{n}(x)μ̆(uφ_{n} − u, vφ_{n})

= lim_{l→∞} lim_{β→∞} \frac{β}{2}∫∫_{A_{l}} χ_{A}(x)(uχ_{A}(x) − u(x) + uχ_{A}(y) − u(y))

× (vχ_{A}(x) − vχ_{A}(y))σ̆_{β}(dx, dy) = 0

since for l sufficiently large and for all β > 0

\frac{β}{2}∫∫_{\{A×G_{l},ρ(x,y)<δ_{l}\}}(uχ_{A}(y) − u(y))(v(x) − vχ_{A}(y))σ̆_{β}(dx, dy) = \frac{β}{2}∫∫_{\{A×G_{l},ρ(x,y)<δ_{l}\}}(uχ_{A}(y) − u(y))(vχ_{A}(y) − v(x))σ̆_{β}(dy, dx)

= 0.

With the same arguments we have

lim_{n→∞} ∫_{X} φ_{n}(x)μ̆(u, vφ_{n} − v) = 0

and the thesis follows.

Step III. We show now that for the sequence {φ_{n}} as before, we have

lim_{n→∞} ∫_{X} φ_{n}(x)μ̆(uφ_{n}, vφ_{n}) + φ_{n}(x)μ̆(vφ_{n}, uφ_{n}) = 0.

By the Lebesgue convergence theorem we have

lim_{n→∞} ∫_{X} φ_{n}(x)μ̆(uφ_{n}, vφ_{n}) + φ_{n}(x)μ̆(vφ_{n}, uφ_{n})

= lim_{l→∞} lim_{β→∞} β∫∫_{A_{l}} χ_{A}(x)(u(x)v(x)χ_{A}(x) − u(y)v(y)χ_{A}(y))σ̆_{β}(dx, dy).

For l, β sufficiently large, we have

β∫∫_{{A×G_{l},ρ(x,y)<δ_{l}}} (u(x)v(x) − u(y)v(y)χ_{A}(y))σ̆_{β}(dx, dy) = β∫∫_{{A×G_{l},ρ(x,y)<δ_{l}}} ( − u(x)v(x) + u(y)v(y))σ̆_{β}(dy, dx)

→ 0

by Step I.

Step IV. By Steps II and III and the Lebesgue convergence theorem, we have

0 = lim_{n→∞} ∫_{X} φ_{n}(x)[μ̆(uφ_{n}, vφ_{n}) + μ̆(vφ_{n}, uφ_{n})]

= lim_{n→∞} ∫_{X} φ_{n}(x)[μ̆(u, v) + μ̆(v, u)]

= ∫_{A} μ̆(u, v) + μ̆(v, u).

We have so proved the thesis for all open subset A of Ω subset of X. By the regularity of measures μ̆ the thesis holds for every Borel subset E of \overline{Ω}. To conclude, we just note that since supp [u] ∩ supp [v] ⊂ \overline{Ω} we have for every Borel subset E of X

∫_{E} μ̆(u, v) = ∫_{E∩\overline{Ω}} μ̆(u, v) = − ∫_{E∩\overline{Ω}} μ̆(v, u) = − ∫_{E} μ̆(v, u)

and the proposition is proved

Remark 3.9. We point out that it is just the locality property of μ̆ which enables to state some general relations holding “in the sense of measure” as consequence of the suitable analogous relations involving the measure of the all space.

Proposition 3.10 (Leibniz property). Let E be a non-symmetric diffusion form and μ(u, v) the measure associated with E by (3.5). Then for any u, v, w ∈ D(E)∩C_{0}(X) the relations

\begin{matrix}μ(uv, w) & = vμ(u, w) + uμ(v, w) \\ μ(w, uv) & = vμ(w, u) + uμ(w, v) \end{matrix} \}  (3.6)

hold in the sense of measure.

Proof. Since the Leibniz property holds for the symmetric part μ̃(u, v) (see [4: Lemma 5.4.2]), it is enough to prove (3.6) for the measure μ̆(u, v). Moreover, by Proposition 3.8 and Remark 3.9, we have just to prove that

∫_{X} μ̆(w, uv)(dx) = ∫_{X} u(x)μ̆(w, v)(dx) + ∫_{X} v(x)μ̆(w, u)(dx).

Actually, by (3.1) and (3.2) one has

∫_{X} μ̆(w, uv)(dx)

= lim_{l→∞} lim_{β→∞} \frac{β}{2}∫∫_{A_{l}}(w(x) + w(y))(u(x)v(x) − u(y)v(y))σ̆_{β}(dx, dy)

= lim_{l→∞} lim_{β→∞} \frac{β}{2}∫∫_{A_{l}} u(x)(w(x) + w(y))(v(x) − v(y))σ̆_{β}(dx, dy)

+ \frac{β}{2}∫∫_{A_{l}} v(y)(w(x) + w(y))(u(x) − u(y))σ̆_{β}(dx, dy)

= ∫_{X} u(x)μ̆(w, v) + ∫_{X} v(x)μ̆(w, u)

where the last equality holds by (3.4) and (3.3)

Proposition 3.11 (Schwarz inequality). Let E be a non-symmetric diffusion form such that (1.1) holds and let μ(u, v) be the measure associated with E by (3.5). We suppose that there exist the two Radon-Nikodym derivatives

μ̃(u, v)(·) = \frac{dμ̃(u, v)}{dm} ∈ L^{1}(X, m)  (3.7)
μ̆(u, v)(·) = \frac{dμ̆(u, v)}{dm} ∈ L^{1}(X, m)  (3.8)

for every u, v ∈ D(E) ∩ C_{0}(X). If φ ∈ L^{2}(X, μ̃(u, u)) and ψ ∈ L^{2}(X, μ̃(v, v)), then φψ is integrable with respect to the total variation of |μ(u, v)| and one has for every u, v ∈ D(E) ∩ C_{0}(X)

∫_{X} |φψ| |μ(u, v)| (dx) ≤ K (∫_{X} |φ|^{2}μ̃(u, u) (dx))^{\frac{1}{2}} (∫_{X} |ψ|^{2}μ̃(v, v) (dx))^{\frac{1}{2}}

where K is the constant that appears in (1.1).

Proof. The proof of the proposition in the symmetric case is proved in [4: Lemma 5.4.3]. By the strong sector condition (1.3) it results that, for every u, v ∈ D(E)∩C_{0}(X),

|E(u, v)| ≤ \frac{K}{2} (E(u, u) + E(v, v)),

that is, by (3.2), (3.7) and (3.8),

|∫_{X} μ(u, v)(x)dm| ≤ \frac{K}{2} ∫_{X} [μ̃(u, u)(x) + μ̃(v, v)(x)]dm.

By the local character of the measures μ(u, v) and μ̃(u, v) it results that

|∫_{E} μ(u, v)(x) dm| ≤ \frac{K}{2} ∫_{E} [μ̃(u, u)(x) + μ̃(v, v)(x)] dm

for every Borel subset E of X. Choosing v = \frac{μ^{\frac{1}{4}} (u,u)}{μ^{\frac{1}{4}} (v,v)}(x)v and u = \frac{μ^{\frac{1}{4}} (v,v)}{μ^{\frac{1}{4}} (u,u)}(x)u we

obtain

|∫_{E} μ(u, v)(x) dm| ≤ K∫_{E} [μ̃^{\frac{1}{2}} (u, u)(x)μ̃^{\frac{1}{2}} (v, v)(x)] dm

for every Borel subset E of X. Thus, we have

|μ(u, v)(x)| ≤ K(μ̃^{\frac{1}{2}} (u, u)(x) μ̃^{\frac{1}{2}} (v, v)(x)).

Then, as φμ̃^{\frac{1}{2}} (u, u) and ψμ̃^{\frac{1}{2}} (v, v) belong to L^{2}(X, m), by Hölder we obtain

∫_{X} |φψ| |μ(u, v)(dx)| ≤ K(∫_{X} |φ|^{2}μ̃(u, u)(x) dm)^{\frac{1}{2}} (∫_{X} |ψ|^{2}μ̃(v, v)(x) dm)^{\frac{1}{2}}

and the proposition is proved

Proposition 3.12 (Chain rule). Let E be a non-symmetric diffusion form and let μ(u, v) be the measure associated with E by (3.5). Let u ∈ D(E)∩C_{0}(X) and f ∈ C^{1}(ℝ), with bounded derivative and f(0) = 0. Then f(u) belongs to D(E)∩C_{0}(X) and the chain rule holds for every v ∈ D(E) ∩ C_{0}(X), i.e.

\begin{matrix}μ(f(u), v) & = f′(u)μ(u, v) \\ μ(v, f(u)) & = f′(u)μ(v, u) \end{matrix} \}  (3.9)

in the sense of measure.

Proof. The first part of the proposition is well known (see [8]), since it only concerns the symmetric part of E. Let us prove equalities (3.9), that is, by Proposition 3.8 and Remark 3.9,

∫_{X} μ(f(u), v) (dx) = ∫_{X} f′(u)μ(u, v) (dx).

Let v ∈ D(E)∩C_{0}(X). Firstly, we can observe that if f(v) = v^{2}, by the Leibniz property, (3.9) holds. Now if f(v) = v^{3}, we can apply the Leibniz rule on v^{2} · v to have (3.9) and in general, by the linearity of the form, we obtain that (3.9) holds when f(v) is a polynomial such that f(0) = 0. Since v ∈ D(E) ∩ C_{0}(X), hence −M ≤ v ≤ M, then by the Weierstrass theorem, we know that for every f ∈ C^{1}([−M, M]) with f(0) = 0 there exists a sequence f_{n} of polynomials such that f_{n}(0) = 0 and f_{n} → f in C^{1}([−M, M]). Then we have

∫_{X} μ(f_{n}(u), v) (dx) = ∫_{X} f′_{n}(u)μ(u, v) (dx).

By the Lebesgue convergence theorem, taking into account that the integrals are taken on the compact set supp [u] where {f′_{n}(u)} is uniformly bounded by a constant, it results that

∫_{X} f′_{n}(u)μ(u, v) (dx) → ∫_{X} f′(u)μ(u, v) (dx) ∀ u, v ∈ D(E) ∩ C_{0}(X).

We shall prove that

∫_{X} μ(f_{n}(u), v) (dx) → ∫_{X} μ(f(u), v) (dx)

hence, by the uniqueness of the limit, the thesis shall follows. Actually, one has

|∫_{X} [μ(f_{n}(u), v) − μ(f(u), v)] (dx)|
= |∫_{X} μ(f_{n}(u) − f(u), v) (dx)|
≤ KE_{1}(f_{n}(u) − f(u), f_{n}(u) − f(u))^{\frac{1}{2}} E_{1}(v, v)^{\frac{1}{2}} .

Since

∫_{X} |f_{n}(u) − f(u)|^{2}dx → 0 as n → ∞

it is enough to prove that

\begin{aligned}E(f_{n}(u) − f(u), f_{n}(u) − f(u)) & = ∫_{X} μ(f_{n}(u) − f(u), f_{n}(u) − f(u))(dx) \\ & → 0 \end{aligned}

as n → ∞. Applying the chain rule on functions f_{n}, we have

∫_{X} μ(f_{n}(u) − f(u), f_{n}(u) − f(u))(dx)

= ∫_{X} μ(f_{n}(u), f_{n}(u))(dx) − 2∫_{X} μ̃(f_{n}(u), f(u))(dx) + ∫_{X} μ(f(u), f(u))(dx)
= ∫_{X}(f′_{n})^{2}(u)μ(u, u)(dx) − 2∫_{X} f′_{n}(u)μ̃(u, f(u))(dx) + ∫_{X} μ(f(u), f(u))(dx).

Taking the limit as n → ∞, by the Lebesgue convergence theorem it follows that

E(f_{n}(u) − f(u), f_{n}(u) − f(u)) → ∫_{X}(f′)^{2}(u)μ(u, u)(dx) − 2∫_{X} f′(u)μ̃(u, f(u))(dx) + ∫_{X} μ(f(u), f(u))(dx) = ∫_{X} μ(f(u), f(u))(dx) − 2∫_{X} μ(f(u), f(u))(dx) + ∫_{X} μ(f(u), f(u))dx = 0

where we have applied the chain rule in the symmetric case. Thus (3.9) is proved

The above Proposition can be also proved for a Lipschitz continuous function f, with f(0) = 0. Then it is simple to prove the truncation rule, that is

μ̃(u^{+}, v) = χ_{{u>0}}μ̃(u, v)

for every u, v ∈ D(E) ∩ C_{0}(X).

4. Example

Let X := Ω be an open subset of ℝ^{n} (n ≥ 3) and let m = dx be the Lebesgue measure on Ω. Letting a_{ij} ∈ L_{loc}^{1}(Ω, dx), we define

ã_{ij} = \frac{1}{2}(a_{ij} + a_{ji}) and ă_{ij} = \frac{1}{2}(a_{ij} − a_{ji}) (1 ≤ i, j ≤ n).

Let us suppose that there exist two constant λ, M > 0 such that

λ|ξ|^{2} ≤ ∑_{i,j=1}^{n} ã_{ij}(x)ξ_{i}ξ_{j} ∀ ξ = (ξ_{1}, …, ξ_{d}) ∈ ℝ^{n}

1062 S. Mataloni

and

|ă_{ij}| ≤ M ∀ 1 ≤ i, j ≤ n.

Let b_{i}, d_{i} ∈ L_{loc}^{n}(Ω, dx) and c ∈ Lloc^{\frac{n}{2}} such that b_{i} − d_{i} ∈ L^{n}(Ω, dx) ∪ L^{∞}(Ω, dx) and

\begin{matrix}∫ (cu + ∑_{i=1}^{n} d_{i} \frac{∂u}{∂x_{i}} ) dx \\ ∫ (cu + ∑_{i=1}^{n} b_{i} \frac{∂u}{∂x_{i}} ) dx \end{matrix} \} ≥ 0 ∀ u ∈ C_{0}^{∞}(Ω), u ≥ 0.

Let us define for u, v ∈ C_{0}^{∞}(Ω)

E(u, v) = ∑_{i,j=1}^{n} ∫ a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} dx + ∑_{i=1}^{n} ∫ d_{i}u\frac{∂v}{∂x_{i}} dx + ∑_{i=1}^{n} ∫ b_{i}v \frac{∂u}{∂x_{i}} dx + ∫ cuv dx.

The closure of (E, C_{0}^{∞}(Ω)) is a regular non-symmetric Dirichlet form on H = L^{2}(X, m) (see [5: Proposition 2.11]).

Let us observe that

∑_{i,j=1}^{n} ∫ a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} dx + ∑_{i=1}^{n} ∫ d_{i}u\frac{∂v}{∂x_{i}} dx + ∑_{i=1}^{n} ∫ b_{i}v \frac{∂u}{∂x_{i}} dx

is not a non-negative definite form. Actually, it results that

\begin{aligned}E(u, v) & = ∑_{i,j=1}^{n} ∫ a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} dx + ∑_{i=1}^{n} ∫ \frac{d_{i} − b_{i}}{2} u\frac{∂v}{∂x_{i}} dx \\ & + ∑_{i=1}^{n} ∫ \frac{b_{i} − d_{i}}{2} v \frac{∂u}{∂x_{i}} dx + ∑_{i=1}^{n} ∫ (c − \frac{1}{2} \frac{∂(b_{i} + d_{i})}{∂x_{i}}) uv dx, \end{aligned}

that is (2.16) holds with j = 0, k = c − \frac{1}{2} ∑_{i=1}^{n} \frac{∂(b_{i}+d_{i})}{∂x_{i}} and

E^{c}(u, v) = ∑_{i,j=1}^{n} ∫ a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} dx + ∑_{i=1}^{n} ∫ \frac{d_{i} − b_{i}}{2} u\frac{∂v}{∂x_{i}} dx + ∑_{i=1}^{n} ∫ \frac{b_{i} − d_{i}}{2} v \frac{∂u}{∂x_{i}} dx.

Moreover, let us observe that Theorems 2.7 and 2.8 hold with j = 0 and

\begin{aligned}G^{c}(u, v) & = ∑_{i,j=1}^{n} ∫ a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} dx + ∑_{i=1}^{n} ∫ (d_{i} − b_{i})u\frac{∂v}{∂x_{i}} dx \\ F^{c}(u, v) & = ∑_{i,j=1}^{n} ∫ a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} dx + ∑_{i=1}^{n} ∫ (b_{i} − d_{i})v \frac{∂u}{∂x_{i}} dx \\ g & = c − ∑_{i=1}^{n} \frac{∂b_{i}}{∂x_{i}} \\ h & = c − ∑_{i=1}^{n} \frac{∂d_{i}}{∂x_{i}} . \end{aligned}

Representation Formulas 1063

In the case that d_{i} = b_{i} and c = \frac{∂d_{i}}{∂x_{i}} = \frac{∂b_{i}}{∂x_{i}}, it results that k = 0. Then

E^{c}(u, v) = ∑_{i,j=1}^{n} ∫ a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} dx

is a non-symmetric diffusion form. This form satisfyes the strong sector condition (1.1) with K = \frac{M}{λ} + 1:

|E(u, v)| ≤ (\frac{M}{λ} + 1)E^{\frac{1}{2}} (u, u)E^{\frac{1}{2}} (v, v).  (4.1)

Let us observe that Theorem 3.4 holds with

μ(u, v)dx = a_{ij}(x)\frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} dx.

For such a measure the Leibniz property and the Chain rule can be easily proved. Moreover, the Schwarz inequality holds:

∫_{Ω} |φψ|| ∑_{i,j=1}^{n} ă_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} |(dx)

≤ ∫_{Ω} ∑_{i,j=1}^{n} |φψ| |ă_{ij}||\frac{∂u}{∂x_{i}} | |\frac{∂v}{∂x_{j}} |(dx)

≤ M ∫_{Ω} (∑_{i=1}^{n} |φ|^{2} |\frac{∂u}{∂x_{i}} |^{2})^{\frac{1}{2}} (∑_{i=1}^{n} |ψ|^{2} |\frac{∂v}{∂x_{i}} |^{2})^{\frac{1}{2}}

≤ M (∑_{i=1}^{n} ∫_{Ω} |φ|^{2} |\frac{∂u}{∂x_{i}} |^{2} dx)^{\frac{1}{2}} (∑_{i=1}^{n} ∫_{Ω} |ψ|^{2} |\frac{∂v}{∂x_{i}} |^{2} dx)^{\frac{1}{2}}

≤ \frac{M}{λ} ( ∑_{i,j=1}^{n} ∫_{Ω} |φ|^{2}a_{ij} \frac{∂u}{∂x_{i}} \frac{∂u}{∂x_{j}} dx)^{\frac{1}{2}} ( ∑_{i,j=1}^{n} ∫_{Ω} |ψ|^{2}a_{ij} \frac{∂v}{∂x_{i}} \frac{∂v}{∂x_{j}} dx)^{\frac{1}{2}} .

Moreover, since the Schwarz inequality holds for the symmetric part, it results that

\begin{matrix}∫_{Ω} |φψ|| ∑_{i,j=1}^{n} a_{ij} \frac{∂u}{∂x_{i}} \frac{∂v}{∂x_{j}} |(dx) \\ ≤ (\frac{M}{λ} + 1) ( ∑_{i,j=1}^{n} ∫_{Ω} |φ|^{2}a_{ij} \frac{∂u}{∂x_{i}} \frac{∂u}{∂x_{j}} dx)^{\frac{1}{2}} ( ∑_{i,j=1}^{n} ∫_{Ω} |ψ|^{2}a_{ij} \frac{∂v}{∂x_{i}} \frac{∂v}{∂x_{j}} dx)^{\frac{1}{2}} \end{matrix}

and \frac{M}{λ} + 1 is the constant that appears in the strong sector condition (4.1).

References

[1] Beurling, A. and J. Deny: Espace de Dirichlet. Acta Math. 99 (1958), 203 – 204.

[2] Beurling, A. and J. Deny: Dirichlet spaces. Proc. Nat. Acad. Sci. USA 45 (1959), 208 – 215.

[3] Dal Maso, G. and A. Garroni: New results on the asymptotic behaviour of Dirichlet problems in perforated domains. Math. & Meth. Appl. Sci. 4 (1994), 373 – 407.

[4] Fukushima, M.: Dirichlet Forms and Markov Processes. Amsterdam - Oxford - New York: North Holland 1980.

[5] Ma, Z. M. and M. Röckner: Introduction to the Theory of (Non-Symmetric) Dirichlet Forms. Berlin: Springer Verlag 1992.

[6] Le Jean, Y.: Measures associees à une forme de Dirichlet. Applications. Bull. Soc. Math. France 106 (1978), 61 – 112.

[7] Mataloni, S. and N. A. Tchou: Limits of relaxed Dirichlet problems involving a non symmetric Dirichlet form. Ann. di Mat. (to appear).

[8] Mosco, U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 (1994), 368 – 421.

[9] Silverstein, M. L.: Symmetric Markov Processes. Lect. Notes Math. 426 (1974).

[10] Silverstein, M. L.: Boundary Theory for Symmetric Markov Processes. Lect. Notes Math. 516 (1976).

Received 08.02.1999; in revised form 16.07.1999