Small time asymptotics of diffusion processes

Small time asymptotics of diffusion processes A.F.M. ter Elst1, Derek W. Robinson2 and Adam Sikora3 arXiv:math/0601350v1 [math.AP] 14 Jan 2006 Abstract We establish the short-time asymptotic behaviour of the Markovian semigroups associated with strongly local Dirichlet forms under very general hypotheses. Our results apply to a wide class of strongly elliptic, subelliptic and degenerate elliptic operators. In the degenerate case the asymptotics incorporate possible non-ergodicity. November 2005 AMS Subject Classification: 35B40, 58J65, 35J70, 35Hxx, 60J60. Home institutions: 1. Department of Mathematics and Computing Science Eindhoven University of Technology P.O. Box 513 5600 MB Eindhoven The Netherlands 3. Department of Mathematical Sciences New Mexico State University P.O. Box 30001 Las Cruces NM 88003-8001, USA 2. Centre for Mathematics and its Applications Mathematical Sciences Institute Australian National University Canberra, ACT 0200 Australia 1 Introduction One of the iconic results in the theory of second-order elliptic operators is Varadhan’s [Var67b] [Var67a] identification of the small time asymptotic limit lim t log Kt (x ; y) = −4−1 d(x ; y)2 t↓0 (1) of the heat kernel K of a strongly elliptic operator on Rd in terms of the intrinsic Riemannian distance d(· ; ·). The small time behaviour was subsequently analyzed at length by Molchanov [Mol75] who extended (1) to a much wider class of operators and manifolds. These results were then analyzed, largely by probabilistic methods, by various authors (see, for example, the Paris lectures [Aze81]). Most of the early results were restricted to non-degenerate operators with smooth coefficients acting on smooth manifolds. Optimal results for the heat flow on Lipschitz Riemannian manifolds were obtained much later by Norris [Nor97]. The asymptotic relation (1) has, however, been established for certain classes of degenerate subelliptic operators by several authors, in particular for sublaplacians constructed from vector fields satisfying Hörmander’s condition for hypoellipticity [BKRR71] [Léa87a] [Léa87b] [KS88]. It is nevertheless evident from explicit examples that (1) fails for large classes of degenerate elliptic operators. Difficulties arise, for example, from non-ergodic behaviour. The problems introduced by degeneracies are illustrated by the operator H = −d cδ d, where d = d/dx and  x2 δ cδ (x) = , (2) 1 + x2 acting on the real line. If δ ∈ [0, 1/2i then the associated diffusion is ergodic. If, however, δ ≥ 1/2 then the diffusion process separates into two processes on the left and right half lines, respectively (see [ERSZ04], Proposition 6.5). The degeneracy of cδ at the origin creates an impenetrable obstacle for the diffusion. Therefore the corresponding kernel K satisfies Kt (x ; y) = 0 for all x < 0, y > 0 and t > 0 and the effective distance between the left and right half lines is infinite. But this behaviour is not reflected by the R y −1/2 Riemannian distance d(x ; y) = | x cδ | which is finite for all δ ∈ [0, 1i. Hence (1) must fail for this diffusion process if δ ∈ [1/2, 1i. More complicated phenomena can occur for degenerate operators in higher dimensions. To be specific let I = {(α, 0)P : α ∈ [−1, 1]} be 2 a bounded one-dimensional interval in R and consider the form h(ϕ) = 2i=1 (∂i ϕ, cδ ∂i ϕ) with D(h) = W 1,2 (R2 ) where cδ (x) = (|x|2I /(1 + |x|2I ))δ and |x|I denotes the Euclidean distance from x ∈ R2 to the interval I. It follows that if δ ≥ 1/2 then the interval I presents an impenetrable obstacle for the corresponding diffusion and the effective configuration space for the process is R2 \I. In particular the appropriate distance for the description of the diffusion is the intrinsic Riemannian distance on R2 \I rather than that on R2 . Although the Riemannian distance on R2 is well-defined if δ ∈ [0, 1i it is not suited to the description of the diffusion if δ ∈ [1/2, 1i. Thus the problem of a deeper understanding of the small time behaviour of the heat kernel associated with degenerate operators consists in part in identifying the appropriate measure of distance. Hino and Ramı́rez [HR03] (see also [Hin02] [Ram01]) made considerable progress in understanding the small time asymptotics of general diffusion processes by examining the problem in the broader context of Dirichlet forms on a σ-finite measure space (X, B, µ) [FOT94] [BH91] [Mos94]. First they consider an integrated version of (1). Set Kt (A ; B) = 1 R R dµ(x) B dµ(y) Kt(x ; y) for measurable subsets A, B ∈ B. Then the pointwise asymptotic estimate (1) leads, under quite general conditions, to a set-theoretic version A lim t log Kt (A ; B) = −4−1 d(A ; B)2 t↓0 (3) for open sets where the distance between the sets A and B is defined in the usual manner with infima. Secondly, if the measure µ is finite then Hino–Ramı́rez establish (3) for the kernel of the semigroup corresponding to a strongly local, conservative, Dirichlet form on L2 (X ; µ) and for bounded measurable sets A and B but with a set-theoretic distance d(A ; B) defined directly in terms of the Dirichlet form. This distance takes values in [0, ∞] and is not necessarily the distance arising from any underlying Riemannian structure. Nevertheless the estimate (3) establishes that it is the correct measure of small time behaviour. A key feature of this formalism is that it allows for the possibility that Kt (A ; B) = 0 for all small t and then d(A ; B) = ∞. The principal disadvantages of the Hino–Ramı́rez result is that it requires (X, B, µ) to be a probability space and the form to be conservative. One of our aims is to remove these restrictions and to derive the estimate (3) for diffusion processes related to a large class of regular, strongly local, Markovian forms on a general measure space (X, B, µ). Our formalism is suited to applications to degenerate elliptic operators. A second aim is to establish conditions which allow one to pass from the estimate (3) to the pointwise estimate (1). We prove a general result which covers a variety of canonical situations. For example, we are able to derive pointwise estimates for subelliptic operators on Lie groups, to give an independent proof of Norris’ result [Nor97] for the Laplace–Beltrami operator on a Lipschitz Riemannian manifold and to discuss Schrödinger-like semigroups with locally bounded potentials. Simultaneous with this work Ariyoshi and Hino [AH05] extended the Hino–Ramı́rez result to strongly local Dirichlet forms on general σ-finite measure spaces. Their tactic is broadly similar to ours although both were developed independently. Both proofs rely on local approximations and limits of the local diffusions and the corresponding distances. The limits are controlled by use of the evolution equation associated with the form. Ariyoshi and Hino refine the estimates on the equations of motion given in [HR03] and [Ram01] whilst our arguments rely on the observation that the corresponding wave equation has a finite speed of propagation [ERSZ04]. The latter property gives rather precise information about the small scale evolution. In Section 2 we establish the general formalism in which we work and give a complete definition of the distance d(A ; B). Then we are able to give a precise statement of our conclusions. Proofs are given in Section 3. In Section 4 we discuss various applications to elliptic and subelliptic differential operators. 2 General formalism We begin by summarizing some standard definitions and results on Markovian forms and Dirichlet forms. As background we refer to the books by Fukushima, Oshima and Takeda [FOT94], Bouleau and Hirsch [BH91] and Ma and Röckner [MR92] and the papers by LeJan [LeJ78] and Mosco [Mos94]. Most of the early results we need are summarized in Mosco [Mos94] together with more recent results, notably on relaxed forms and convergence properties. 2 Let X be a connected locally compact separable metric space equipped with a positive Radon measure µ, such that supp µ = X. We consider forms and operators on the real Hilbert space L2 (X) and all functions in the sequel are assumed to be real-valued. Moreover, all forms are assumed to be symmetric and densely defined but not necessarily closed or even closable. We adopt the definition of Markovian form given in [FOT94], page 4, or Mosco [Mos94], page 374. Then a Dirichlet form is a closed Markovian form. A Dirichlet form E on L2 (X) is called regular if there is a subset of D(E) ∩ Cc (X) which is a core of E, i.e., which is dense in D(E) with respect to the natural norm ϕ 7→ (E(ϕ) + kϕk22 )1/2 , and which is also dense in C0 (X) with respect to the supremum norm. There are several non-equivalent kinds of locality for forms or Dirichlet forms [BH91] [FOT94] [Mos94]. In general we will call any positive quadratic form h strongly local if h(ψ, ϕ) = 0 for all ϕ, ψ ∈ D(h) with supp ϕ and supp ψ compact and ψ constant on a neighbourhood of supp ϕ. Following Mosco [Mos94] we refer to a strongly local regular Dirichlet form as a diffusion. There is a possibly stronger version of the locality condition given in [BH91], Section I.5. A Dirichlet form E is called [BH]-local if E(ψ, ϕ) = 0 for all ϕ, ψ ∈ D(E) and a ∈ R such that (ϕ + a1)ψ = 0. It follows, however, by [BH91] Remark I.5.1.5 and Proposition I.5.1.3 (L0 ) ⇒ (L2 ), that strong locality and [BH]-locality are equivalent for regular Dirichlet forms. Therefore diffusions are [BH]-local. The Dirichlet form E is called conservative if 1 ∈ D(E) and E(1) = 0. The condition 1 ∈ D(E) of course requires that 1 ∈ L2 (X), i.e., µ(X) < ∞, so it is of limited applicability. Let E be a general Dirichlet form on L2 (X). First, for all ψ ∈ D(E) ∩ L∞ (X) define (E) Iψ : D(E) ∩ L∞ (X) → R by (E) Iψ (ϕ) = E(ψ ϕ, ψ) − 2−1 E(ψ 2 , ϕ) . (E) If no confusion is possible we drop the suffix and write Iψ (ϕ) = Iψ (ϕ). If ϕ ≥ 0 it follows that ψ 7→ Iψ (ϕ) is a Markovian form with domain D(E) ∩ L∞ (X) which satisfies the key properties 0 ≤ IF ◦ψ (ϕ) ≤ Iψ (ϕ) ≤ kϕk∞ E(ψ) (4) for all ϕ, ψ ∈ D(E) ∩ L∞ (X) with ϕ ≥ 0 and all normal contractions F (see [BH91], Proposition I.4.1.1). Hence |Iψ1 (ϕ) − Iψ2 (ϕ)| ≤ Iψ1 −ψ2 (ϕ)1/2 Iψ1 +ψ2 (ϕ)1/2 ≤ E(ψ1 − ψ2 )1/2 E(ψ1 + ψ2 )1/2 kϕk∞ (5) for all ϕ, ψ1 , ψ2 ∈ D(E) ∩ L∞ (X) with ϕ ≥ 0. This form is referred to as the truncated form by Roth [Rot76], Theorem 5. A regular Dirichlet form has a canonical representation originating with Beurling and Deny [BD58] and in its final form by LeJan [LeJ78] (see [Mos94], Section 3e or [FOT94], Section 3.2). If E is strongly local, i.e., if it is a diffusion, this representation takes the simple form Z E(ψ) = dµψ (x) X (E) for all ψ ∈ D(E) where the µψ (= µψ ) are positive Radon measures. The measures are uniquely determined by the identity Z Iψ (ϕ) = dµψ ϕ X 3 for all ϕ, ψ ∈ D(E) ∩ L∞ (X) and the continuity property kµψ1 − µψ2 k ≤ E(ψ1 − ψ2 )1/2 E(ψ1 + ψ2 )1/2 for all ψ1 , ψ2 ∈ D(E) as a consequence of (5) (see [LeJ78], Propositions 1.4.1 and 1.5.1 or [FOT94], Section 3.2 and Lemma 3.2.3). The measure µψ is usually referred to as the energy measure. If E and F are two Dirichlet forms which are [BH]-local and satisfy E ≤ F then (E) (F ) Iψ (ϕ) ≤ Iψ (ϕ) (6) for all ϕ, ψ ∈ D(F ) ∩ L∞ (X) with ϕ ≥ 0 (see, for example, [ERSZ05] Proposition 3.2). Let E be a diffusion. Define D(E)loc as the vector space of (equivalent classes of) all measurable functions ψ: X → C such that for every compact subset K of X there is a (E) ψ̂ ∈ D(E) with ψ|K = ψ̂|K . Since E is regular and [BH]-local one can define Ibψ = Ibψ : D(E) ∩ L∞,c (X) → R by Ibψ (ϕ) = Iψ̂ (ϕ) for all ψ ∈ D(E)loc ∩ L∞ (X) and ϕ ∈ D(E) ∩ L∞,c (X) where ψ̂ ∈ D(E) ∩ L∞ (X) is such that ψ|supp ϕ = ψ̂|supp ϕ . Next, for all ψ ∈ D(E)loc ∩ L∞ (X), define |||Ibψ ||| = sup{ |Ibψ (ϕ)| : ϕ ∈ D(E) ∩ L∞,c (X), kϕk1 ≤ 1 } ∈ [0, ∞] . Now, for all ψ ∈ L∞ (X) and measurable sets A, B ⊂ X, introduce dψ (A ; B) = sup{ M ∈ R : ψ(x) − ψ(y) ≥ M for a.e. x ∈ A and a.e. y ∈ B } = ess inf ψ(x) − ess sup ψ(y) ∈ h−∞, ∞] . x∈A y∈B Recall that ess sup ψ(y) = inf{ m ∈ R : |{y ∈ B : ψ(y) > m}| = 0 } y∈B = min{ m ∈ [−∞, ∞i : |{y ∈ B : ψ(y) > m }| = 0} ∈ [−∞, ∞i and ess inf x∈A ψ(x) = − ess supx∈A −ψ(x). Finally define d(A ; B) = d(E) (A ; B) = sup{ dψ (A ; B) : ψ ∈ D0 (E) } , where D0 (E) = {ψ ∈ D(E)loc ∩ L∞ (X) : |||Ibψ ||| ≤ 1} as in [ERSZ05]. A similar definition was given by Hino and Ramı́rez [HR03] (see also [Stu98]), but since they considered probability spaces the introduction of D(E)loc was unnecessary. Regularity of the form was also unnecessary. If, however, we were to replace D(E)loc by D(E) in the definition of d(A ; B) then, since ψ ∈ L2 (X), one would obtain dψ (A ; B) ≤ 0 for all measurable sets A, B ⊂ X with |A| = |B| = ∞ and this would give d(A ; B) = 0. On the other hand the definition with D(E)loc is not useful unless D(E) contains sufficient bounded functions with compact support. Hence regularity of some sort is essential. The distance determines the following L2 -off diagonal bounds, or Davies–Gaffney bounds on the semigroup associated to a diffusion by Theorem 1.2 in [ERSZ05]. 4 Theorem 2.1 Let E be a diffusion over X. If S (E) denotes the semigroup generated by the self-adjoint operator H on L2 (X) associated with E and if A and B are measurable subsets of X then (E) 2 −1 (E) |(ψ, St ϕ)| ≤ e−d (A;B) (4t) kψk2 kϕk2 for all ψ ∈ L2 (A), ϕ ∈ L2 (B) and t > 0 with the convention e−∞ = 0. Let h be a positive quadratic form on L2 (X). Then h is not necessarily closable but there exists a largest closed quadratic form, denoted by h0 , which is majorized by h, i.e., D(h) ⊆ D(h0 ) and h0 (ϕ) ≤ h(ϕ) for all ϕ ∈ D(h). The form h0 is referred to in the literature on discontinuous media as the relaxation of h (see [Mos94], Section 1.e). It can be understood in various ways. Simon (see [Sim78], Theorems 2.1 and 2.2) established that h can be decomposed uniquely as a sum h = hr + hs of two positive forms with D(hr ) = D(h) = D(hs ) and hr the largest closable form majorized by h. Simon refers to hr as the regular part of h. Then h0 = hr , the closure of hr . Simon also proved that D(h0 ) consists of those ϕ ∈ L2 (X) for which there is a sequence ϕn ∈ D(h) such that limn→∞ ϕn = ϕ in L2 (X) and lim inf n→∞ h(ϕn ) < ∞. Moreover, h0 (ϕ) equals the minimum of all lim inf n→∞ h(ϕn ), where the minimum is taken over all ϕ1 , ϕ2 , . . . ∈ D(h) such that limn→∞ ϕn = ϕ in L2 (X). (See [Sim77], Theorems 2 and 3.) Note that if D is a subspace of L2 (X) which is dense in D(h) then (h|D )0 = h0 . Moreover, if h and k are two positive quadratic forms with h ≤ k then h0 ≤ k0 . The relaxation can also be understood by approximation. If h, h1 , h2 , . . . are closed positive quadratic forms on L2 (X) and H, H1 , H2 , . . . the corresponding positive self-adjoint operators then we write h = r.limn→∞ hn if H1 , H2 , . . . converges to H in the strong resolvent sense, i.e., if (I + H)−1 = limn→∞ (I + Hn )−1 strongly. It follows from [Mos94], Theorem 2.4.1, that h = r.limn→∞ hn if and only if h(ϕ) ≤ lim inf n→∞ hn (ϕn ) for all ϕ, ϕ1 , ϕ2 , . . . ∈ L2 (X) with limn→∞ ϕn = ϕ weakly in L2 (X) and, in addition, for all ϕ ∈ L2 (X) there exists a sequence ϕ1 , ϕ2 , . . . ∈ L2 (X) such that limn→∞ ϕn = ϕ strongly in L2 (X) and h(ϕ) = lim inf n→∞ hn (ϕn ). If h1 , h2 , . . . are closed positive quadratic forms on L2 (X) such that h1 ≥ h2 ≥ . . . then it follows from a result of Kato [Kat80], Theorem VIII.3.11, that the corresponding sequence H1 , H2 , . . . of operators converges in the strong resolvent sense to a positive selfadjoint operator H. If h is the form corresponding to H, then h = r.limn→∞ hn and h is the largest closed form which is majorized by hn for all n ∈ N (Simon [Sim77], Theorem 3.2). Now let h be a positive quadratic form on L2 (X) and l a closed positive quadratic form such that h ≤ λ l, for some λ > 0, and with D(l) dense in D(h). Then the forms hε = h+ε l, with ε > 0, are all closed positive forms, with domain D(l), since ε l ≤ hε ≤ (λ + ε) l. But ε 7→ hε is monotonically decreasing if ε ↓ 0 and it follows from the results of Kato and Mosco cited in the foregoing paragraphs that r.limε↓0 hε = h0 where h0 is the relaxation of h. This characterization justifies the notation h0 . Note that h0 is independent of the particular l used in this construction which is akin to the viscosity method of partial differential equations. So h0 could well be called the viscosity closure of h. If h is the form of a pure second-order elliptic operator in divergence form on Rd and l is the form of the Laplacian then the condition h ≤ λ l corresponds to uniform boundedness of the coefficients. Throughout the remainder of this paper we fix a strongly local regular Dirichlet form l on L2 (X), i.e., a diffusion, satisfying the following property. 5 Condition L The function d(e) : X × X → [0, ∞] defined by (l) d(e) (x ; y) = sup{|ψ(x) − ψ(y)| : ψ ∈ D(l)loc ∩ Cb (X) and |||Ibψ ||| ≤ 1} is a metric, the topology induced by this metric equals the original topology on X and the balls B (e) (x ; r) defined by the metric d(e) are relatively compact for all x ∈ X and r > 0. The first result of this paper is given by the following. Theorem 2.2 Let l be a diffusion on L2 (X) satisfying Condition L. Further let h be a positive form over X such that h ≤ λ lD(l)∩L∞ (X) for some λ > 0 and h|D(l)∩L∞ (X) is strongly local and Markovian. Then the relaxation h0 of h is a diffusion. Moreover, if A, B ⊂ X are relatively compact and measurable then lim t log(1A , St (h0 ) t↓0 1B ) = −4−1 d(h0 ) (A ; B)2 (7) where S (h0 ) denotes the semigroup associated with h0 . Note that if S (h0 ) has an integrable kernel K (h0 ) then Z Z (h0 ) (h ) (h ) dµ(x) dµ(y) Kt 0 (x ; y) = Kt 0 (A ; B) ( 1A , S t 1B ) = A B so (7) corresponds to the integrated version (3) of Varadhan’s result (1). Next note that it follows by choosing h equal to l that one has the following conclusion. Corollary 2.3 If l is a diffusion on L2 (X) satisfying Condition L, then lim t log(1A , St 1B ) = −4−1 d(l) (A ; B)2 (l) t↓0 for all relatively compact measurable A, B ⊂ X. One application of the corollary is to strongly elliptic operators in divergence form on R with bounded coefficients. Then d(e) (· ; ·) is the corresponding Riemannian distance and d(l) (A ; B) = d(e) (A ; B) if A and B are open and non-empty, where we set d d(e) (A ; B) = inf inf d(e) (x ; y) x∈A y∈B for general non-empty A, B ⊆ X. There is also a weaker version of Theorem 2.2 for sets which are possibly not relatively compact. In the formulation of this result we let PA denote the orthogonal projection from L2 (X) onto L2 (A). Corollary 2.4 Assume the conditions of Theorem 2.2. Then (h0 ) lim t log kPA St t↓0 PB k2→2 = −4−1 d(h0 ) (A ; B)2 for all measurable A, B ⊂ X. 6 It is possible to transform the set-theoretic bounds on the semigroup into pointwise bounds but this requires some additional assumptions which are satisfied for large classes of non-degenerate elliptic operators (see Section 4). In general it is possible that ψ ∈ D(l)loc ∩L∞ with |||Ibψ ||| < ∞ but ψ is not continuous. Therefore one can have d(l) (A ; B) > d(e) (A ; B) even for non-empty open sets A and B. This behaviour is illustrated by the one-dimensional example given in the introduction with δ ∈ [1/2, 1i in (2). This explains the origin of the first assumption in the next theorem. The second assumption is typically a consequence of a parabolic version of the Harnack inequality. Theorem 2.5 Let l be a diffusion on L2 (X) satisfying Condition L. Assume I. D0 (l) ⊆ C(X), and, II. the semigroup S (l) has a continuous kernel K and there are ν, ω, T ∈ h0, ∞i such that (e) 2 −1 Ks (x ; y) ≤ Kt (x ; z) (ts−1 )ν eω(1+d (y;z) (t−s) ) for all 0 < s < t ≤ T and x, y, z ∈ X. Then lim t log Kt (x ; y) = −4−1 d(e) (x ; y)2 t↓0 for all x, y ∈ X. Applications of Theorem 2.5 are discussed in Section 4. 3 Proof of the theorems First we derive several useful results for general diffusions. Lemma 3.1 Let E be a diffusion on L2 (X). If A, B are measurable with |A|, |B| < ∞ and lim t log(1A , St 1B ) = −4−1 d(E) (A ; B)2 (E) t↓0 then d(E) (A ; B) is the supremum of all r ≥ 0 for which there are M, t0 > 0 such that (1A , St 1B ) ≤ Me−(4t) −1 r 2 (E) for all t ∈ h0, t0 ]. Proof Let s denote the supremum. Then the Davies–Gaffney bounds of Theorem 2.1 give d(A ; B) ≤ s. If d(A ; B) < s then there are r ∈ hd(A ; B), si and M, t0 > 0 such that −1 2 (1A , St 1B ) ≤ Me−(4t) r for all t ∈ h0, t0 ]. Then lim t log(1A , St 1B ) ≤ −4−1 r 2 < −4−1 d(A ; B)2 t↓0 which gives a contradiction. ✷ Lemma 3.2 Let E be a diffusion on L2 (X). Let ψ1 , ψ2 ∈ D(E)loc ∩ L∞ . 7 I. II. (E) (E) (E) If ϕ ∈ D(E) ∩ L∞,c with ϕ ≥ 0 then Ibψ1 ∧ψ2 (ϕ) ≤ Ibψ1 (ϕ) + Ibψ1 (ϕ). (E) (E) (E) |||Ibψ1 ∨ψ2 ||| ≤ |||Ibψ1 ||| ∨ |||Ibψ2 |||. Proof Let µψ be the energy measure associated with E. It follows from (2.10) in [BM95] that µψ1 ∧ψ2 = 1{x∈X:ψ1 (x)<ψ2 (x)} µψ1 + 1{x∈X:ψ1 (x)>ψ2 (x)} µψ2 for all ψ1 , ψ2 ∈ D(l) ∩ L∞ . Hence Z Z Z Iψ1 ∧ψ2 (ϕ) = dµψ1 ∧ψ2 ϕ = dµψ1 ϕ 1{x∈X:ψ1 (x)<ψ2 (x)} + dµψ2 ϕ 1{x∈X:ψ1 (x)>ψ2 (x)} ≤ Z dµψ1 ϕ + Z dµψ2 ϕ = Iψ1 (ϕ) + Iψ1 (ϕ) for all ϕ ∈ D(E) ∩ L∞ with ϕ ≥ 0. Now Statement I of the lemma follows. The proof of Statement II is similar. ✷ Lemma 3.3 Let E be a diffusion on L2 (X). Let ψ1 , ψ2 , . . . : X → [0, ∞i be measurable and assume that ψn ∧ N ∈ D0 (E) for all n, N ∈ N. Define ψ: X → [0, ∞] by ψ = supn∈N ψn . Then ψ ∧ N ∈ D0 (E) for all N ∈ N. Proof Since ϕ1 ∨ ϕ2 ∈ D0 (E) for all ϕ1 , ϕ2 ∈ D0 (E) by Lemma 3.2.II we may assume that ψ1 ≤ ψ2 ≤ . . .. Let N ∈ N. We may also assume that ψn ≤ N for all n ∈ N by (4). Let K ⊂ X compact. Since E is regular there exist χ, χ̃ ∈ D(E) ∩ Cc (X) such that 0 ≤ χ ≤ χ̃ ≤ 1, χ|K = 1 and χ̃|supp χ = 1. It follows from Lemma 3.2.I and the strong locality of E that E(ψn ∧ Nχ) = Iψn ∧N χ (χ̃) ≤ Ibψn (χ̃) + IN χ (χ̃) ≤ kχ̃k1 + N 2 E(χ) uniformly for all n ∈ N. Moreover, kψn ∧ Nχk2 ≤ N kχk2 uniformly for all n ∈ N. Hence the sequence ψ1 ∧ Nχ, ψ2 ∧ Nχ, . . . is bounded in D(E). Therefore there exists a subsequence such that limk→∞ ψnk ∧ Nχ exists weakly in D(E) and limk→∞ ψnk ∧ Nχ exists almost everywhere. By definition of ψ one has limk→∞ ψnk ∧ Nχ = ψ ∧ Nχ almost everywhere. Hence ψ ∧ Nχ ∈ D(E). In particular ψ ∈ D(E)loc ∩ L∞ . Moreover, it follows from Statement (a) on page 269 of [Sim77] that Ibψ (ϕ) = Iψ∧N χ (ϕ) ≤ lim inf Iψnk ∧N χ (ϕ) k→∞ = lim inf Ibψnk (ϕ) ≤ lim inf |||Ibψnk ||| kϕk1 ≤ kϕk1 k→∞ k→∞ for all ϕ ∈ D(E) ∩ L∞,c with supp ϕ ⊂ K and ϕ ≥ 0. So ψ ∈ D0 (E). ✷ Lemma 3.4 Let ES be a diffusion on L2 (X). Let X1 ⊂ X2 ⊂ . . . be measurable subsets of X such that X = ∞ n=1 Xn . If A, B are measurable subsets of X then d(E) (A ; B) = lim d(E) (A ∩ Xn ; B) = inf d(E) (A ∩ Xn ; B) . n→∞ n∈N 8 Proof Obviously d(A ∩ X1 ; B) ≥ d(A ∩ X2 ; B) ≥ . . . ≥ d(A ; B). Hence limn→∞ d(A ∩ Xn ; B) = inf n→∞ d(A ∩ Xn ; B) ≥ d(A ; B). Let M ∈ [0, ∞i and suppose that M ≤ inf n∈N d(A ∩ Xn ; B). Let ε > 0. Then by Lemma 4.2.III in [ERSZ05] for all n ∈ N there exists a ψn ∈ D0 (E) such that 0 ≤ ψn ≤ M, ψn |B = 0 and ψn |A∩Xn ≥ M − ε. Set ψ = supn∈N ψn . Then ψ ∈ D0 (E) by Lemma 3.3. Moreover, ψ|A ≥ M − ε and ψ|B = 0. So M − ε ≤ dψ (A ; B) ≤ d(A ; B) . Since this is valid for all M ∈ [0, ∞i with M ≤ inf n∈N d(A ∩ Xn ; B) and all ε > 0 it follows that d(A ; B) ≥ inf n∈N d(A ∩ Xn ; B) and the proof of the lemma is complete. ✷ Next we develop some structural results which depend on the domination property assumed in Theorem 2.2. Initially we do not need to assume that l satisfies Condition L. We fix from now on a diffusion l over X and set A = D(l) ∩ L∞ (X). Define Cl as the cone of all positive quadratic forms h over X with A ⊆ D(h) satisfying 1. h ≤ λ l|A for some λ > 0, 2. A is dense in D(h), and, 3. h|A is strongly local and Markovian. Note that we do not assume that the forms h are closed or even closable. The first lemma shows that there is no confusion possible for h(ϕ) if ϕ ∈ D(l)\A under a mild condition that is satisfied if h ∈ Cl . Lemma 3.5 Let h be a positive quadratic form over X with D(h) = A and suppose there exists a λ > 0 such that h ≤ λ l|A. Then there exists a unique positive quadratic form h̃ over X with D(h̃) = D(l) and h̃|A = h. This form h̃ then satisfies h̃ ≤ λ l. Proof The existence and the second part of the lemma are easy. So it remains to prove the uniqueness. Let h̃, ĥ be two positive quadratic form over X with D(h̃) = D(l) = D(ĥ) and h̃|A = h = ĥ|A. Let ε > 0. Set h̃ε = (1 + ε)−1 (h̃ + ελ l) and ĥε = (1 + ε)−1 (ĥ + ελ l) Then D(h̃ε ) = D(l) = D(ĥε ) and h̃ε (ϕ) = ĥε (ϕ) for all ϕ ∈ A. Moreover, h̃ε (ϕ) ≤ λ l(ϕ) for all ϕ ∈ A. Let ϕ ∈ D(l). Since A is dense in D(l) there are ϕ1 , ϕ2 , . . . ∈ A such that limn→∞ (l(ϕ − ϕn ) + kϕ − ϕn k2 ) = 0. Since h̃ε is closed and h̃|A ≤ l|A it follows from [Kat80] Theorem VI.1.12, that h̃ε (ϕ) = limn→∞ h̃ε (ϕn ). But similarly ĥε (ϕ) = limn→∞ ĥε (ϕn ). Therefore h̃ε (ϕ) = ĥε (ϕ). This is valid for all ε > 0. Hence h̃(ϕ) = limε↓0 h̃ε (ϕ) = . . . = ĥε (ϕ). ✷ Corollary 3.6 If h ∈ Cl then there is a unique h̃ ∈ Cl such that D(h̃) = D(l) and h̃|A = h|A. It also follows from Lemma 3.5 that for a positive quadratic form h over X with D(h) ⊆ D(l) one can give a characterization of the condition h ∈ Cl . 9 Corollary 3.7 If h is a quadratic form over X with D(l) ⊆ D(h) and if λ > 0 then the following are equivalent. I. h ≤ λ l|A and A is dense in D(h). II. h ≤ λ l and D(l) is dense in D(h). Next we begin the analysis of the relaxations of the forms in Cl . Lemma 3.8 Let h ∈ Cl . Then the relaxation h0 of h is a regular Dirichlet form with D(l) ⊆ D(h0 ). Moreover, every core for the form l is a core for h0 . Proof By assumption the algebra A is dense in D(h). Therefore (h|A)0 = h0 . Since h|A is Markovian it follows from [Mos94], Corollary 2.8.2, that h0 is a Dirichlet form. Let hAr be the regular part of h|A as in [Sim78]. Then A = D(hAr ). Hence A is dense in D(hAr ). But h0 = (h|A)0 is the closure hAr of the regular part hAr of h|A. So A is a core for h0 . Since h ≤ λ l|A it follows that h0 ≤ λ (l|A)0 = λ l. But by the above D(l) is a core for h0 . It then easily follows that any core for l is also a core for h0 . Finally, since D(l) ∩ Cc (X) is dense in C0 (X) and D(l) ∩ Cc (X) ⊂ D(h0 ) ∩ Cc (X) it follows that D(h0 ) ∩ Cc (X) is dense both in D(h0 ) and in C0 (X). ✷ Next set hε = h̃ + ε l for all ε > 0 and h ∈ Cl , where h̃ is the unique form given by Corollary 3.6. Then D(hε ) = D(l). Corollary 3.9 If h ∈ Cl then hε is a diffusion for all ε > 0. Moreover, r.limε↓0 hε = h0 . Proof If h ≤ λ l|A then ε l ≤ hε ≤ (λ + ε) l. Since l is closed it follows that hε is closed. Therefore hε = (hε )0 is a regular Dirichlet form by Lemma 3.8. Moreover, A is a core for hε and hε |A is strongly local. Therefore hε is strongly local by [BH91], Remark I.5.1.5 and Proposition I.5.1.3 (L0 ) ⇒ (L2 ). If h̃ is the unique form as in Corollary 3.6 then r.lim hε = (h̃)0 = (h̃|A)0 = (h|A)0 = h0 ε↓0 where we used that A is dense in D(h̃) and in D(h). ✷ Next we derive a crude L2 off-diagonal bound. If E is a Dirichlet form over X then we denote by S = S (ce) the semigroup generated by the self-adjoint operator associated with E. Lemma 3.10 Let h ∈ Cl and λ > 0 with h ≤ λ l|A. If A, B ⊂ X are measurable then (h0 ) |(ψ, St −1 d(l) (A;B)2 ϕ)| ≤ e−(4λt) kψk2 kϕk2 for all t > 0, ϕ ∈ L2 (A) and ψ ∈ L2 (B). Proof For all ε > 0 the form hε is a diffusion with hε ≤ (λ + ε) l and D(hε ) = D(l) by Corollary 3.9. So (h ) ((λ+ε)l) (l) Iψ ε (ϕ) ≤ Iψ (ϕ) = I(λ+ε)1/2 ψ (ϕ) 10 (h ) for all ϕ, ψ ∈ D(l)∩L∞ (X) with ϕ ≥ 0 by [ERSZ05], Proposition 3.2. Therefore |||Ibψ ε ||| ≤ (l) |||Ib(λ+ε)1/2 ψ ||| for all ψ ∈ D(l)loc ∩ L∞ (X) = D(hε )loc ∩ L∞ (X). So (λ + ε)−1/2 d(l) (A ; B) ≤ d(hε ) (A ; B) for all measurable A, B ⊂ X. The form hε is regular by Corollary 3.9. Then it follows from Theorem 2.1, that (hε ) |(ψ, St −1 d(hε ) (A;B)2 ϕ)| ≤ e−(4t) −1 d(l) (A;B)2 kψk2 kϕk2 ≤ e−(4(λ+ε)t) kψk2 kϕk2 (hε ) for all t > 0, measurable A, B ⊂ X, ϕ ∈ L2 (A) and ψ ∈ L2 (B). Since limε↓0 St strongly, as a consequence of strong resolvent convergence, the lemma follows. (h0 ) = St ✷ To exploit the L2 off-diagonal bound we need an estimate for d(l) (A ; B). One can obtain a simple estimate if l satisfies Condition L. Therefore from now on we always assume that l satisfies Condition L. Recall that if A, B ⊂ X with A 6= ∅ and B 6= ∅ then d(e) (A ; B) = inf x∈A inf y∈B d(e) (x ; y). Lemma 3.11 I. If U, V ⊂ X are open and non-empty then d(l) (U ; V ) ≥ d(e) (U ; V ). II. If K1 , K2 ⊂ X are compact with K1 ∩ K2 = ∅ then d(l) (K1 ; K2 ) > 0. Proof The first statement is easy. In order to prove the second one, we use regularity of the metric space X. There are non-empty open subsets U1 , U2 and closed subsets F1 , F2 such that K1 ⊂ U1 ⊂ F1 , K2 ⊂ U2 ⊂ F2 and F1 ∩ F2 = ∅. Then d(l) (K1 ; K2 ) ≥ d(l) (U1 ; U2 ) ≥ d(e) (U1 ; U2 ) ≥ d(e) (F1 ; F2 ) > 0 , from which the lemma follows. ✷ We next examine local approximations for the Markovian forms h ∈ Cl defined by truncation. We adopt the definition used for Dirichlet forms and demonstrate that the domination property allows one to deduce many of the standard properties for the relaxations. If h ∈ Cl and Φ ∈ A with Φ ≥ 0 then the quadratic form hΦ is defined by D(hΦ ) = A and hΦ (ϕ) = h(Φϕ, ϕ) − 2−1 h(Φ, ϕ2 ) (h) for all ϕ ∈ A. This is well-defined since A is an algebra. Moreover, hΦ (ϕ) = Iϕ (Φ) if h is a Dirichlet form. Now the truncation hε Φ of the Dirichlet form hε satisfies   (hε ) 0 ≤ hε Φ (ϕ) = Iϕ (Φ) ≤ kΦk∞ hε (ϕ) = kΦk∞ h(ϕ) + ε l(ϕ) for all ϕ ∈ A and ε > 0, where we have used (4). But hε Φ (ϕ) = hΦ (ϕ) + ε lΦ (ϕ). Hence in the limit ε ↓ 0 one deduces that 0 ≤ hΦ (ϕ) ≤ kΦk∞ h(ϕ) ≤ λ kΦk∞ l(ϕ) (8) for all ϕ ∈ A. One deduces in a similar fashion that hΦ satisfies the basic Markov property: ϕ ∈ A implies 0 ∨ ϕ ∧ 1 ∈ A and hΦ (0 ∨ ϕ ∧ 1) ≤ hΦ (ϕ). Moreover, hΦ is strongly local. Hence hΦ ∈ Cl . Note that in general hΦ is not a Dirichlet form as it is not necessarily closed and possibly not closable, even if h is a Dirichlet form. The relaxation, or viscosity closure, hΦ 0 = (hΦ )0 is, however, a regular Dirichlet form by Lemma 3.8 and the next lemma establishes that it is even a diffusion if supp Φ is compact. 11 Lemma 3.12 Let h ∈ Cl and Φ ∈ A with 0 ≤ Φ ≤ 1 and supp Φ compact. Then hΦ 0 ∈ Cl and hΦ 0 is a diffusion. (h ) Moreover, S (hΦ 0 ) leaves L2 (supp Φ) invariant and St Φ 0 ϕ = ϕ for all ϕ ∈ L2 ((supp Φ)c ) and t > 0. Proof For brevity write S = S (hΦ 0 ) . It follows from the foregoing that hΦ ∈ Cl . Therefore hΦ 0 is a regular Dirichlet form by the first statement of Lemma 3.8 and D(l) is dense in D(hΦ 0 ) by the second statement of the lemma. Next we establish the localization properties of hΦ 0 and the semigroup S. Since l is regular there are χ1 , χ2 , . . . ∈ D(l) ∩ Cc (X) such that limn→∞ χn = 1supp Φ in L2 (X) and for all n ∈ N there is a neighbourhood U of supp Φ with χn |U = 1. Then hΦ (χn ) = h(Φ, χn ) − 2−1 h(Φ, χ2n ) = 0 because h is strongly local. Moreover hΦ 0 (1supp Φ ) ≤ lim inf hΦ (χn ) = 0 . n→∞ Therefore 1supp Φ ∈ D(hΦ 0 ) and hΦ 0 (1supp Φ ) = 0. Hence St 1supp Φ = 1supp Φ for all t > 0 and S leaves L2 (supp Φ) invariant. (Cf. [ERSZ04] Lemma 6.1.) Similarly, if ϕ ∈ A then hε Φ ((1 − χn )ϕ) = −2−1 hε (Φ, (1 − χn )2 ϕ2 ) = 0 for all n ∈ N and ε > 0 by [BH]-locality of hε . So hΦ ((1 − χn )ϕ) = limε↓0 hε Φ ((1 − χn )ϕ) = 0 and hΦ 0 (1(supp Φ)c ϕ) ≤ lim inf hΦ ((1 − χn )ϕ) = 0 . n→∞ Now let ϕ ∈ L2 ((supp Φ)c ). Since A is dense in L2 (X) there are ϕ1 , ϕ2 , . . . ∈ A such that limn→∞ ϕn = ϕ in L2 (X). Then limn→∞ 1(supp Φ)c ϕn = ϕ in L2 (X). Since hΦ 0 is lower semicontinuous one deduces that hΦ 0 (ϕ) ≤ lim inf n→∞ hΦ 0 (1(supp Φ)c ϕ) = 0. So ϕ ∈ D(hΦ 0 ) (h ) and hΦ 0 (ϕ) = 0. Hence St Φ 0 ϕ = ϕ for all ϕ ∈ L2 ((supp Φ)c ) and t > 0. Finally we prove that hΦ 0 is strongly local. First, if hΦ ≤ λ l|A with λ > 0 then −1 d(l) (A ;B)2 |(ψ, St ϕ)| ≤ e−(4µt) kψk2 kϕk2 for all measurable A, B ⊂ X, t > 0, ϕ ∈ L2 (A) and ψ ∈ L2 (B) by Lemma 3.10 and (8), where µ = λ kΦk∞ + 1. Secondly, let ϕ, ψ ∈ D(hΦ 0 ) with supp ϕ and supp ψ compact and suppose there exists a neighbourhood U of supp ϕ such that ψ|U = 1. There exists a χ ∈ D(l) ∩ Cc (X) such that K = supp χ is compact and χ|supp ψ∪supp Φ = 1. Then hΦ 0 (χ) = 0 since χ = 1supp Φ + χ 1(supp Φ)c . Therefore St χ = χ for all t > 0. Then (ψ, ϕ) = (χ, ϕ) = (χ, St ϕ) for all t > 0. Hence −1 d(l) (K\U ;supp ϕ)2 t−1 |(ψ, (I − St )ϕ)| = t−1 |(χ − ψ, St ϕ)| ≤ t−1 e−(4µt) kχ − ψk2 kϕk2 for all t > 0. But d(l) (K\U; supp ϕ) > 0 by Condition L and Lemma 3.11. So |hΦ 0 (ψ, ϕ)| = lim t−1 |(ψ, (I − St )ϕ)| = 0 . t↓0 Thus hΦ 0 is strongly local. This completes the verification that hΦ 0 is a diffusion and hΦ 0 ∈ Cl . ✷ 12 If h ∈ Cl and Φ ∈ A with supp Φ compact and 0 ≤ Φ ≤ 1 write I (Φ) = I (hΦ 0 ) and d(Φ) = d(hΦ 0 ) . Further let HΦ 0 and S (Φ) denote the operator and semigroup associated with the form hΦ 0 . Moreover, let H0 denote the operator associated with the form h0 . It will be clear from the context which h is involved. Proposition 3.12 establishes that hΦ 0 is a localization of h: the corresponding semigroup S Φ leaves L2 (supp Φ) invariant. Next we consider the distance corresponding to the generator of the restriction of S Φ to L2 (supp Φ) and its relation to the distance corresponding to hΦ 0 . Lemma 3.13 Let h ∈ Cl and Φ ∈ A with 0 ≤ Φ ≤ 1 and supp Φ compact. Define the form ĥ on L2 (supp Φ) by D(ĥ) = {ϕ|supp Φ : ϕ ∈ D(hΦ 0 )} and ĥ(ϕ|supp Φ ) = hΦ 0 (ϕ1supp Φ ) for all ϕ ∈ D(hΦ 0 ). Then d(Φ) (A ; B) = d(ĥ) (A ∩ supp Φ ; B ∩ supp Φ) for all measurable A, B ⊂ X. Proof Note that the form ĥ is a conservative [BH]-local Dirichlet form. Moreover, D(ĥ)loc = D(ĥ). If ϕ, ψ ∈ D(hΦ 0 ) ∩ L∞ then Iψ1supp Φ (ϕ) = hΦ 0 (ψ 1supp Φ ϕ, ψ 1supp Φ ) − 2−1 hΦ 0 (ϕ, ψ 2 1supp Φ ) = Iψ|supp Φ (ϕ|supp Φ ) (Φ) (ĥ) (Φ) (ĥ) since hΦ 0 (1(supp Φ)c ϕ) = 0. So |||Ibψ1supp Φ ||| = |||Ibψ|supp Φ ||| for all ψ ∈ D(hΦ 0 )loc ∩ L∞ . Let M ∈ [0, d(Φ) (A ; B)] ∩ R and ε > 0. By [ERSZ05], Lemma 4.2.III, there exists a ψ ∈ D0 (hΦ 0 ) such that ψ|B = 0 and ψ|A ≥ M − ε. Then ψ|supp Φ ∈ D0 (ĥ). Therefore M − ε ≤ dψ (A ; B) ≤ dψ|supp Φ (A ∩ supp Φ ; B ∩ supp Φ) ≤ d(ĥ) (A ∩ supp Φ ; B ∩ supp Φ) . So d(Φ) (A ; B) ≤ d(ĥ) (A ∩ supp Φ ; B ∩ supp Φ). Conversely, let M ∈ [0, d(ĥ) (A ∩ supp Φ ; B ∩ supp Φ)] ∩ R and ε > 0. By another application of Lemma 4.2.III in [ERSZ05] there exists a ψ̃ ∈ D0 (ĥ) such that ψ̃|B∩supp Φ = 0 and ψ̃|A∩supp Φ ≥ M − ε. There exists a ψ ∈ D(hΦ 0 ) such that ψ|supp Φ = ψ̃. Then (Φ) ψ 1supp Φ ∈ D0 (hΦ 0 ). Moreover, Ib1(supp Φ)c ∩A = 0 by the last part of Lemma 3.12. Let τ = (M − ε)1(supp Φ)c ∩A + ψ 1supp Φ . Then τ ∈ D0 (hΦ 0 )loc by [ERSZ05], Lemma 3.3.I, τ |B = 0 and τ |A ≥ M − ε. Therefore M − ε ≤ dτ (A ; B) ≤ d(Φ) (A ; B) and d(ĥ) (A ∩ supp Φ ; B ∩ supp Φ) ≤ d(Φ) (A ; B). ✷ Now we can make the first key deduction in the proof of the theorem. One can apply the results of Hino–Ramı́rez [HR03] to the form ĥ in Lemma 3.13. Corollary 3.14 Let h ∈ Cl and Φ ∈ A with 0 ≤ Φ ≤ 1 and supp Φ compact. Let A, B ⊂ X be measurable with A, B ⊆ supp Φ. Then lim t log(1A , St 1B ) = −4−1 d(Φ) (A ; B)2 . (Φ) t↓0 13 Proof This follows from Lemma 3.13 and [HR03], Theorem 1.1. ✷ The next idea is to take a limit over the localizations Φ as Φ increases monotonically to the identity function. This involves analyzing both the limit of the semigroups S (Φ) and of the distances d(Φ) . It is not difficult using arguments of monotonicity to deduce that the strong limit of the semigroups S (Φ) and the pointwise limit of the distances d(Φ) exist. But we also have to identify the limits with the semigroup and distance corresponding to the relaxation and to control simultaneously the small time behaviour. The key to this analysis is the observation that the associated wave equations have a finite speed of propagation (see [ERSZ04], Proposition 3.2). If A ⊂ X with A 6= ∅ and r > 0 set e (e) (A ; r) = { x ∈ X : d(e) (x ; A) ≤ r } , B e (e) (A ; r) is closed since x 7→ d(e) (x ; A) where d(e) (x ; A) = inf{|x − a| : a ∈ A}. Note that B is continuous. Since hΦ ≤ kΦk∞ h|A by (8) one has hΦ 0 ≤ kΦk∞ (h|A)0 = kΦk∞ h0 , so clearly D(h0 ) ⊂ D(hΦ 0 ) if Φ ∈ D(l) ∩ L∞ with Φ ≥ 0. Proposition 3.15 Let h ∈ Cl , λ > 0 and Φ ∈ A and suppose that h ≤ λ l|A, 0 ≤ Φ ≤ 1 and supp Φ is compact. Further let A ⊂ X measurable, Ω ⊂ X open with ∅ = 6 A ⊂ Ω and suppose Φ|Ω = 1. Then 1/2 1/2 cos(tH0 )ϕ = cos(tHΦ 0 )ϕ (9) for all t ∈ R with |t| ≤ λ−1/2 d(e) (A ; Ωc ) and ϕ ∈ L2 (A). Moreover, if ψ ∈ D(hΦ 0 ) and supp ψ ⊂ Ω then ψ ∈ D(h0 ) and h0 (ψ) = hΦ 0 (ψ). The proof of the proposition relies on two lemmas. Lemma 3.16 Let h ∈ Cl be a diffusion, λ > 0 with h ≤ λ l|A, and let H denote the corresponding positive self-adjoint operator. If A ⊂ X is open with A 6= ∅ then e (e) (A ; λ1/2 |t|)) cos(tH 1/2 )L2 (A) ⊂ L2 (B and for all t ∈ R\{0}. e (e) (A ; λ1/2 |t|)) (tH 1/2 )−1 sin(tH 1/2 )L2 (A) ⊂ L2 (B Here and in the sequel the operator formally denoted by (tH 1/2 )−1 sin(tH 1/2 ) is properly defined by spectral theory, even if H = 0. Proof It follows from Lemmas 3.10 and 3.11.I that (h) −1 d(e) (A ;B)2 |(ψ, St ϕ)| ≤ e−(4λt) kψk2 kϕk2 for all t > 0, non-empty open B ⊂ X, ϕ ∈ L2 (A) and ψ ∈ L2 (B). Therefore (ψ, cos(tH 1/2 )ϕ) = 0 for all non-empty open B ⊂ X, ϕ ∈ L2 (A), ψ ∈ L2 (B) and t ∈ R with |t| ≤ λ−1/2 d(e) (A ; B) e (e) (A ; r)c) and ϕ ∈ L2 (A) then by [ERSZ04], Lemma 3.3. Hence if r > 0, ψ ∈ Cc (B 14 (ψ, cos(tH 1/2 )ϕ) = 0 for all |t| ≤ λ−1/2 r. This implies the first identity. The second identity follows from the observation that Z t 1/2 −1 1/2 −1 (ψ, (tH ) sin(tH )ϕ) = t ds (ψ, cos(sH 1/2 )ϕ) 0 combined with the first identity. ✷ Lemma 3.17 Let h ∈ Cl be a diffusion and let H denote the corresponding positive selfadjoint operator. If A ⊂ X is open then D(H) ∩ L2 (A) = L2 (A). Proof We may assume that A 6= ∅. Let λ > 0 and suppose that h ≤ λ l|A. Let ϕ ∈ Cc (A) with ϕ 6= 0. For all t > 0 set ϕt = (t2 H)−1 sin2 (tH 1/2 )ϕ. Then ϕt ∈ D(H) and e (e) (supp ϕ ; 2λ1/2 |t|)) for all t > 0 by Lemma 3.16. Hence ϕt ∈ L2 (A) if t > 0 ϕt ∈ L2 (B is small enough. Finally, limt↓0 ϕt = ϕ by spectral theory. So Cc (A) ⊂ D(H) ∩ L2 (A) and the lemma follows. ✷ The conclusion of the last lemma is very useful since it shows that the operator domain contains abundant functions with compact support. Proof of Proposition 3.15 Fix ε > 0. We begin by comparing the actions of the operators Hε and HΦ ε corresponding to the diffusions hε and hΦ ε associated with h and hΦ (see Corollary 3.9). First assume that A is an open subset of X and let r ∈ h0, (λ + ε)−1/2 d(e) (A ; Ωc )i. Fix ϕ ∈ L2 (A). Then it follows from Lemma 3.16 applied to hε that e (e) (A ; (λ + ε)1/2 |t|)) ⊂ L2 (B e (e) (A ; (λ + ε)1/2 r)) ⊂ L2 (Ω) cos(tHε1/2 )ϕ ∈ L2 (B for all t ∈ h−r, ri. 1/2 If in addition ϕ ∈ D(h ε ) = D(l) then cos(tHε )ϕ ∈ D(l). Thus if one sets ϕn = 1/2 (−n) ∨ cos(tHε )ϕ ∧ n for all n ∈ N one has ϕn ∈ D(l) ∩ L∞ (X) = A and supp ϕn ⊂ e (e) (A ; (λ + ε)1/2 r). Since l is regular there exists a χ ∈ A with 0 ≤ χ ≤ 1, supp χ ⊂ Ω B and χ|Be (e) (A;(λ+ε)1/2 r) = 1. Let ψ ∈ A. Then χ ψ ∈ A and hΦ ε (ψ, ϕn ) = hΦ ε (χ ψ, ϕn ) by locality. But hΦ ε (χ ψ, ϕn ) = hΦ (χ ψ, ϕn ) + ε l(χ ψ, ϕn ) = h(χ ψ, ϕn ) + ε l(χ ψ, ϕn ) = hε (χ ψ, ϕn ) by the definition of hΦ and the assumption Φ|Ω = 1 . Therefore hΦ ε (ψ, ϕn ) = hε (χ ψ, ϕn ) = hε (ψ, ϕn ) where the second identity follows by another use of locality. Then the limit n → ∞ gives hΦ ε (ψ, cos(tHε1/2 )ϕ) = hε (ψ, cos(tHε1/2 )ϕ) for all ψ ∈ A, ϕ ∈ D(hε ) ∩ L2 (A) and t ∈ h−r, ri by [FOT94], Theorem 1.4.2(iii). Let ϕ ∈ D(Hε ) ∩ L2 (A) and t ∈ h−r, ri. Because A is a core for hΦ ε one has hΦ ε (ψ, cos(tHε1/2 )ϕ) = hε (ψ, cos(tHε1/2 )ϕ) = (ψ, cos(tHε1/2 )Hε ϕ) 15 1/2 for all ψ ∈ D(hΦ ε ). Since the form hΦ ε is closed it follows that cos(tHε )ϕ ∈ D(HΦ ε ) and HΦ ε cos(tHε1/2 )ϕ = cos(tHε1/2 )Hε ϕ (10) for all t ∈ h−r, ri and ϕ ∈ D(Hε ) ∩ L2 (A). Next let ϕ ∈ D(Hε ) ∩ L2 (A) and for all t ∈ R define 1/2 χt = cos(tHε1/2 )ϕ − cos(tHΦ ε )ϕ . Clearly kχt k2 ≤ 2 kϕk2 for all t ∈ R. Let ψ ∈ D(HΦ ε ). Then τ : t 7→ (ψ, χt ) is twice differentiable and τ (0) = τ ′ (0) = 0. Moreover, Z t Z t1 (ψ, χt ) = dt1 dt2 τ ′′ (t2 ) 0 = Z 0 t dt1 0 =− Z t1 0 Z t dt1 0 Z   1/2 dt2 (HΦ ε ψ, cos(t2 HΦ ε )ϕ) − (ψ, cos(t2 Hε1/2 )Hε ϕ) t1 dt2 (HΦ ε ψ, χt2 ) 0 for all t ∈ h−r, ri, where T we used (10) in the last step. Now assume that ψ is a bounded n n n vector for HΦ ε , i.e., ψ ∈ ∞ n=1 D(HΦ ε ) and there exists a b > 0 such that kHΦ ε ψk2 ≤ b for all n ∈ N. Then by iteration one deduces that Z t Z t2n−1 dt1 . . . dt2n (HΦn ε ψ, χt2n ) |(ψ, χt )| = 0 0 2n −1 ≤ 2 |t| (2n)! kHΦn ε ψk2 kϕk2 ≤ 2 |t|2n (2n)!−1 bn kϕk2 for all n ∈ N and t ∈ h−r, ri. Taking the limit n → ∞ it follows that (ψ, χt ) = 0 for all bounded vectors ψ for HΦ ε and t ∈ h−r, ri. Since the bounded vectors for HΦ ε are dense 1/2 1/2 in L2 by spectral theory one deduces that cos(tHΦ ε )ϕ = cos(tHε )ϕ for all t ∈ h−r, ri and ϕ ∈ D(Hε ) ∩ L2 (A). Hence 1/2 cos(tHΦ ε )ϕ = cos(tHε1/2 )ϕ for all t ∈ h−r, ri and ϕ ∈ L2 (A) by Lemma 3.17, since A is open. 1/2 1/2 Let ϕ ∈ L2 (A). Then cos(tHε )ϕ = cos(tHΦ ε )ϕ for all ε > 0 and for all t with |t| < (λ + ε)−1/2 d(e) (A ; Ωc ). Therefore 1/2 1/2 1/2 cos(tH0 )ϕ = lim cos(tHε1/2 )ϕ = lim cos(tHΦ ε )ϕ = cos(tHΦ 0 )ϕ ε↓0 ε↓0 for all |t| < λ−1/2 d(e) (A ; Ωc ) by strong resolvent convergence of Hε to H0 and of HΦ ε to HΦ 0 (see [RS72], Theorem VIII.20 and Corollary 3.9). Then the first statement of the proposition follows if A is open. If A is not open then for all δ ∈ h0, d(e) (A ; Ωc )i one can apply the above to the open set B (e) (A ; δ) = {x ∈ X : d(e) (x ; A) < δ} and deduce that (9) is valid for all ϕ ∈ L2 (A) and |t| < λ−1/2 d(e) (B (e) (A ; δ) ; Ωc). Since limδ↓0 d(e) (B (e) (A ; δ) ; Ωc ) = d(e) (A ; Ωc ) by the triangle inequality the first statement of the proposition follows for general A. 16 The last statement of the proposition follows since 1/2 D(h0 ) = { ϕ ∈ L2 (X) : sup 2 t−2 (ϕ, (I − cos(tH0 ))ϕ) < ∞ } t∈h0,1] 1/2 and h0 (ϕ) = limt↓0 2 t−2 (ϕ, (I − cos(tH0 ))ϕ) for all ϕ ∈ D(h0 ), with a similar expression for hΦ 0 . ✷ The proposition has three very useful corollaries. The first corollary establishes the first statement of Theorem 2.2. Corollary 3.18 If h ∈ Cl then h0 is a diffusion. In particular, h0 ∈ Cl . Proof By Lemma 3.8 it remains to show that h0 is strongly local. Let ϕ, ψ ∈ D(h0 ) with supp ϕ and supp ψ compact and ψ is constant on a neighbourhood of supp ϕ. Since l is regular there exist Φ ∈ A and an open Ω ⊂ X such that supp(ϕ ± ψ) ⊂ Ω, 0 ≤ Φ ≤ 1, supp Φ is compact and Φ|Ω = 1. Then hΦ 0 (ψ, ϕ) = 0 by Lemma 3.12. But h0 (ϕ ± ψ) = hΦ 0 (ϕ ± ψ) by the last part of Proposition 3.15. Therefore h0 (ψ, ϕ) = hΦ 0 (ψ, ϕ) = 0 by polarization and symmetry. ✷ Corollary 3.19 Adopt the assumptions of Proposition 3.15. If ψ ∈ D(hΦ 0 ) ∩ L∞ with (h ) (Φ) supp ψ ⊂ Ω then |||Ibψ 0 ||| = |||Ibψ |||. Proof Since l is regular there exists a τ ∈ D(l) ∩ L∞ such that 0 ≤ τ ≤ 1, τ |supp ψ = 1 and supp τ ⊂ Ω. Let ϕ ∈ D(hΦ 0 )loc ∩ L∞,c . Then ϕ τ ∈ D(hΦ 0 ) ∩ L∞ and ϕ τ ∈ D(h0 ) by the last part of Proposition 3.15. Hence by strong locality and Proposition 3.15 one has (Φ) (Φ) Ibψ (ϕ) = Iψ (ϕ τ ) = hΦ 0 (ψ ϕ τ, ψ) − 2−1 hΦ 0 (ϕ τ, ψ 2 ) (h ) = h0 (ψ ϕ τ, ψ) − 2−1 h0 (ϕ τ, ψ 2) = Ibψ 0 (ϕ τ ) . So (Φ) (h ) (h ) (h ) (Φ) |Ibψ (ϕ)| ≤ |||Ibψ 0 ||| kϕ τ k1 ≤ |||Ibψ 0 ||| kϕk1 and |||Ibψ ||| ≤ |||Ibψ 0 |||. The opposite inequality is similar. ✷ The third corollary establishes that the semigroups S (Φ) converge strongly to the semigroup associated with the relaxation h0 . Corollary 3.20 Adopt the assumptions of Proposition 3.15. Then (h0 ) kSt −1 r 2 (Φ) ϕ − St ϕk2 ≤ 2 e−(4t) kϕk2 for all t > 0 and ϕ ∈ L2 (A), where r = λ−1/2 d(e) (A ; Ωc ). Proof This follows from the identity (h ) St 0 ϕ = (πt) −1/2 Z ∞ −1 s2 ds e−(4t) 0 and Proposition 3.15. 1/2 cos(sH0 )ϕ (11) ✷ Corollary 3.20 gives good control over the semigroups S (Φ) as Φ → 1 and indirectly it gives control over the distances. We first deduce that the distances d(Φ) (A ; B) become independent of the choice of Φ as Φ → 1 with A and B fixed. 17 Lemma 3.21 Let h ∈ Cl , λ > 0 and Φ, Ψ ∈ A and assume that with h ≤ λ l|A, 0 ≤ Φ, Ψ ≤ 1 and supp Φ, supp Ψ compact. Further let A, B ⊂ X measurable, Ω ⊂ X open with ∅ 6= A ⊂ Ω, B ⊂ supp Φ ∩ supp Ψ and suppose Φ|Ω = Ψ|Ω = 1. If d(Φ) (A ; B) ∨ d(Ψ) (A ; B) ≤ λ−1/2 d(e) (A ; Ωc ) then d(Φ) (A ; B) = d(Ψ) (A ; B). Proof It follows from Proposition 3.15 that 1/2 1/2 1/2 cos(tHΦ 0 )ϕ = cos(tH0 )ϕ = cos(tHΨ 0 )ϕ for all t ∈ R with |t| ≤ λ−1/2 d(e) (A ; Ωc ) and ϕ ∈ L2 (A). Hence it follows as in the proof of Corollary 3.20 that −1 2 (Φ) (Ψ) kSt ϕ − St ϕk2 ≤ 2 e−(4t) r kϕk2 for all t > 0 and ϕ ∈ L2 (A), where r = λ−1/2 d(e) (A ; Ωc ). In particular, |(1B , St 1A ) − (1B , St 1A )| ≤ 2 e−(4t) (Φ) −1 r 2 (Ψ) |A|1/2 |B|1/2 and (1B , St 1A ) ≤ (1B , St 1A ) + 2 e−(4t) (Φ) −1 r 2 (Ψ) −1 d(Ψ) (A;B)2 ≤ e−(4t) |A|1/2 |B|1/2 −1 r 2 |A|1/2 |B|1/2 + 2 e−(4t) |A|1/2 |B|1/2 −1 d(Ψ) (A;B)2 ≤ 3 |A|1/2 |B|1/2 e−(4t) for all t > 0, where we used the Davies–Gaffney bounds of Theorem 2.1 in the second step. It follows by Corollary 3.14 and Lemma 3.1 that d(Φ) (A ; B) ≥ d(Ψ) (A ; B). The converse inequality is valid by a similar argument. Hence the distances are equal. ✷ Identification of the limit of the distances corresponding to the localizations relies on the construction of certain cut-off functions and this construction is dependent on the topological assumption. The subsequent argument follows similar reasoning in [BM95], Section 3, and [Stu95], Appendix A. Lemma 3.22 Let y ∈ X and define ψ: X → [0, ∞i by ψ(x) = d(e) (x ; y). Then ψ ∧ N ∈ D0 (l) for all N ∈ N. Proof Since X is separable are y1 , y2 , . . . ∈ X such that {yk : k ∈ N} dense is in S∞ there (e) X. Let n ∈ N. Then X = k=1 B (yk ; n−1 ). Let k ∈ N. By definition of d(e) there exists a ψnk ∈ D0 (l) ∩ Cb (X) such that ψnk (y) − ψnk (yk ) ≥ d(e) (y ; yk ) − n−1 . Then ψnk (x) + d(e) (x ; y) ≤ ψnk (x) − ψnk (yk ) + ψnk (yk ) + d(e) (y ; yk ) − d(e) (y ; yk ) + d(e) (x ; y) ≤ d(e) (x ; yk ) + ψnk (y) + n−1 + d(e) (x ; yk ) ≤ ψnk (y) + 3 n−1 for all x ∈ B(yk ; n−1 ). Set ψ̃nk = (ψnk (y) − ψnk ) ∨ 0. Then ψ̃nk ∈ D0 (l). Since ψnk (y) − ψnk (x) ≤ d(e) (x ; y) for all x ∈ X one has 0 ≤ ψ̃nk (x) ≤ d(e) (x ; y) for all x ∈ X. Moreover, if x ∈ B(yk ; n−1 ) then ψ̃nk (x) ≥ ψnk (y) − ψnk (x) ≥ d(e) (x ; y) − 3 n−1 . Hence ψ = supn∈N supk∈N ψ̃nk and the lemma follows from Lemma 3.3. 18 ✷ Corollary 3.23 If K ⊂ X is compact and ε > 0 then there exists a ψ ∈ D(l) ∩ Cc (X) (l) such that 0 ≤ ψ ≤ 1, ψ|K = 1 and |||Ibψ ||| ≤ ε. Proof We may assume that K 6= ∅. Let y ∈ K. There exists an M > 0 such that d(e) (x ; y) ≤ M for all x ∈ K. Define ψ(x) = 0 ∨ (N − ε1/2 d(e) (x ; y)) ∧ 1 where N = 1 + ε1/2 M. Then the corollary follows from Lemma 3.22. ✷ At this point we are prepared to prove the second statement in Theorem 2.2. The first step in the proof is to identify the limit of the distances corresponding to the localizations as Φ increases to the identity. Lemma 3.24 Let h ∈ Cl . Fix x ∈ X. Let Φ1 , Φ2 , . . . ∈ A be such that 0 ≤ Φ1 ≤ Φ2 ≤ . . . ≤ 1, suppose Φn |B(e) (x;n) = 1 and supp Φn is compact for all n ∈ N. Let A, B ⊂ X be relatively compact and measurable. Then d(h0 ) (A ; B) = d(Φn ) (A ; B) for all large n ∈ N. Proof One has d(h0 ) (A ; B) ≤ d(Φn+1 ) (A ; B) ≤ d(Φn ) (A ; B) for all n ∈ N by (6), Corollary 3.23 and Proposition 5.1 of [ERSZ05]. Hence d(h0 ) (A ; B) ≤ inf d(Φn ) (A ; B) = lim d(Φn ) (A ; B) . n→∞ n∈N Let M ∈ [0, ∞i and suppose that M ≤ inf n∈N d(Φn ) (A ; B). Let ε ∈ h0, 1]. By Corollary 3.23 there exists a χ ∈ D(l) ∩ Cc (X) such that 0 ≤ χ ≤ 1, χ|A∪B = 1 and (l) |||Ibχ ||| ≤ ε2 . There is an n ∈ N such that supp χ ⊂ B (e) (x ; n). Then M ≤ d(Φn ) (A ; B), so by [ERSZ05], Lemma 4.2.III, there exists a ψ ∈ D0 (hΦn 0 ) such that 0 ≤ ψ ≤ M, ψ|B = 0 and ψ|A ≥ M − ε. Then χ ψ ∈ D(hΦn 0 ) ∩ L∞ . Moreover, it follows from Corollary 3.19 and [ERSZ05], Lemma 3.3.II, that (h ) (Φ ) (Φ ) |||Ibχ ψ0 ||| = |||Ibχ ψn ||| ≤ (1 + ε)kχk2∞ |||Ibψ n ||| + (1 + ε−1 )kψk2∞ |||Ibχ(Φn ) ||| ≤ (1 + ε) + (1 + ε−1 )M 2 λ |||Ibχ(l) ||| ≤ 1 + N ε where N = 1 + 2M 2 λ and λ > 0 is such that h ≤ l|A. So (1 + N ε)−1/2 χ ψ ∈ D0 (h0 ). Therefore (1 + N ε)−1/2 (M − ε) ≤ d(1+N ε)−1/2 χ ψ (A ; B) ≤ d(h0 ) (A ; B) and M ≤ d(h0 ) (A ; B). Thus d(h0 ) (A ; B) = inf d(Φn ) (A ; B) = lim d(Φn ) (A ; B) . n→∞ n∈N (12) If d(h0 ) (A ; B) = ∞ then obviously d(h0 ) (A ; B) = d(Φn ) (A ; B) for all n ∈ N. Alternatively, if d(h0 ) (A ; B) < ∞ then the lemma follows from Lemma 3.21 together with (12). ✷ At this point we have control over the limits of the semigroups S (Φ) and the distances d(Φ) associated with the truncated forms. Proof of Theorem 2.2 Since, by hypothesis, h ∈ Cl and l satisfies Condition L the foregoing results are applicable. Let λ > 0 be such that h ≤ λ l|A. It follows from the Davies–Gaffney bounds Theorem 2.1 that t log(1A , St (h0 ) 1B ) ≤ −4−1 d(h0 ) (A ; B)2 19 for all t > 0. If d(h0 ) (A ; B) = ∞ then (7) is obviously valid. So we may assume that d(h0 ) (A ; B) < ∞. Then in particular A 6= ∅. Fix x ∈ X. Let Φ1 , Φ2 , . . . ∈ A be such that 0 ≤ Φ1 ≤ Φ2 ≤ . . . ≤ 1, Φn |B(e) (x;n) = 1 and supp Φn is compact for all n ∈ N. Let ε > 0. By Lemma 3.24 there exists an n ∈ N such that d(h0 ) (A ; B) = d(Φn ) (A ; B) and r 2 = λ−1 d(e) (A ; B (e) (x; n)c )2 > d(h0 ) (A ; B)2 + 8ε. Then −1 2 (h ) (Φ ) |(1A , St 0 1B ) − (1A , St n 1B )| ≤ 2 e−(4t) r |A|1/2 |B|1/2 for all t > 0 by Corollary 3.20. Moreover, by Corollary 3.14 there exists a t0 > 0 such that t log(1A , St (Φn ) 1B ) ≥ −4−1 d(Φn ) (A ; B)2 − ε for all t ∈ h0, t0]. Then ( 1A , S t (h0 ) 1B ) ≥ (1A , St(Φn ) 1B ) − 2e−(4t) −1 r 2 −1 d(Φn ) (A;B)2 ≥ e−(4t) |A|1/2 |B|1/2 −1 e−εt −1 2 − 2 e−(4t) r |A|1/2 |B|1/2   −εt−1 1/2 1/2 −(4t)−1 d(h0 ) (A;B)2 −εt−1 1− 2e |A| |B| ≥e e for all t ∈ h0, t0 ], where we used d(h0 ) (A ; B)2 + 4ε − r 2 ≤ −4ε in the last step. There is a −1 t1 > 0 such that 2e−εt1 |A|1/2 |B|1/2 ≤ 4−1 . Then (1A , St (h0 ) and t log(1A , St (h0 ) 1B ) ≥ 2−1 e−(4t) −1 d(h0 ) (A;B)2 −1 e−εt 1B ) ≥ −4−1 d(h0 ) (A ; B)2 − ε − t log 2 for all t ∈ h0, t0 ∧ t1 ]. This completes the proof of Theorem 2.2. ✷ Proof of Corollary 2.4 We may assume that |A| = 6 0 and |B| = 6 0. If ϕ, ψ ∈ L2 (X) then PA ψ ∈ L2 (A) and PB ϕ ∈ L2 (B). So by the Davies–Gaffney bounds of Theorem 2.1 one deduces that |(ψ, PA St PB ϕ)| = |(PA ψ, St PB ϕ)| −1 ≤ e−(4t) d(A;B)2 −1 kPA ψk2 kPB ϕk2 ≤ e−(4t) d(A;B)2 kψk2 kϕk2 . Hence lim sup t log kPA St PB k2→2 ≤ −4−1 d(A ; B)2 . t↓0 So it remains to show that − 4−1 d(A ; B)2 ≤ lim inf t log kPA St PB k2→2 . t↓0 (13) Let A0 ⊆ A and B0 ⊆ B be measurable and relatively compact and assume that |A0 | > 0 and |B0 | > 0. Then (1A0 , St 1B0 ) = (1A0 , PA St PB 1B0 ) ≤ |A0 |1/2 kPA St PB k2→2 |B0 |1/2 20 for all t > 0. So lim inf t log kPA St PB k2→2 t↓0   −1 ≥ lim inf t log(1A0 , St 1B0 ) − 2 t log(|A0 | |B0|) t↓0 = −4−1 d(A0 ; B0 )2 by Theorem 2.2. Now fix x ∈ X and choose A0 = A ∩ B (e) (x ; n) and B0 = B ∩ B (e) (x ; n) with n ∈ N large enough. Then lim inf t log kPA St PB k2→2 ≥ − lim 4−1 d(A ∩ B (e) (x ; n) ; B ∩ B (e) (x ; n))2 n→∞ t↓0 = −4−1 d(A ; B)2 where the equality follows from Lemma 3.4. This proves (13) and the corollary. ✷ Now we turn to the proof of Theorem 2.5. First we use the continuity assumption 2.5.I of the theorem to identify the distance. Lemma 3.25 If D0 (l) ⊂ C(X) then d(l) (A ; B) = d(e) (A ; B) for all non-empty open subsets A, B ⊂ X. Proof Let M ∈ [0, d(l) (A ; B)] ∩ R and ε > 0. Then by [ERSZ05], Lemma 4.2.III, there exists a ψ ∈ D0 (l) such that ψ|B = 0 almost everywhere and ψ|A ≥ M − ε almost everywhere. Since D0 (l) ⊂ C(X) it follows that ψ ∈ D(l)loc ∩ Cb (X). Moreover, since A and B are open one has ψ|B = 0 and ψ|A ≥ M − ε pointwise. So |ψ(x) − ψ(y)| ≥ M − ε for all x ∈ A and y ∈ B. It follows that d(e) (A ; B) ≥ M − ε and d(e) (A ; B) ≥ d(l) (A ; B). The converse inequality is trivial (see Lemma 3.11.I). ✷ Proof of Theorem 2.5 Let x0 , y0 ∈ X and ε ∈ h0, 1]. Choose A = B (e) (x0 ; ε) and B = B (e) (y0 ; ε). Let t ∈ h0, 4−1T ]. Then by assumption II of Theorem 2.5 and symmetry of the kernel −1 ) Kt (x0 ; y0) ≤ K(1+ε)t (x0 ; y) (1 + ε)ν eω(1+εt −1 ) ≤ K(1+ε)2 t (x ; y) (1 + ε)2ν e2ω(1+εt for all x ∈ A and y ∈ B. So Kt (x0 ; y0 ) ≤ |A| −1 |B| −1 Z A dx Z −1 ) dy K(1+ε)2 t (x ; y) (1 + ε)2ν e2ω(1+εt B −1 ) = |A|−1 |B|−1 (1 + ε)2ν e2ω(1+εt (1A S(1+ε)2 t 1B ) . (l) Therefore t log Kt (x0 ; y0 ) ≤ t log(1A S(1+ε)2 t 1B ) − t log(|A| |B|) + 2 ν t log(1 + ε) + 2 ω t + 2 ω ε (l) for all t ∈ h0, 4−1T ]. Since |A|, |B| > 0 one deduces from Corollary 2.3 that lim sup t log Kt (x0 ; y0) ≤ −4−1 (1 + ε)−2d(l) (A ; B) + 2 ω ε = −4−1 (1 + ε)−2 d(e) (A ; B) + 2 ω ε t↓0 21 where we used Lemma 3.25 in the last step. But d(e) (x0 , y0 ) − 2 ε ≤ d(e) (B (e) (x0 ; ε) ; B (e) (y0 ; ε)) ≤ d(e) (x0 , y0 ) + 2 ε by the triangle inequality. So taking the limit ε ↓ 0 one establishes that lim sup t log Kt (x0 ; y0) ≤ −4−1 d(e) (x0 , y0 ) . t↓0 Since, by an analogous argument, −1 ) Kt (x0 ; y0) ≥ K(1−ε)2 t (x ; y) (1 + ε)−2ν e−3ω(1+εt for all t ∈ h0, T ], ε ∈ h0, 1/2i, x ∈ B (e) (x0 ; ε) and y ∈ B (e) (y0 ; ε) one deduces similarly that lim inf t log Kt (x0 ; y0 ) ≥ −4−1 d(e) (x0 , y0) . t↓0 So limt↓0 t log Kt (x0 ; y0 ) = −4−1 d(e) (x0 , y0) and Theorem 2.5 follows. 4 ✷ Applications Theorems 2.2 and 2.5 have a broad range of applications to second-order, divergence-form, elliptic operators both degenerate and non-degenerate. First we discuss the application to operators on Rd . Let cij be bounded real-valued measurable functions on Rd and assume that the d × dmatrix C = (cij ) is symmetric and positive-definite almost-everywhere. Define h by h(ϕ) = d X (∂i ϕ, cij ∂j ϕ) i,j=1 where ∂i = ∂/∂xi and D(h) = W 1,2 (Rd ). Then h is a positive form which is strongly local and Markovian. It is not in general closed or even closable. A characterization of closable forms in one dimension can be found in [FOT94], Theorem 3.1.6, and sufficient conditions in higher dimensions are given in [FOT94], Section 3.1 (see also [RW85] or [MR92], Chapter II). Nevertheless, h ≤ λ l where λ denotes the essential supremum of the matrix norms kC(x)k and l is the form of the usual Laplacian on Rd , i.e., D(l) = W 1,2 (Rd ) and d X k∂i ϕk22 = k∇ϕk22 l(ϕ) = i=1 for all ϕ ∈ D(l). First, consider the case of strongly elliptic operators, i.e., assume that C ≥ µ I almost everywhere, with µ > 0. Then one has λ l ≥ h ≥ µ l and the form h is closed on W 1,2 (Rd ). It follows readily that h is a diffusion satisfying Condition L and the distance d(e) is the Riemannian distance corresponding to the metric C −1 . Moreover, Conditions I and II of Theorem 2.5 are valid. The latter condition follows from the parabolic Harnack inequality (see, for example, [SC02], Chapter 5 and in particular Corollary 5.4.6). Therefore Varadhan’s small time asymptotic result (1) follows for each strongly elliptic operator by Corollary 2.3 and Theorem 2.5. 22 Secondly, consider degenerate operators. Thus C ≥ 0 almost everywhere but this is the only coercivity condition. Nevertheless, h is a strongly local Markovian form and h ≤ λ l|A. Therefore Theorem 2.2 is directly applicable to h with no further assumptions on the coefficients. It then follows that h0 is strongly local and the asymptotic identification (7) is valid. Further detail on the asymptotic behaviour requires more detailed analysis of the set-theoretic distance d(h0 ) (A ; B). In the degenerate situation the set Uh = {x ∈ Rd : kC(x)k = 0} may have non-zero measure and the diffusion associated with the relaxation h0 occurs on the closure of the complement Rd \Uh . In fact the effective space of the diffusion can be smaller since sets of capacity zero act as obstacles [RS05]. The properties of the distance d(h) (A ; B) can be understood for non-degenerate, or weakly degenerate, elliptic operators. In particular it can be analyzed for subelliptic operators in quite general situations. Let M be a Riemannian manifold and X1 , . . . , Xn smooth vector fields on M. Each such vector field X defines a closed linear partial differential operator, also denoted by X, on L2 (M). Now consider the form h(ϕ) = d X kXi ϕk22 = k |∇ϕ| k22 i=1 T with D(h) = ni=1 D(Xi ) and ∇ϕ = (X1 ϕ, . . . , Xn ϕ). The form is automatically closed regular and strongly local. Next assume the form is densely-defined and that the vector fields satisfy the Hörmander condition of order r. It follows readily that ψ ∈ D0 (h) if and only if |∇ψ| ∈ L∞ (M) and |∇ψ|2 ≤ 1 almost everywhere. But it then follows from [RS76], Theorem 17, that ψ is locally Lipschitz. In fact it has a Lipschitz derivative of order 1/r. Consequently, D0 (l) ⊆ C(M). Thus the first assumption of Theorem 2.5 is verified in quite general circumstances. The second assumption is also verifiable in many situations as it is a consequence of a parabolic Harnack inequality. As a specific illustration we give an application to subelliptic operators on Lie groups. Proposition 4.1 Let X1 , . . . , Xn be right invariant vector fields on a Lie group G which satisfy the Hörmander condition, i.e., the vector fields generate the Lie algebra of G. For all i, j ∈ {1, . . . , n} let cij = cji ∈ L∞ (G) be real valued and assume there is a µ > 0 such that (cij (g)) ≥ µ I for almost every g ∈ G. Define the quadratic form h on L2 (G) by h(ϕ) = n X (Xi ϕ, cij Xj ϕ) i,j=1 T with form domain D(h) = ni=1 D(Xi ). Then the form h is a diffusion satisfying Condition L and the assumptions of Theorem 2.5 are satisfied. Hence lim t log Kt (g1 ; g2) = −4−1 d(g1 ; g2)2 t↓0 for all g1 , g2 ∈ G, where K is the kernel of the semigroup generated by the operator H associated to the form h and d(g1 ; g2 ) = sup{|ψ(g1 ) − ψ(g2 )| : ψ ∈ D(h)loc ∩ Cb (G) and |||Ibψ ||| ≤ 1} . 23 (14) Proof It is straightforward to check that h is a diffusion. Note also that h is bounded above and below by multiples of the form l(ϕ) = n X kXi ϕk22 i=1 associated with the sublaplacian given by the vector fields. Therefore the distances d(e) for l and d (defined by (14)) for h are equivalent. Condition L and the first assumption of Theorem 2.5 for the form l and then also for the form h follow since the vector fields satisfy the Hörmander condition. The second assumption of the theorem follows from the parabolic Harnack inequality (again see [SC02], Chapter 5 and in particular Corollary 5.4.6, or the first remark following Theorem 4.4 on page 35–36 in [SC92]). ✷ Next we note that Norris’ result on the small time behaviour of the Laplace–Beltrami operator can also be deduced from our results. Let M be a d-dimensional Lipschitz Riemannian manifold with Borel measure µ which under some chart is locally equivalent with the Lebesgue measure. Further let l denote the Dirichlet form Z l(ϕ) = dµ |∇ϕ|2 M where ∇ denotes the usual gradient and D(l) = W 1,2 (M). The corresponding self-adjoint operator on L2 (M) is the Laplace–Beltrami operator for M. Using a local coordinate (l) 1,2 chart one easily argues that ψ ∈ Wloc ∩ L∞ and |||Ibψ ||| ≤ 1 if and only if ψ ∈ W 1,∞ and k∇ψk∞ ≤ 1. In particular D0 (l) ⊆ C(M). Since the Riemannian distance on M is given by d(x ; y) = sup{|ψ(x) − ψ(y)| : ψ ∈ W 1,∞ , k∇ψk∞ ≤ 1} it follows that Condition L is satisfied. Moreover, the parabolic Harnack inequality is valid (and is used in the proof of Norris (see [Nor97], page 87)). Therefore, by the above reasoning, one can apply Theorem 2.5 to establish lim t log Kt (x ; y) = −4−1 d(x ; y)2 t↓0 for all x, y ∈ M. Note that as in Norris’ argument this proof requires neither smoothness nor completeness of M. Finally we note that the asymptotic estimates can be extended to operators with lower order terms by various arguments such as perturbation theory. For example, if h is a Dirichlet form and v the form of a real, positive, bounded multiplication operator V then the Trotter product formula (h+v) St (h) (v) = lim (St/n St/n )n . n→∞ Therefore one deduces from positivity of the semigroups that (h) (h+v) St e−tkV k∞ ≤ St (h) ≤ St (15) 1B ) = lim t log(1A , St(h) 1B ) (16) for all t > 0. Hence lim t log(1A , St (h+v) t↓0 t↓0 24 for all relatively compact measurable subsets A and B. Thus the small time asymptotics is independent of v. Next, if h is a regular Dirichlet for and v is the form of the operator of multiplication by a real, positive, locally bounded measurable function V , then obviously h + v is densily defined. But in addition h+v is closed by [Sim78] Theorem 4.1, since the closed forms h+vn converge monotonically upwards to h + v, where vn denotes the form of the operator of multiplication by S the bounded function V 1Xn and X1 ⊂ X2 ⊂ . . . are measurable subsets of X with X = ∞ n=1 Xn . These observations extend to the following result. Proposition 4.2 Let h be a positive form satisfying the hypotheses of Theorem 2.2 and with D(h) = D(l). Further, let v be the form of the operator of multiplication by a real, positive, locally bounded measurable function V . Then (h + v)0 = h0 + v = r.lim (hε + v) ε→0 and lim t log(1A , St ((h+v)0 ) t↓0 1B ) = −4−1 d(h0 ) (A ; B)2 for all relatively compact measurable subsets A and B. Proof First, h0 + v is closed, densely-defined and h0 + v ≤ h + v ≤ hε + v for all ε > 0, where in the second inequality we use the additional assumption D(h) = D(l). (Recall that by definition D(hε ) = D(l).) Therefore h0 + v ≤ (h + v)0 ≤ ĥ where ĥ = r.limε↓0 (hε + v). We now establish the first statement of the proposition by proving that ĥ = h0 +Sv. Let Ω1 ⊂ Ω2 ⊂ . . . be open relatively compact subsets of X such that X = ∞ n=1 Ωn . For all n ∈ N let vn denote the form of the operator of multiplication by the bounded (h ) (h ) function V 1Ωn . Let ε > 0. Set hε,n = hε + vn for all n ∈ N. Then St ε,n ≤ St ε for all t > 0 by (15) and ( 1A , S t (hε,n ) 1B ) ≤ (1A , St(hε ) 1B ) ≤ e−(λ+ε) −1/2 d(e) (A;B)2 (4t)−1 |A|1/2 |B|1/2 for all non-empty open A, B ⊂ X and t > 0 where the second bound follows by Lemmas 3.10 and 3.11.I and λ > 0 is such that h ≤ λ l|A. Hence the positive self-adjoint operator Hε,n associated with hε,n has a finite speed of propagation. Explicitly 1/2 (ψ, cos(tHε,n )ϕ) = 0 for all non-empty open A, B ⊂ X, all ϕ ∈ L2 (A), ψ ∈ L2 (B) and all t ∈ R with |t| ≤ (λ + ε)−1/2 d(e) (A ; B) by [ERSZ04], Lemma 3.3, and the assumptions on h. Then arguing as in the proof of Proposition 3.15 one deduces that 1/2 1/2 cos(tHε,n )ϕ = cos(tHε,m )ϕ for all n, m ∈ N, measurable A ⊂ X, ϕ ∈ L2 (A) and t ∈ R with m ≥ n, ∅ 6= A ⊂ Ωn and |t| ≤ (λ + ε)−1/2 d(e) (A ; Ωcn ). The proof is a repetition of the arguments used to prove Proposition 3.15 but now one also uses locality of vn and vm . Therefore using the representation (11) one obtains an estimate (hε,n ) kSt (hε,m ) ϕ − St −1 r 2 n ϕk2 ≤ 2 e−(4t) 25 kϕk2 for all n, m ∈ N, measurable A ⊂ X, ϕ ∈ L2 (A) and t > 0 with m ≥ n, ∅ 6= A ⊂ Ωn , where rn = (λ + ε)−1/2 d(e) (A ; Ωcn ). Now if m → ∞ then hε,m converges monotonically (h ) (h +v) upward to hε + v. Hence St ε,m converges strongly to St ε by [Sim78], Theorem 3.1. (hε,n ) (hε +v) (ĥ) (h +v ) But St converges strongly to St as ε ↓ 0 and St converges strongly to St 0 n as ε ↓ 0 because r.limε↓0 hε,n = h0 + vn . Combining these observations one deduces that (h0 +vn ) kSt −1 r 2 n (ĥ) ϕ − St ϕk2 ≤ 2 e−(4t) kϕk2 (17) for all n ∈ N, measurable A ⊂ X, ϕ ∈ L2 (A) and t > 0 with ∅ = 6 A ⊂ Ωn . Finally r.limn→∞ (h0 + vn ) = h0 + v by monotone convergence and as rn → ∞ as n → ∞ one (h +v) (ĥ) (h +v) (ĥ) concludes that St 0 ϕ = St ϕ for all ϕ ∈ L2,c (X) and t > 0. Thus St 0 = St for all t > 0. Hence ĥ = h0 + v. Finally we deduce from (15) that (1A , St 1B ) = (1A , St (ĥ) (h0 +v) 1B ) ≤ (1A , St(h0 ) 1B ) ≤ e−d (h0 ) (A;B)2 (4t)−1 |A|1/2 |B|1/2 for all non-empty open A, B ⊂ X and t > 0, where the last bound uses Theorem 2.1. Finally the proof of the proposition is completed by repeating the arguments used above in the proof of Theorem 2.2 with S (h0 ) replaced by S (ĥ) and S (Φ) replaced by S (h0 +vn ) . The equality (16) and inequality (17) replace Corollaries 3.14 and 3.20. ✷ Note that the identification (h + v)0 = h0 + v shows that (h + v)0 is local in the sense of [FOT94]. Acknowledgements This work was supported by the Australian Research Council (ARC) Discovery Grant DP 0451016. The greater part of this work was carried out during a visit of the first named author to the Australian National University. The work was completed whilst the first and second named authors were visiting the Centre International de Rencontres Mathématiques at the Université de la Méditerranée at Luminy. 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