Heat Kernels for Non-symmetric Non-local Operators

We survey the recent progress in the study of heat kernels for a class of non-symmetric non-local operators. We focus on the existence and sharp two-sided estimates of the heat kernels and their connection to jump diffusions.


Introduction
Second order elliptic differential operators and diffusion processes take up, respectively, an central place in the theory of partial differential equations (PDE) and the theory of probability. There are close relationships between these two subjects. For a large class of second order elliptic differential operators L on R d , there is a diffusion process X on R d associated with it so that L is the infinitesimal generator of X, and vice versa. The connection between L and X can also be seen as follows. The fundamental solution (also called heat kernel) for L is the transition density function of X. For example, when where (a ij (x)) 1≤i,j≤d is a d × d symmetric matrix-valued continuous function on R d that is uniformly elliptic and bounded, and b(x) = (b 1 (x), . . . , b d (x)) is a bounded R d -valued function, there is a unique diffusion X = {X t , t ≥ 0; P x , x ∈ R d } on R d that solves the martingale problem for (L, C 2 c (R d )). That is, for every x ∈ R d , there is a unique probability measure P x on the space C([0, ∞); R d ) of continuous functions on R d so that P x (X 0 = x) = 1 and for every f ∈ C 2 c (R d ), Lf (X s )ds * Research partially supported by NSF Grant DMS-1206276. is a P x -martingale. Here X t (ω) = ω(t) is the coordinate map on C([0, ∞); R d ). It is also known that (X, P x ) is the unique weak solution to the following stochastic differential equation where W t is an n-dimensional Brownian motion and σ(x) = a(x) 1/2 is the symmetric square root matrix of a(x) = (a ij (x)) 1≤i,j≤d . When a is Hölder continuous, it is known that L has a jointly continuous heat kernel p(t, x, y) with respect to the Lebesgue measure on R d that enjoys the following Aronson's estimate (see Theorem 4.3 below): there are constants c k > 0, k = 1, · · · , 4, so that for t > 0 and x, y ∈ R d . As many physical and economic systems exhibit discontinuity or jumps, in-depth study on non-Gaussian jump processes are called for. See for example, [5,29,34,40] and the references therein. The infinitesimal generator of a discontinuous Markov process in R d is no longer a differential operator but rather a non-local (or, integro-differential) operator. For instance, the infinitesimal generator of an isotropically symmetric α-stable process in R d with α ∈ (0, 2) is a fractional Laplacian operator c ∆ α/2 := −c (−∆) α/2 . During the past several years there is also many interest from the theory of PDE (such as singular obstacle problems) to study non-local operators; see, for example, [8,42] and the references therein. Quite many progress has been made in the last fifteen years on the development of the De Giorgi-Nash-Moser-Aronson type theory for non-local operators. For example, Kolokoltsov [35] obtained two-sided heat kernel estimates for certain stable-like processes in R d , whose infinitesimal generators are a class of pseudo-differential operators having smooth symbols. Bass and Levin [3] used a completely different approach to obtain similar estimates for discrete time Markov chain on Z d , where the conductance between x and y is comparable to |x − y| −n−α for α ∈ (0, 2). In [17], two-sided heat kernel estimates and a scale-invariant parabolic Harnack inequality (PHI in abbreviation) for symmetric α-stable-like processes on d-sets are obtained. Recently in [18], two-sided heat kernel estimates and PHI are established for symmetric non-local operators of variable order. The De Giorgi-Nash-Moser-Aronson type theory is studied very recently in [19] for symmetric diffusions with jumps. We refer the reader to the survey articles [10,28] and the references therein on the study of heat kernels for symmetric non-local operators. However, for non-symmetric non-local operators, much less is known. In this article, we will survey the recent development in the study of heat kernels for non-symmetric non-local operators. We will concentrate on the recent progress made in [25,26] and [13]. In Section 5 of this paper, we summarize some other recent work on heat kernels for non-symmetric non-local operators. We also take this opportunity to fill a gap in the proof of [25, (3.20)], which is (3.23) of this paper. The proof in [25] works for the case |x| ≥ t 1/α . In Section 3, a proof is supplied for the case |x| ≤ t 1/α . In fact, a slight modification of the original proof for [25,Theorem 2.5] gives a better estimate (3.20) than (3.23).
In this survey, we concentrate on heat kernel on the whole Euclidean spaces and on the work that the authors are involved. We will not discuss Dirichlet heat kernels in this article.

Lévy process
A Lévy process on R d is a right continuous process X = {X t ; t ≥ 0} having left limit that has independent stationary increments. It is uniquely characterized by its Lévy exponenent ψ: (2.1) Here for x ∈ R d , the subscript x in the mathematical expectation E x and the probability P x means that the process X t starts from x. The Lévy exponent ψ admits a unique decomposition: The Lévy measure Π(dz) has a strong probabilistic meaning. It describes the jumping intensity of X making a jump of size z. Denote by {P t ; t ≥ 0} the transition semigroup of X; that is, For an integrable function f , its Fourier transform is defined to be f (ξ) = R d e iξ·x f (x)dx. Then we have by (2.1) and Fubini's theorem, If we denote the infinitesimal generator of {P t ; t ≥ 0} (or X) by L, then Hence −ψ(−ξ) is the Fourier multiplier (or symbol) for the infinitesimal generator L of X. One can derive a more explicit expression for the generator L: for f ∈ C 2 c (R d ), When b = 0, Π = 0 and (a ij ) = I d×d the identity matrix, that is when ψ(ξ) = |ξ| 2 , X is a Brownian motion in R d with variance 2t and infinitesimal generator ∆ : is a normalizing constant so that ψ(ξ) = |ξ| α , X is a rotationally symmetric α-stable process in R d , whose infinitesimal generator is the fractional Laplacian ∆ α/2 := −(−∆) α/2 . Unlike Brownian motion case, explicit formula for the transition density function of symmetric α-stable processes is not known except for a very few cases. However we can get its two-sided estimates as follows. It follows from (2.1) that under P 0 , (i) AX t has the same distribution as X t for every t > 0 and rotation A (an orthogonal matrix); (ii) for every λ > 0, X λt has the same distribution as λ 1/α X t . Let p(t, x) be the density function of X t under P 0 ; that is, Then p(t, x) is a function of t and |x| and p(t, x) = t −d/α p(1, t −1/α x). Using Fourier's inversion, one gets lim (See Pólya [39] (2.5) Here for a, b ∈ R, a ∧ b := min{a, b}, and for two functions f, g, f ≍ g means that f /g is bounded between two positive constants.
In real world, almost every media we encounter has impurities so we need to consider statedependent stochastic processes and state-dependent local and non-local operators. Intuitively speaking, we need to consider processes and operators where ψ(ξ) is dependent on x; that is, ψ(x, ξ). If one uses Fourier multiplier approach (2.3), one gets pseudo differential operators. The connection between pseudo differential operators and Markov processes has been nicely exposited in N. Jacob [30]. In this survey, we take (2.4) as a starting point but with a ij (x), b(x) and Π(x, dz) being functions of x ∈ R d . That is, We will concentrate on the case where Π(x, dz) = κ(x,z) |z| d+α dz for some α ∈ (0, 2) and a measurable function κ(x, z) on R d × R d satisfying for any x, y, z ∈ R d , and for some β ∈ (0, 1), 3 Stable-like processes and their heat kernels In this section, we consider the case where a ij = 0, b = 0 and Π(x, dz) = κ(x,z) |z| d+α dz; that is, The non-local operator L of (3.1) typically is not symmetric, as oppose to non-local operator given by in the distributional sense. Here c(x, y) is a symmetric function that is bounded between two positive constants. The operator L is the infinitesimal generator of the symmetric α-stable-like process studied in Chen and Kumagai [17], where it is shown that L has a jointly Hölder continuous heat kernel that admits two-sided estimates in the same form as (2.5).
The following result is recently established in [25].

Approach
We now sketch the main idea behind the proof of Theorem 3.1.
To emphasize the dependence of L in (3.1) on κ, we write it as L κ . For each fixed y ∈ R d , we consider Lévy process (starting from 0) with Lévy measure Π y (dz) = κ(y,z) |z| d+α dz, and denote its marginal probability density function and infinitesimal generator by p y (t, x) and L κ(y) , respectively.
We use Levi's idea and search for heat kernel p(t, x, y) for L κ with the following form: with function q(s, z, y) be determined below. We want Formally, It follows from (3.13) that q(t, x, y) should satisfy Thus for the construction and the upper bound heat kernel estimates of p(t, x, y), the main task is to solve q(t, x, y), and to make the above argument rigorous. We use Picard's iteration to solve (3.15). For n ≥ 1, define Then it can be shown that converges absolutely and locally uniformly on (0, 1] × R d × R d . Moreover, q(t, x, y) is jointly continuous in (t, x, y) and has the following upper bound estimate We then need to address the following issues.
(i) Show that p(t, x, y) constructed through (3.14) and (3.17) is non-negative, has the property R d p(t, x, y)dy = 1 and satisfies the Chapman-Kolmogorov equation.
(ii) The kernel p(t, x, y) has the claimed two-sided estimates, and derivative estimates.
This requires detailed studies on the kernel p κ α (t, x − y) for the symmetric Lévy process with Lévy measure κ(z) |z| d+α dz, including its fractional derivative estimates, and its continuous dependence on κ(z), which will be outlined in the next two subsections.

Upper bound estimates
Key observation: For any symmetric function κ(z) with κ 0 ≤ κ(z) ≤ κ 1 , let κ(z) := κ(z) − κ 0 2 . Since the Lévy process with Lévy measure κ(z) |z| d+α dz can be decomposed as the independent sum of Lévy processes having respectively Lévy measures κ(z) |z| d+α dz and κ 0 /2 |z| d+α dz, we have Thus the gradient and fractional derivative estimates on p κ(z) α (t, x) can be obtained from those on p κ 0 /2 (t, x). On the other hand, it follows from [17] First one can establish that for where B denotes the usual β-function.
Next we establish the continuous dependence of p κ(z) α (t, y) on the symmetric function κ(z). Let κ(z) and κ(z) be two symmetric functions that are bounded between κ 0 and κ 1 . Then for every 0 < γ < α/4, there is a constant c > 0 so that the following estimates hold for all t ∈ (0, 1] and We take this opportunity to fill a gap in the proof of [25, (3.20)]. The proof there works only for |x| ≥ t 1/α and t ∈ (0, 1], as in this case, by [25, (2.2)], which gives (3.23) by line 8 on p.284 of [25]. On the other hand, one deduces by the inverse Fourier transform that x). In fact, by a slight modification of the original proof given in [25] for (3.23), we can get estimate (3.20). Indeed, by the symmetry of L κ(z) and L κ(z) , Hence by (3.18) and [25, (2.28)], The same proof as that for [25, Theorem 2.5] but using (3.20) instead of (3.23) then gives (3.21)- (3.22). Since From these estimates, one can establish the first part ((i)-(iv)) of the Theorem 3.1 as well as

Lower bound estimates
The upper bound estimates in Theorem 3.1 are established by using analytic method, while the lower bound estimate in Theorem 3.1 are obtained mainly by probabilistic argument. From (i)-(iv) of Theorem 3.1, we see that P t f (x) := R d p(t, x, y)f (y)dy is a Feller semigroup. Hence, it determines a Feller process (Ω, F, (P x ) x∈R d , (X t ) t≥0 ).
We first claim the following.
Theorem 3.2 Let F t := σ{X s , s ≤ t}. Then for each x ∈ R d and every Lf (X s )ds is an F t -martingale. (3.25) In other words, P x solves the martingale problem for (L, C 2 b (R d )). Thus P x in particular solves the martingale problem for (L, C ∞ c (R d )).
ds. Then we have by (3.4) in Theorem 3.1 that Since P t f also satisfies the equation (3.26) The desired property (3.25) now follows from (3.26) and the Markov property of X. ✷ Theorem 3.2 allows us to derive a Lévy system of X by following an approach from [15]. It is easy to see from (3.25) where and Here X i,c is the continuous local martingale part of the semimartingale X i . Now suppose that A and B are two bounded closed subsets of R d having a positive distance from each other. Let f ∈ C ∞ c (R d ) with f = 0 on A and f = 1 on B. Let M f be defined as in (3.25). Clearly, so L can be rewritten as We get by (3.25)-(3.29) and (3.31), By taking a sequence of functions f n ∈ C ∞ c (R d ) with f n = 0 on A, f n = 1 on B and f n ↓ ½ B , we get that, for any is a martingale with respect to P x . Thus, Using this and a routine measure theoretic argument, we get For a set K ⊂ R d , denote Let B(x, r) be the ball with radius r and center x. We need the following lemma (see [2,17]).
By the definition of f r , we have On the other hand, by the definition of L, we have for λ > 0, where s 1 is the sphere area of the unit ball. Substituting this into (3.34), we get Choosing first λ large enough and then R large enough yield the desired estimate. ✷ We can now proceed to establish the lower bound heat kernel estimate (3.11). By Lemma 3.4, there is a constant λ ∈ (0, 1 2 ) such that for all t ∈ (0, 1), In view of the estimate (3.24), it remains to consider the case that |x − y| > 3t 1/α . Using (3.35) and the Lévy system of X, P x (X λt ∈ B(y, t 1/α )) ≥ P x X hits B(y, t 1/α /2) before λt and then travels less than distance t 1/α /2 for at least λt units of time This proves that for every x, y ∈ R d and t ≤ 1.
Observe that by (3.5) and (3.11), the term t (t 1/α +|x−y|) d+α in (3.36) is comparable to p κ α (t, x, y) and to p κ (t, x, y). So the error bound (3.36) is also a relative error bound, which is good even in the region when |x − y| is large.
For uniformly elliptic divergence form operators L and L on R d , pointwise estimate on |p(t, x, y)− p(t, x, y)| and the L p -operator norm estimates on P t − P t are obtained in Chen, Hu, Qian and Zheng [14] in terms of the local L 2 -distance between the diffusion matrix of L and L. Recently, Bass and Ren [4] obtained strong stability result for symmetric α-stable-like non-local operators of (3.3), with error bound expressed in terms of the L q -norm on the function c(x) := sup y∈R d |c(x, y) − c(x, y)|.

Applications to SDE driven by stable processes
Suppose that σ(x) = (σ ij (x)) 1≤i,j≤d is a bounded continuous d × d-matrix-valued function on R d that is non-degenerate at every x ∈ R d , and Y is a (rotationally) symmetric α-stable process on R d for some 0 < α < 2. It is shown in Bass and Chen [1, Theorem 7.1] that for every x ∈ R d , SDE has a unique weak solution. (Although in [1] it is assumed d ≥ 2, the results there are valid for d = 1 as well.) The family of these weak solutions forms a strong Markov process {X, P x , x ∈ R d }.

42)
and Then the strong Markov process X formed by the unique weak solution to SDE (3.38) has a jointly continuous transition density function p(t, x, y) with respect to the Lebesgue measure on R d , and there is a constant C ≥ 1 that depends only on (d, α, β, λ 0 , λ 1 ) so that for every t ∈ (0, 1] and x, y ∈ R d . Moreover, p(t, x, y) enjoys all the properties stated in the conclusions of Theorem 3.1 with κ 0 = A(d, −α)λ d+α and κ 2 = κ 2 (d, λ 0 , λ 1 , λ 2 ).
The following strong stability result for SDE (3.38) is a direct consequence of Corollary 3.6 and (3.41).

Diffusion with jumps
In this section, we consider non-local operators that have both elliptic differential operator part and pure non-local part: where , Here a(x) := (a ij (x)) 1≤i,j≤d is a d × d-symmetric matrix-valued measurable function on R d , b(x) : R d → R d and κ(x, z) : R d × R d → R are measurable functions, and α ∈ (0, 2). For convenience, we assume d ≥ 2. Throughout this section, we impose the following assumptions on a and κ: (H a ) There are c 1 > 0 and β ∈ (0, 1) such that for any x, y ∈ R d , and for some c 2 ≥ 1, Note that when κ(x, z) is a positive constant function, for some constant c > 0. A function f defined on R d is said to be in Kato class K 2 if f ∈ L 1 loc (R d ) and Let q(t, x, y) be the fundamental solution of {L a ; t ≥ 0}; see Theorem 4.3 below for more information. Since L can be viewed as a perturbation of L a by L b,κ := b · ∇ + L κ , heuristically the fundamental solution (or heat kernel) p(t, x, y) of L should satisfy the following Duhamel's formula: for all t > 0 and x, y ∈ R d , The following is a special case of the main results in [13], where the corresponding results are also obtained for time-inhomogeneous operators. and x, y ∈ R d , (4.8) (ii) (C-K equation) For all s, t > 0 and x, y ∈ R d , we have R d p(s, x, y)p(t, y, z)dy = p(s + t, x, y). (4.9) (iii) (Gradient estimate) For any T > 0, there exist constants C 1 , λ 1 > 0 such that for t ∈ (0, T ] and x, y ∈ R d , (4.10) (iv) (Conservativeness) For any t > 0 and x ∈ R d , R d p(t, x, y)dy = 1.
(v) (Generator) Define P t f (x) = R d p(t, x, y)f (y)dy. Then for any f ∈ C 2 b (R d ), we have Define m κ = inf x∈R d essinf z∈R d k(x, z).

Theorem 4.2 ([13, Theorem 1.3])
If κ is a bounded function satisfying (H κ ) and that for each then p(t, x, y) ≥ 0. Furthermore, if m κ > 0, then for any T > 0, there are constants C 1 , λ 2 > 0 such that for any t ∈ (0, T ] and x, y ∈ R d , (4.13) We have by Theorems 4.1 and 4.2 that when κ ≥ 0, then there is a conservative Feller process X = {X t , t ≥ 0; P x , x ∈ R d } having p(t, x, y) as its transition density function with respect to the Lebesgue measure. It follows from (4.11) that X is a solution to the martingale problem for (L, C 2 b (R d )). When a is the identity matrix, b = 0 and κ(x, z) is a positive constant, L = ∆ + c∆ α/2 for some positive constant c > 0. In this case, the corresponding Markov process X is a symmetric Lévy process that is the sum of a Brownian motion W and an independent rotationally symmetric α-stable process Y . Thus the heat kernel p(t, x, y) for L is the convolution of the transition density function of W and Y . In this case, its two-sided bounds can be obtained through a direct calculation. Indeed such a computation is carried out in Song and Vondraček [43].
Symmetric diffusions with jumps corresponding to symmetric non-local operators on R d with variable coefficients of the the following form have been studied in [19]: c(x, y) |x − y| d+α dy, (4.14) where a(x) := (a ij (x)) 1≤i,j≤d is a d × d-symmetric matrix-valued measurable function on R d , c(x, y) is a symmetric measurable function on R d × R d that is bounded between two positive constants, and α ∈ (0, 2). Clearly, when a(x) is the identity matrix and c(x, y) is a positive constant, the above non-local operator is ∆ + c 0 ∆ α/2 for some c 0 > 0. Among other things, it is established in Chen and Kumagai [19] that the symmetric non-local operator L of (4.14) has a jointly Hölder continuous heat kernel p(t, x, y) and there are positive constants c i , 1 ≤ i ≤ 4 so that for all t > 0 and x, y ∈ R d . It is easy to see that for each fixed T > 0, the two-sided estimates (4.15) on (0, T ] × R d × R d is equivalent to When a is the identity matrix and b = 0, the results in Theorems 4.1 and 4.2 have been obtained recently in [47] for κ(x, z) that is symmetric in z.

Approach
The approach in [13] is to treat L as L a under lower order perturbation b · ∇ + L κ , and thus one can construct the fundamental solution for L from that of L a through Duhamel's formula.
Using Theorem 4.3, one can show that p n (t, x, y) is well defined and that ∞ n=0 p n (t, x, y) converges locally uniformly to some function p(t, x, y), and that p(t, x, y) is the unique solution stated in Theorem 4.1. The positivity (4.12) of Theorem 4.2 can be established by using Hille-Yosida-Ray theorem and Courrége's first theorem.
The Gaussian part in the lower bound estimate on p(t, x, y) in Theorem 4.2 is obtained from the near diagonal lower bound estimate on p(t, x, y) and a chaining ball argument, while the pure jump part in the lower bound estimate on p(t, x, y) is obtained by using a probabilistic argument through the Lévy system, similar to that in Section 3.

Application to SDE
Let σ(x) be a d × d-matrix valued function on R d that is uniformly elliptic and bounded, and each entry σ ij is β-Hölder continuous on R d , b ∈ K 2 and σ a bounded d × d-matrix valued measurable function on R d . Suppose X solves the following stochastic differential equation: where W is a Brownian motion on R d and Y is a rotationally symmetric α-stable process on R d . By Itô's formula, the infinitesimal generator L of X is of the form L a +b·∇+L κ with a(x) = σ(x)σ(x) * and So by Theorems 4.1 and 4.2, X has a transition density function p(t, x, y) satisfying the properties there. If in addition, σ is uniformly elliptic, then for any T > 0, for t ∈ (0, T ] and x, y ∈ R d .

Other related work
In this section, we briefly mention some other recent work on heat kernels of non-symmetric nonlocal operators.
Using a perturbation argument, Bogdan and Jakubowski [7] constructed a particular heat kernel (also called fundamental solution) q b (t, x, y) for operator 2) and b is a function on R d that is in a suitable Kato class. It is based on the following heuristics: q b (t, x, y) of L b can be related to the fundamental solution p(t, x, y) of L 0 = ∆ α/2 , which is the transition density of the rotationally symmetric α-stable process Y , by the following Duhamel's formula: Applying the above formula recursively, one expects that is a fundamental solution for L b , where q b 0 (t, x, y) := p(t, x, y) and for k ≥ 1, It is shown in [7] that the series in (5.2) converges absolutely and, for every T > 0, such defined q b (t, x, y) is a conservative transition density function and is comparable to p(t, x, y) on (0, T ] × Recall that p(t, x, y) has two-sided estimate (2.5). In [22], Chen and Wang showed that the Markov process X t having q b (t, x, y) as its transition density function is the unique solution to the martingale problem (L b , C 2 b (R d ); moreover, it is the unique weak solution to the following stochastic differential equation: where Y t is the rotationally symmetric α-stable process on R d . Dirichlet heat kernel estimate for L b in a bounded C 1,1 open set has been obtained in [15]. In [31,32], Kim and Song extended results in [7,16] to ∆ α/2 +µ·∇, where µ = (µ 1 , . . . , µ d ) are signed measures in suitable Kato class. These work can be regarded as heat kernels for fractional Laplacian under gradient perturbation. Heat kernel estimates for relativistic stable processes and for mixed Brownian motions and stable processes with drifts have recently been studied in [23] and [12], respectively. See [11] for drift perturbation of subordinate Brownian motion of pure jump type and its heat kernel estimate. While in [49], Xie and Zhang considered the critical operator L b := a∆ 1/2 + b · ∇, where for some 0 < c 0 < c 1 , a : R d → [c 0 , c 1 ] and b : R d → R d are two Hölder continuous functions. They established two-sided estimates for the heat kernel of L b by using Levi's method as described in Subsection 3.1.
In the same spirit, Wang and Zhang in [45] considered more general fractional diffusion operators over a complete Riemannian manifold perturbed by a time-dependent gradient term, and showed two-sided estimates and gradient estimate of the heat kernel. More precisely, let M be a d-dimensional connected complete Riemannian manifold with Riemannian distance ρ. Let ∆ M be the Laplace-Beltrami operator. Suppose that the heat kernel p(t, x, y) of ∆ M with respect to the Riemannian volume dx exists and has the following two-sided estimates: c 1 t −d/2 e −c 2 ρ(x,y) 2 /t ≤ p(t, x, y) ≤ c 3 t −d/2 e −c 4 ρ(x,y) 2 /t , t > 0, x, y ∈ M, (5.3) and gradient estimate |∇ x p(t, x, y)| ≤ c 5 t −(d+1)/2 e −c 4 ρ(x,y) 2 /t , where ∇ x denotes the covariant derivative. Let P t be the corresponding semigroup, that is, Notice that when α = 2 and f is time-independent, K α is the same as in (4.5).
The following result is shown in [45].
In this survey, we mainly concentrate on the quantitive estimates of the heat kernels of nonsymmetric nonlocal operators. For derivative formula of the heat kernel associated with stochastic differential equations with jumps, we refer the interested reader to [50,48,46]. For other results on the existence and smoothness of heat kernels or fundamental solutions for non-symmetric jump processes or non-local operators under Hörmander's type conditions, see [41,37] for the studies of linear Ornstein-Uhlenbeck processes with jumps, and [51,52,53] and the references therein for the studies of general stochastic differential equations with jumps. We will not survey these results since the arguments in the above references are mainly based on the Malliavin calculus and thus belong to another topic.