An integrated version of Varadhan's asymptotics for lower-order perturbations of strong local Dirichlet forms

The studies of Ram\'irez, Hino-Ram\'irez, and Ariyoshi-Hino showed that an integrated version of Varadhan's asymptotics holds for Markovian semigroups associated with arbitrary strong local symmetric Dirichlet forms. In this paper, we consider non-symmetric bilinear forms that are the sum of strong local symmetric Dirichlet forms and lower-order perturbed terms. We give sufficient conditions for the associated semigroups to have asymptotics of the same type.


Introduction
Let (E, B, µ) be a σ-finite measure space and (E 0 , D) a symmetric strong local Dirichlet form on the L 2 space of (E, B, µ). Let {T 0 t } t>0 denote the semigroup associated with (E 0 , D), and set P 0 t (A, B) = A T 0 t 1 B dµ for t > 0 and A, B ∈ B with positive and finite measure. In [1], the following small-time asymptotic estimate for {T 0 t } t>0 was proved as a generalization of results from previous work [16,8,9]: (1.1) Here, d(A, B) is the intrinsic distance between A and B, which can be determined from only (E 0 , D) (see [1, p. 1241] or Definition 2.6 below for details). Similar small-time asymptotics of transition densities have been studied extensively. These are usually called Varadhan-type estimates, in reference to [19]. In particular, that the estimate holds was proved in [15] for a class of symmetric and uniform elliptic diffusion processes on Lipschitz manifolds. This is one of the most general results. Asymptotics of the form (1.1) can be considered as an integrated version of Varadhan's asymptotics. The purpose of this paper is to extend the formula (1.1) to a class of non-symmetric bilinear forms. Specifically, we first assume that (E 0 , D) mentioned above is expressed as where D is a first-order derivation operator taking values in a separable Hilbert space H. Our main object is to obtain small-time asymptotics for a non-symmetric form (E, D) given by the sum of E 0 and the lower-order perturbed term E (b, Df ) H g dµ+ E (c, Dg) H f dµ+ E V f g dµ (see (2.4)). When the perturbed term is small relative to E 0 , the form (E, D) becomes a lowerbounded bilinear form and has an associated positivity-preserving semigroup {T t } t>0 on L 2 (E, µ). For measurable sets A and B having positive and finite µ-measure, let P t (A, B) = A T t 1 B dµ, as before. We study the conditions on b, c, and V that suffice for the semigroup {T t } t>0 to have the same integrated Varadhan's asymptotics as {T 0 t } t>0 . That is, for What kind of restrictions should we impose on b, c, and V to guarantee the validity of (1.2)? It is reasonable to expect that (1.2) would hold if they were sufficiently smaller than E 0 in terms of quadratic forms. From another perspective, we can make a probabilistic argument, exemplified in the following typical case. Let (E, H, µ) be an abstract Wiener space, and suppose that (E 0 , D) and (E, D) are defined as where D denotes the H-derivative in the Malliavin calculus, b is an H-valued measurable function on E, and D 1,2 is the first-order L 2 -Sobolev space on E. If exp(γ|b| 2 H ) is µ-integrable for some γ > 8, then by using the logarithmic Sobolev inequality, we can prove that (E, D) is well-defined as a lower-bounded bilinear form and that there exists a corresponding semigroup {T t } t>0 on L 2 (E, µ) (see Example 5.5). Moreover, {T t } t>0 has a probabilistic representation as where ({X t } t≥0 , {P x } x∈E ) is the Ornstein-Uhlenbeck process associated with (E 0 , D) and {M t } t≥0 is a martingale additive functional suitably associated with b (see, e.g., [7]). Note, in particular, that the quadratic variation of M is given by M t = t 0 |b(X s )| 2 H ds. From Hölder's inequality, for measurable sets A and B with positive µ-measure, where p > 1 and q is the conjugate exponent of p. The first term of the right-hand side is P 0 t (A, B) 1/p . The second term is dominated by µ(A) 1/2q . The third term is estimated by Jensen's inequality: By the exponential integrability of |b| 2 H , the right-hand side is equal to which is finite for sufficiently small positive values of t and converges to µ(A) as t → 0. Combining these estimates and the asymptotics with respect to (E 0 , D) and letting p → 1, we obtain the upper estimate This kind of probabilistic argument is applicable to more general situations, by using a generalized Cameron-Martin-Maruyama-Girsanov formula (see, e.g., [12,17,5]). The exponential integrability condition imposed above is not exactly consistent with smallness in the sense of quadratic forms. Indeed, in the estimate of (1.3), we used the fact that exp(γ|b| 2 H ) is µ-integrable for only some γ > 0. Therefore, it is reasonable to consider two types of smallness-smallness in term of quadratic forms and in terms of some exponential integrability-in describing the conditions sufficient for (1.2).
In this paper, we introduce conditions that take the observation above into consideration (see conditions (B.2) A,B , (B.2 ′ ), and Proposition 2.14) and prove the upper estimate under their assumptions (Theorem 2.10). Moreover, we prove that the lower estimate holds under minimal assumptions on b, c, and V along with the assumption of the validity of the upper estimate (Theorem 2.11). Combining these two results gives sufficient conditions for the integrated Varadhan estimates. As in the previous studies [16,9,1], the proof is purely analytic and only a measurable structure is imposed on the state space. We remark that even for b = c, that is, even with (E, D) as a symmetric form, our results are new.
Because the proof is long, we briefly explain the broad ideas of the proof here. The upper estimate (Theorem 2.10) is proved in the spirit of Davies-Gaffney's method. In previous works [16,9,1], they define σ(t) = E (e αw T t 1 B ) 2 dµ for given α > 0 and w with |Dw| H ≤ 1 µ-a.e., and deduce the key differential inequality σ ′ (t) ≤ α 2 σ(t). Solving this inequality and optimizing it with respect to α and w yields the desired estimate. Under the assumptions of our theorem, however, the perturbed terms cannot be controlled. Instead, we define σ in the form σ(t) = E (e αw T t 1 B ) p(t) dµ, where p(t) = q − St with q > 2 and S > 0 being chosen suitably. Since {T t } t>0 can be extended to a semigroup on L p (µ) for p near 2 in our setting, σ(t) is finite for small t. The variable exponent p(t) means that the derivative of σ involves an extra logarithmic term, which suppresses the influence on b and c. The price to pay for this is that the resulting differential inequality is coarser, in the form σ ′ (t) ≤ (1 + ε)α 2 σ(t) + Cσ(t) max{0, − log σ(t)}. Fortunately, the extra logarithmic term has no influence on the Varadhan-type estimate. Introducing a variable exponent is a standard technique for estimating heat kernel densities (see, e.g., [4]), but (unlike in such a context) p(t) is taken to be a decreasing function in this study. The definition of σ shown above is valid when µ is a finite measure; in general cases, we further need to modify the definition of σ (see (3.15) and (3.6)) to avoid some technical obstacles. For this reason, we need a series of quantitative estimates, which makes the proof long.
The proof of the lower estimate is based on previous studies [16,9,1], but the argument is more complicated due to the perturbed terms and the fact that the semigroup {T t } t>0 preserves positivity but is not Markovian. We will outline the proof by the following formal argument. We see the function u t = −t log T t 1 B satisfies the relation where L 0 is the generator of {T 0 t } t>0 . (This identity corresponds to (4.2) in the actual argument.) If we assume for argument that the left-hand side converges to 0 as t → 0 and the last term of the right-hand side is negligible, then the limit u 0 of u t (if it exists) will satisfy |Du 0 | 2 H = 2u 0 . What is actually obtained is an inequality of the form |Du 0 | 2 H ≤ 2u 0 , which implies |D √ 2u 0 | H ≤ 1. Furthermore, u 0 = 0 µ-a.e. on B should be satisfied. Under these conditions, we have the formal inequality by the definition of d, which is close to the lower-side estimate. Several difficulties, such as that u t is not necessarily bounded, make it hard to justify this procedure directly. To cope with problems such as the integrability (or not) of various terms and the existence (or not) of limits, we introduce a nice truncating function φ and bump functions {χ k }, and consider Note that these cut-off functions are slightly different from those in [1,9] in order to deal with the lack of the Markov property of {T t } t>0 . This modification results in an increasing number of terms in the quantitative estimates as the proof progresses, which makes the proof longer and more technical than without the modification.
This paper organized as follows. In Section 2, we introduce a framework and state the main theorems. Section 3 provides the proof of the upper estimate (Theorem 2.10). Section 4 provides the proof of the lower estimate (Theorem 2.11). In the last section, we prove some auxiliary propositions, discuss the conditions imposed on the theorems, and show some typical examples.

Framework
Let (E, B, µ) be a σ-finite measure space, and H a real separable Hilbert space. The inner product and norm of H will be denoted by (·, ·) H and | · | H , respectively. The set of all real-valued measurable functions on E is denoted by L 0 (µ), where two functions are identified if they coincide on µ-a.e. For p ∈ [1, ∞], the real L p space on (E, B, µ) is denoted by L p (µ), and its norm by · p . The L 2 space of H-valued measurable functions on (E, B, µ) is denoted by L 2 (µ; H), and its norm by · 2 .
Let D be a dense subspace of L 2 (µ), and D be a closed linear operator from L 2 (µ) to L 2 (µ; H) with domain D. We assume that D has the following derivation property: For arbitrary functions f 1 , f 2 , . . . , f m ∈ D and C 1 functions F on R m with bounded first-order derivatives and F (0) = 0, F (f 1 , . . . , f m ) belongs to D and Then, a bilinear form (E 0 , D) on L 2 (µ), defined by is a Dirichlet form on L 2 (µ). Moreover, this bilinear form has a strong local property: For any f ∈ D and C 1 -functions F, G on R with bounded first-order derivatives such that the For other equivalent statements, see [3,Proposition I.5.1.3], where this property is called a local property. For l ≥ 0, we write E 0 l (f, g) for E 0 (f, g) + l E f g dµ. We also use E 0 (f ) and E 0 l (f ) to denote E 0 (f, f ) and E 0 l (f, f ), respectively. The space D becomes a Hilbert space with the inner product (f, g) → E 0 1 (f, g). The following proposition is fundamental.
If a function f ∈ D is constant µ-a.e. on a set A ∈ B, then Df = 0 µ-a.e. on A.
For A ∈ B, we set We follow [1] in introducing the concept of measurable nests and related function spaces. (i) For every k ∈ N, there exists h k ∈ D such that h k ≥ 1 µ-a.e. on E k . (ii) The set ∞ k=1 D E k is dense in D. Remark 2.3. We note that measurable nests exist, from [1, Lemma 3.1]. For every k ∈ N, we have µ(E k ) < ∞ because of condition (i). By condition (ii), Df is defined as an H-valued measurable function on E by Df = Df k on E k , where f k ∈ D and f k = f µ-a.e. on E k . From Proposition 2.1, Df is well-defined up to µ-equivalence.
This definition is consistent with [1, Definition 2.6], which considers more general situations. The function space D 0 ({E k }) does not depend on the choice of {E k } ∞ k=1 , from [1, Proposition 3.9]; we therefore denote it as D 0 . We now define the intrinsic distance between two sets as follows.
Definition 2.6 (see [1, p. 1241]). For A, B ∈ B with positive µ measures, we define where the essential infimum ess inf and essential supremum ess sup are taken with respect to µ.
We introduce the concept of a distance-like function d B from the set B and quote a result from [1]. For B ∈ B and N ≥ 0, define For x, y ∈ R, we let x ∨ y and x ∧ y denote max{x, y} and min{x, y}, respectively.  To introduce the perturbation terms, let b and c be H-valued measurable functions on E, and let V be a real measurable function on E. From here, we always assume the following minimal requirement.
(A.2) There exist η ∈ [0, 1), θ ≥ 0, ω ≥ 0, and l ≥ 0 such that, for every f, g ∈ ∞ k=1 D E k ,b , It follows that Therefore, E(·, ·) extends continuously to a bilinear form on D and the bilinear form D × D ∋ (f, g) → E θ (f, g) := E(f, g) + θ E f g dµ is a coercive closed form on L 2 (µ). Thus, a strongly continuous semigroup {T t } t>0 exists on L 2 (µ) and some closed operator (L, Dom(L)) on L 2 (µ) associated with (E, D) satisfies E(f, g) = − E (Lf )g dµ for f ∈ Dom(L) and g ∈ D. In particular, T t can be given as where D * denotes the adjoint operator of D. Confirming that (E θ , D) satisfies the condition (S) in [14,Proposition 1.2] by an argument similar to [14, Proof of Theorem 2.2] and applying [14,Theorem 1.5], we see that {T t } t>0 is positivity preserving. That is, T t f ≥ 0 µ-a.e. if f ≥ 0 µ-a.e. In general, {T t } t>0 is not necessarily Markovian. LetT t denote the adjoint operator of T t on L 2 (µ). Then, {T t } t>0 is also a positivity-preserving semigroup and is associated with a bilinear form (Ê, D) defined bŷ Let B 0 denote the set of all sets A ∈ B such that 0 < µ(A) < ∞. For A, B ∈ B 0 , define If d(A, B) = ∞, then the situation is simple, with the proof of the following proposition given in Section 5. From this proposition, it is sufficient to consider the case when d(A, B) < ∞. For this case, we need some extra assumptions to begin. Let log ± x denote 0 ∨ (± log x) for x ≥ 0. Then, Accordingly, In particular, we have the following theorem.
Proposition 2.14. Suppose that b and c are decomposed into b = b 1 + b 2 and c = c 1 + c 2 such that b 1 , b 2 , c 1 , c 2 are measurable and the following hold.
The proof is based on a simple application of a type of Hausdorff-Young inequality. We provide the proof in Section 5 together with a discussion of other sufficient conditions.

L p property of semigroups
The following proposition is interesting in its own right, as well as being used in the proof of Theorem 2.10. Although claims of the kind made by the proposition have been studied in many papers (e.g., [11,18,5] and the references therein), we give a proof for completeness since our framework is slightly different from that used in other proofs.

Preliminary estimates
In this subsection, we provide several quantitative estimates used in the proof of Theorem 2.10.
Proof. (i) and (v): Straightforward from the definitions.
(ii): The first and last inequalities are easy to prove. Since the second inequality also holds.
(iv): This follows by integrating each term of the inequality in (ii).

Lemma 3.4.
For any x ≥ 0 and y ≥ 2, the following hold.
Proof. Since g R (x, y) = g R (R, y) for x ≥ R, the term g R (x, y)x −ε is always non-increasing for x ≥ R. It therefore suffices to consider only x in [0, R]. We may additionally assume that ε ∈ (0, 1/2). Define It suffices to prove that there exists some y 0 (ε) > 2 such that, for (x, y) and thus, (3.12) holds for this case.
Thus, (i) and (ii) hold. Combining (i) and Lemma 3.4(i) gives (iii). From (i) and (ii), which proves (iv). Next, we prove (v). From (ii) and Lemma 3.4(iii), we have Then, The second inequality of (v) follows from Lemma 3.3(iv). Last, we prove (vi). The inequality holds for x = 0 by direct computation. Let x > 0. By using (ii) and Lemma 3.4(i), we have
We fix t ∈ (0, t 0 ] and estimate the first term of (3.16). Recall the measurable nest {E k } ∞ k=1 in Assumption 2.8. Take a sequence of functions {w k } ∞ k=1 such that for every k ∈ N, w k ∈ D E k ,b , w k = w µ-a.e. on E k and 0 ≤ w k ≤ N µ-a.e. There also exist functions for each k and the sequence u (k) converges to u t in D as k → ∞. By considering 0 ∨ (u (k) ∧ u t ) (and the Cesàro means if necessary) we may assume that 0 ≤ u (k) ≤ u t µ-a.e. for every k and lim k→∞ u (k) = u t µ-a.e. Let F (k) = e αw k u (k) ∈ D E k ,b for each k. Then, Proof. Since u (k) converges to u t in D as k → ∞, proving that the sequence {h R (F (k) , p(t))e αw k } ∞ k=1 is bounded in D suffices. Boundedness in L 2 (µ) is straightforward. For each k, We note that g R is a bounded function, that |e αw k − 1| ≤ e |α|N , that h R (F (k) , p(t)) = 0 on E \ E k , that |Dw k | H ≤ 1 on E k and that which is bounded in k. From these estimates, the first and third terms of (3.19) are bounded in k. Moreover, Lemma 3.3(ii) and the inequality 2 < 2(q − 1) < q 0 together imply that {h R (F (k) , p(t))} ∞ k=1 is bounded in L 2 (µ). Thus, the second term of (3.19) is also bounded in k, which completes the proof.  (3.20) We provide an upper estimate of the right-hand side. Let G (k) R = ρ R (F (k) , p(t)) for k ∈ N. For the moment, we omit p(t) from the notation and write, for example, h R (F (k) ) instead of h R (F (k) , p(t)). We have Using (3.14), we have (from Lemma 3.4(i) and Lemma 3.8(iv)) (from Lemma 3.8(vi) and (iii)) Moreover, when α > 0, dµ.
This inequality implies (3.28) We can confirm that so that (3.28) implies σ(t) ≤ exp U{χ(log σ(0)) + t} . For the proof of Theorem 2.10, we assume that d(A, B) > 0 because otherwise the assertion is trivial. Define for t > 0 and α > 0. Both σ 1 (t, α) and σ 2 (t, α) have the same kind of estimates as (3.29). Indeed, for the estimate of σ 2 , the discussion in the previous subsection is applied with b, c, and α replaced by c, b, and −α, respectively. The only term that requires care is I 2 , but the estimate (3.21) is unchanged by this replacement. From the Cauchy-Schwarz inequality and Lemma 3.3(ii), We also have We remark that U and χ depend on α (see (3.26) and ( Uχ(x) = x for x ∈ R, in view of (3.27). In particular, U = O(t −2 ) as t → 0. We also remark that lim ε→0 β(ε) = 1. Then, we obtain Therefore, Also, for t small enough that σ 2 (0, N/t) < 1, (3.34) By combining (3.32), (3.33), and (3.34), Letting ε → 0, we obtain (2.8), which finishes the proof of Theorem 2.10. Remark 3.11. (i) As seen from the proof, when we can let γ = 0 in (B.2) A,B , the L panalysis is not necessary and the proof becomes much simpler. (ii) If (E, B, µ) is a finite measure space, then we can define σ, σ 1 , and σ 2 as and use the inequality in place of (3.15), (3.30), and (3.31). This change makes the proof of Theorem 2.10 shorter and simpler since the fine estimates in Section 3.2 are not necessary.

Cutoff functions and their properties
We turn to the lower-side estimate and prove Theorem 2.11. In Section 2.1 of [9], some nice concave functions are introduced as cutoff functions. Because our semigroup {T t } t>0 does not have the Markov property in general, we need to modify these functions to be suitable. First, we take a real-valued function g on R satisfying the following properties: • g is an odd and bounded C 3 -function; These conditions imply that lim x→∞ g(x) = L, lim x→−∞ g(x) = −L for some L > 1 and that the convergence is monotone. Note that g is concave on [−1, ∞).
Define our main cutoff functions at level K > 0 by From the conditions on g, we have the following properties: To simplify the notation, we omit explicit indication of the dependency on K for most of this section. For example, we write φ instead of φ K whenever the value of K is clear from the context. The monotonicities of φ, Φ, (C.3), and (C.5) guarantee that φ, Φ, and Ψ can be extended to continuous functions on [−∞, ∞], and these extensions use the same symbols. The following estimates result: Indeed, this inequality is trivial when x ≤ M, and when We also introduce a functionΦ K on R, defining it aŝ This function is concave and 1-Lipschitz on R.
Proof. The proof here follows the proof of Lemma 2.1 in [9]. Compute From this and (C.2), F has a nonnegative second derivative. If f n → f weakly in L 2 (µ), F (f n ) ∈ L 2 (µ) for each n, and F (f n ) has a subsequence that converges to some functionF weakly in L 2 (µ), thenF ≤ F (f ).
. From Ξ(0) = 0, the boundedness of the derivative of Ξ on [0, ∞), and u δ t − e δ t = −tΞ(T t f ), we conclude that u δ t − e δ t ∈ D. We prove the identity (4.2). First, The first term on the right-hand side is computed as Combining the identities (4.3), (4.4), and (4.5), it holds that By using the identity (ρ, Here, we have These estimates and Lemma 4.5 together imply the claim.
Proof. For k ∈ N, there exists an h k ∈ D such that h k ≥ 1 on E k . Take a function g k from ∞ n=1 D En such that E 0 the Cesàro means of a certain subsequence of k=1 Y k and define η n = 0 ∨ 2(g 1 ∨· · ·∨g n ) ∧1 . Then, Z n ⊂ E n and ∞ n=1 D Zn is dense in D. Moreover, 0 ≤ η n ≤ 1 on E, η n = 1 on Z n and η n ∈ ∞ k=1 D E k . Take a strictly increasing sequence {m(n)} ∞ n=1 such that η n ∈ D E m(n) for every n ∈ N and definê E k = ∅, χ k = 0 for 1 ≤ k < m(1), E k = Z n , χ k = η n for m(n) ≤ k < m(n + 1), n = 1, 2, . . . .
We may also assume, by taking a further subsequence if necessary, that there exist Φ 0 ,Φ 0 , andΨ 0 in L ∞ (µ) such that Φ t n ′ → Φ 0 ,Φ t n ′ →Φ 0 ,Ψ t n ′ →Ψ 0 both in the weak-L 2 (μ) sense and in the weak * -L ∞ (µ) sense. Here,μ is an arbitrary fixed finite measure on E such that µ and µ are mutually absolutely continuous, and L ∞ (µ) is regarded as the dual space of L 1 (µ). We remark that these functions depend on K. Because of this, it is more precise to write φ K t andΦ K 0 instead of φ t andΦ 0 .
is true for some β > 1 for every K > 0 and every limitφ K 0 , then µ-a.e. on {d B < N}.
From Lemma 4.14 and (C.4), we haveΦ 0 = d 2 B /2 on {d 2 B /2 ≤ K} ∩ {d B < N} =: Y K,N , independently of the choice of subsequence. In particular,Φ t 1 Y K,N converges to (d 2 B /2)1 Y K,N weakly in L 2 (μ) as t → 0. Furthermore, we have the following SinceΦ K is concave and 1-Lipschitz, the first term is estimated as follows:

Proof of auxiliary propositions and examples
We prove Propositions 2.9 and 2.14.