Two-sided Estimates on the Density of the Feynman-Kac Semigroups of Stable-like Processes

In this paper we establish two-sided estimates for the density of the Feynman-Kac semigroups of stable-like processes with potentials given by signed measures belonging to the Kato class. We also provide similar estimates for the densities of two other kinds of Feynman-Kac semigroups of stable-like processes.


Introduction
Suppose that X = (X t , P x ) is a Brownian motion on R d and that V is a function on R d belonging to the Kato class of X, i. e., a function satisfying the condition It is well known (see [11] for instance) that the Feynman-Kac semigroup {T V t : t ≥ 0} with potential V has a transition density q V (t, x, y) with respect to the Lebesgue measure and that q V has both an upper and a lower Gaussian estimates, that is there exist positive constants c 1 , c 2 , c 3 , c 4 such that for all (t, x, y) ∈ (0, ∞) × R d × R d . This result can be easily generalized (see [2] for instance) to the case when V is replaced by a signed measure satisfying )|µ|(dy)ds = 0 and t 0 V (X s )ds is replaced by the continuous additive functional A µ t of X associated with µ. Now suppose that α ∈ (0, 2) and that X = (X t , P x ) is a symmetric α-stable process on R d . The question that we are going to address in this paper is the following: can one establish two-sided estimates for the density of the Feynman-Kac semigroup of the symmetric α-stable process X? As far as we know, this question has not been addressed in the literature. The proof of (1.1) in [11] and [2] can not be adapted to the case of discontinuous stable processes. It seems that, to answer the question above, one has to use some new ideas. In this paper, we are going to tackle the question above by adapting an idea used in [13] and [14] to establish heat kernel estimates for diffusions to the present case. Actually, instead of dealing with symmetric stable processes, we are going to deal with the more general stable-like processes introduced in [4].
The content of this paper is organized as follows. In section 2, we first recall the definition of the Kato class with respect to symmetric α-stable processes and some basic facts about stable-like processes, and then we present some preliminary results on Feynman-Kac semigroups. In section 3 we establish two-sided estimates on the density of Feynman-Kac semigroups with potentials given by measures belonging to the Kato class. In the last section we deal with two other kinds of Feynman-Kac semigroups of stable-like processes. The first kind consists of Feynman-Kac semigroups given by purely discontinuous additive functionals, and the second kind consists of Feynman-Kac semigroups given by continuous additive functionals of zero energy.
In this paper we will use the following convention on the labeling of constants. The values of the constants M 1 , M 2 , · · · will remain the same throughout this paper, while the values of the constants C 1 , C 2 , · · · might change from one appearance to the next. The labeling of the constants C 1 , C 2 , · · · starts anew in the statement of each result.

Kato Class and Basic Properties of Feynman-Kac Semigroups
In this paper we will always assume that α ∈ (0, 2). We will use X 0 = {X 0 t , P 0 x } to denote a symmetric α-stable process in R d whose transition density p 0 (t, x, y) = p 0 (t, x − y) with respect to the Lebesgue measure satisfies It is known (see [3]) that there exist positive constants M 1 < M 2 such that for all x, y ∈ R d . When α < d, the process X 0 is transient and its potential density G 0 (x, y) = G 0 (x − y) is given by The Dirichlet form (E 0 , F) of X 0 is given by .
For any function V on R d and t > 0, we define By a signed measure we mean in this paper the difference of two nonnegative measures at most one of which can have infinite total mass. For any signed measure on R d , we use µ + and µ − to denote its positive and negative parts, and |µ| = µ + + µ − its total variation. For any t > 0, we define Definition 2.1 We say that a function V on R d belongs to the Kato class K d,α if lim t↓0 N V (t) = 0. We say that a signed Radon measure µ on R d belongs to the Kato class K d,α if lim t↓0 N µ (t) = 0.
Rigorously speaking a function V in K d,α may not give rise to a signed measure µ in K d,α since it may not give rise to a signed measure at all. However, for the sake of simplicity we use the convention that whenever we write that a signed measure µ belongs to K d,α we are implicitly assuming that we are covering the case of all the functions in K d,α as well.
The following result is well known, see [1] and [12] for instance.
We assume from now on that m is a measure on R d given by for some positive constants M 3 < M 4 . We will fix a symmetric function c(x, y) on R d × R d which is bounded between two fixed positive constants. If for any u, v ∈ F we define . It is shown in [4] that, associated with this Dirichlet form, there is an m-symmetric Hunt process X = {X t , P x } on R d which can start from any point x ∈ R d . We will use {M t ; t ≥ 0} to denote the natural filtration of X.
The process X is called an α-stable-like process in [4]. It is also shown in [4] that the process X admits a transition density p(t, x, y) with respect to m and that p is jointly continuous on (0, ∞) × R d × R d and satisfies the condition A + t and A − t being the positive continuous additive functionals of X associated with µ + and µ − respectively.
Furthermore, if µ ∈ K d,α and A t is the continuous additive functional of X associated with µ, then Proof. We omit the details. 2 In the sequel, whenever we have a signed measure µ ∈ K d,α , we will use A µ t to denote the continuous additive functional of X associated with µ. Using Khas'minskii's lemma (see Lemma 2.6 of [2]), we can easily show the following Lemma 2.3 Suppose that µ ∈ K d,α and A µ t is the continuous additive functional of X associated with µ. There exist positive constants C 1 and C 2 , depending on µ only via the rate at which N µ (t) goes to zero, such that The meaning of the phrase "depending on µ only via the rate at which N µ (t) goes to zero" will become clear in the proof of Theorem 3.3. It roughly means that if w(t) is a increasing function on (0, ∞) with lim t→0 w(t) = 0, then there exist positive constants C 1 and C 2 such that for any signed measure µ with For any µ ∈ K d,α , we define the Feynman-Kac semigroup {T µ t : t ≥ 0} with potential µ by When µ is given by µ(dx) = U (x)dx for some function U , we will sometimes write T µ t as T U t . The following result is well known, see [12] and [5].
Using an argument similar to that of the proof of Theorem 3.1 in [2], we can show the following Theorem 2.5 For any µ ∈ K d,α , there exists a function q µ (t, x, y) such that 1. q µ is jointly continuous on (0, ∞) × R d × R d ; 2. there exist positive constants C 1 and C 2 depending on µ only via the rate at which N µ (t) goes to zero such that x, z)q µ (s, z, y)m(dz) = q µ (t + s, x, y) for all t, s > 0 and (x, y) ∈ R d × R d ; 5. q µ (t, x, y) is symmetric in x and y; 6. if f is a bounded function continuous at x ∈ R d , then Proof. We omit the details. 2 Corollary 2.6 For any µ ∈ K d,α , the function q µ in the theorem above satisfies the equation Proof. Since for any t > 0 for all (t, x) ∈ (0, ∞)×R d and all bounded functions f on R d . Now the conclusion of the corollary follows easily from the Markov property, Fubini's theorem and the two theorems above. 2 When the measure µ is given by µ(dx) = U (x)dx form some function U , we will sometimes write q µ as q U .

Two-sided Estimates for Densities of Local Feynman-Kac transforms
In this section we shall establish two-sided estimates for the densities of Feynman-Kac semigroups with potentials belonging to K d,α . The following elementary lemma will play an important role.
Proof. We prove this lemma by looking at all the different cases.
In the first case we assume that a ≥ b. In this case we have c < b ≤ a < d, so the left and right hand sides of (3.1) are both equal to 1. Thus (3.1) is valid in this case.
In the second case we assume that c ≤ a ≤ b. We further divide this case into two subcases. In the first subcase we assume that c ≤ a ≤ b ≤ d. In this subcase the left hand side of (3.1) is equal to a b and the right hand side is equal to 1. In the second subcase we assume that c ≤ a < d ≤ b. In this subcase the left hand side of (3.1) is equal to a b and the right hand side is equal to d b . Thus (3.1) is valid in this case.
In the third case we assume that a ≤ c. We further divide this case into three subcases. In the first subcase we assume that a ≤ c < b ≤ d. In this subcase the left hand side of (3.1) is equal to a b and the right hand side is equal to a c . In the second subcase we assume that a ≤ c ≤ d ≤ b. In this subcase the left hand side of (3.1) is equal to a b and the right hand side is equal to ad bc ≥ a b . In the third subcase we assume that a < d ≤ c < b. In this subcase the left and right hand sides of (3.1) are both equal to ad bc . Thus (3.1) is also valid in this case. 2 The following lemma is similar to Lemma 3.1 of [14] and is crucial in establishing the main estimates of this paper.

Thus we have
By a similar argument we get

Consequently we have
Theorem 3.3 For any µ ∈ K d,α , there exists a positive constant T , depending on µ only via the rate at which N µ (t) goes to zero, such that for some constants C 1 and C 2 depend only on M 5 and for all (t, x, y) Proof. For (t, x, y) ∈ (0, ∞) × R d × R d , we define I n (t, x, y) recursively for n ≥ 0 by I 0 (t, x, y) = p(t, x, y), We claim that there exists a positive constant T , depending on µ only via the rate at which N µ (t) goes to zero, such that for all n ≥ 1 and (t, x, y) (3.3) We will prove this claim by induction. In fact, for n = 1, we have Applying Lemma 3.2 we get that there exists a constant c 1 > 0 depending only on d and α such that Take T > 0 small enough so that Obviously, this T depends on µ only via the rate at which N µ (t) goes to zero and Thus the claim above is valid for n = 1. Now suppose that the claim is valid for n. Then we have

Feynman-Kac Semigroups Given by Discontinuous Additive Functionals and Continuous Additive Functionals of Zero Energy
We first deal with a class of Feynman-Kac semigroups given by a purely discontinuous additive functional. To do this, we need to recall a definition and introduce some notations.
We say that F belongs to the class J d,α if F is bounded, vanishing on the diagonal, and the function It is easy to see from the definition above that if F ∈ J d,α , then e −F is also in J d,α .
The process X has a Lévy system (N, H) given by H t = t and for every x ∈ R d and t > 0.
For any F belonging to J d,α , we put We can define the following so-called non-local Feynman-Kac semigroup This semigroup has been studied in [12] and [6].
Theorem 4.1 Suppose that F ∈ J d,α is a symmetric function. The semigroup {S F t , t ≥ 0} admits a density k F (t, x, y) with respect to m and that k F is jointly continuous on (0, ∞) × R d × R d . Furthermore, there exist positive constants C 1 , C 2 , C 3 and C 4 such that Let L ρ t be the solution of the following SDE: It follows from the Doleans-Dade formula that where the last equality is shown on page 487 of [7]. L ρ t is a nonnegative local martingale and therefore a supermartingale multiplicative functional of X. Therefore by Theorem 62.19 of [10] L ρ t defines a family of probability measures {P x , x ∈ E} by dP x = L ρ t dP x on M t . We will usẽ X = (X t ,P x ) denote this new process. Put ν(dx) = ρ 2 (x)m(dx). It follows from [7] thatX is a ν-symmetric Hunt process on R d whose Dirichlet form (Ẽ,F) on L 2 (R d , ν) is given byF = F andẼ (u, u) = R d R d ρ(x)ρ(y)c(x, y)(u(x) − u(y)) 2 |x − y| d+α m(dx)m(dy), u ∈ F.

Remark 4.3
Of course, one can combine Theorems 3.4, 4.1 and 4.2 into one theorem about the density of Feynman-Kac semigroup given by additive functionals involving all three components: a continuous part with finite variation, a continuous part of zero energy and a purely discontinuous part. We leave this to the reader.