Apr 16, 2018 - some weak regularity assumptions, the limit of the solutions satisfies a ...... Proof. The existence of solution of the martingale problem is in [20, ...
Homogenization of Periodic Linear Nonlocal Partial Differential Equations Qiao Huang∗, Jinqiao Duan†, Renming Song‡ April 16, 2018 Abstract We study the “periodic homogenization” for a class of linear nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, associated to stochastic differential equations driven by multiplicative isotropic α-stable L´evy noise for 1 < α < 2. Our homogenization method is probabilistic. It turns out that, under some weak regularity assumptions, the limit of the solutions satisfies a nonlocal partial differential equation with constant coefficients, associated to a symmetric α-stable L´evy process. AMS 2010 Mathematics Subject Classification: 35B27, 35R09, 45K05. Keywords and Phrases: Periodic homogenization, nonlocal PDEs, nonlocal Poisson equations, multiplicative stable noise, Zvonkin’s transform, strong well-posedness, Feller processes, Feynman-Kac formula.
1
Introduction
The goal of this paper is to study the limit behavior, as → 0, of the solution u : Rd → R of the following nonlocal partial differential equation (PDE) of parabolic-type, ( 1 ∂u (t, x) = Lα u (t, x) + α−1 e x + g x u (t, x), t > 0, x ∈ Rd , ∂t (1.1) u (0, x) = u0 (x), x ∈ Rd , where 1 < α < 2 and the linear operator Lα is a nonlocal integro-differential operator of L´evy-type with rapidly oscillating coefficients given by Z h x x i α L w(x) := w x+σ , y − w(x) − σ i , y ∂i w(x)1B (y) ν α (dy) Rd \{0} 1 x x + α−1 bi + ci ∂i w(x), x ∈ Rd . ∗
Center for Mathematical Sciences, Huazhong University of Science and Technology, Wuhan, Hubei 430074, P.R. China. Email: hq932309@hust.edu.cn † Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA. Email: duan@iit.edu ‡ Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA. Email: rsong@illinois.edu
1
where ν α (dy) := |y|dy d+α . In this paper, we use Einstein’s convention that the repeated indices in a product will be summed automatically. For notational simplicity, we introduce the linear operator Aσ,α defined by Z σ,ν α A w(x) := w(x + σ(x, y)) − w(x) (1.2) Rd \{0} α i d − σ (x, y)∂i w(x)1B (y) ν (dy), x ∈ R . For a function f on Rd (or F on Rd × Rd ), we denote f (x) := f x (or F (x, y) := F x , y ). Then 1 α σ ,ν α L = A + α−1 b + c · ∇. (1.3) The original probabilistic approach to the homogenization of local linear second order parabolic partial differential operators is presented in [5, Chapter 3], which is based on the ergodic theorem, the Feynman-Kac formula and the functional central limit theorem. They applied a technique of removing the singular drift, which is now known as Zvonkin’s transform, appeared originally in [34]. By now, there are lots of literature concerning the α homogenization of second order local PDEs, i.e., the case of replacing the operator Aσ ,ν in (1.3) by a second order partial differential operator with singular coefficients. Two different scales of spatial variables involved in the coefficients have been considered in [4], by using the nonlinear Feynman-Kac formula in the context of backward stochastic differential equations (SDEs). In [22], the authors allowed the singular coefficients to be time-dependent and rapidly oscillating in time with a different scale in contrast to the spatial variable. The paper [12] dealt with the case when the second order coefficient matrix can be degenerate, using the existence of a spectral gap and Malliavin’s calculus. The work in this paper is highly motivated by [12, 21]. There are also some literature for the homogenization of nonlocal PDEs or SDEs with jumps involved. We refer the reader to [2, 25, 31] for the periodic homogenization results of some kinds of nonlocal operators involving stable-like terms or convolution type kernels. The methods used in these papers are all analytic. The probabilistic study of homogenization of periodic stable-like processes in pure jump or jump-diffusion case can be found in [9, 29]. The homogenization in random medium is slightly different from the periodic case, see [28] for related results for jump-diffusion processes in random medium. In paper [10], the author considered the homogenization of SDEs driven by multiplicative stable processes, where the multiplicative coefficient σ is linear in the second variable in the sense that σ(x, y) = σ0 (x)y, with σ0 three-times continuously differentiable. In the present paper, we generalize his results to the general multiplicative case, and the coefficients only need to possess some H¨older or Lipschitz continuity. This will give rise to several difficulties both in analytic and probabilistic aspects. We also use the homogenization results of SDEs to study the homogenization of the nonlocal PDEs with singular coefficients involved. We denote by C k (Cbk ) with integer k ≥ 0 the space of (bounded) continuous functions possessing (bounded) derivatives of orders not greater than k. We shall explicitly write out the domain if necessary. Denote by Cb (Rd ) := Cb0 (Rd ), it is a Banach space with the supremum norm kf k0 = supx∈Td |f (x)|. The space Cbk (Rd ) is a Banach space endowed with P the norm kf kk = kf k0 + kj=1 k∇⊗j f k. We also denote by C 1− the class of all Lipschitz 2
continuous functions. For a non-integer γ > 0, the H¨older spaces C γ (Cbγ ) are defined as the bγc subspaces of C bγc (Cb ) consisting of functions whose bγc-th order partial derivatives are locally H¨older continuous (uniformly H¨older continuous) with exponent γ − bγc. These two spaces C γ and Cbγ obviously coincide when the underlying domain is compact. The sapce Cbγ (Rd ) is a Banach space endowed with the norm kf kγ = kf kbγc + [∇bγc f ]γ−bγc , where the seminorm [·]γ 0 with 0 < γ 0 < 1 is defined as [f ]γ 0 :=
|f (x) − f (y)| . |x − y|γ 0 x,y∈Rd ,x6=y sup
In the sequel, the torus Td := Rd /Zd will be used frequently. Denote by D := D(R+ ; Td ) the space of all Td -valued c`adl`ag functions on R+ , equipped with the Skorokhod topology. We shall always identify the periodic function on Rd of period 1 with the same function restricted on the torus Td = Rd /Zd . This allows us to regard the space C k (Td ) (C γ (Td )) as a sub-Banach space of Cbk (Rd ) (Cbγ (Rd )). By Br we means the open ball in Rd centering at the origin with radius r > 0, we shall omit the subscript when the radius is one. The capital letter C denotes a finite positive constant whose value may vary from line to line. We also use the notation C(· · · ) to emphasize the dependence on the quantities appearing in the parentheses.
2
Preliminaries and general assumptions
Let (Ω, F, P, {Ft }t≥0 ) be a filtered probability space endowed with a Poisson random measure N α on (Rd \{0})×R+ with jump intensity measure ν α (dy) = |y|dy d+α , where 1 < α < 2. ˜ the associated compensated Poisson random measure, that is, N ˜ α (dy, ds) := Denote by N N α (dy, ds) − ν α (dy)ds. We assume that the filtration {Ft }t≥0 satisfies the usual conditions. Let Lα = {Lαt }t≥0 be a d-dimensional isotropic α-stable L´evy process given by Z tZ Z tZ α ˜ (dy, ds) + Lt = yN yN (dy, ds). 0
B\{0}
0
Bc
Given > 0, x ∈ Rd , consider the following: x, x, x, Xt− Xt− Xt− 1 x, α dXt = b +c dt + σ , dLt , α−1
X0x, = x,
(2.1)
or more precisely, Xtx,
x, x, Xs− Xs− = x+ b +c ds α−1 0 x, x, Z tZ Z tZ Xs− Xs− α ˜ (dy, ds) + + σ ,y N σ , y N α (dy, ds), c 0 B\{0} 0 B Z t
1
where the coefficients b, c, σ(·, y) are periodic, for each y ∈ Rd , of periodic one in each component. The shorthand notation for the stochastic differential term in (2.1) is due to [19]. 3
˜ tx, := 1 Xx, Define X α t . It is easy to check that x, x, x, x, α−1 α ˜ ˜ ˜ ˜ ˜ dXt = b(Xt− ) + c(Xt− ) dt + σ Xt− , dLt ,
˜ 0x, = x , X
(2.2)
d ˜ αt } := { 1 Lαα t } = {Lαt } by virtue of the selfsimilarity. We shall also consider the where {L “limit” equation, namely α x x x ˜ 0x = x. ˜ ˜ ˜ ˜ (2.3) dXt = b(Xt− )dt + σ Xt− , dLt , X
˜ x, to include X ˜ x, For notational simplicity, we shall allow the parameter to be zero in X x,0 x ˜ := X ˜ . i.e., X ˜ x, , X ˜ x of (2.2) and (2.3) as Td -valued proIn the sequel, we will regard the solutions X cesses, by mapping all trajectories of the processes on Rd to the torus Td , via the canonical ˜ x, and quotient map π : Rd → Rd /Zd . Then the periodicity of the coefficients implies that X ˜ are well-defined stochastic processes on Td . X Now we list some general assumptions for the nonlocal PDE (1.1) and the SDE (2.1). All these assumptions are assumed to hold in the sequel unless otherwise specified. Assumption H1. The functions b, c, e, g, u0 are all periodic of period 1 in each component. For every y ∈ Rd , the function x → σ(x, y) is periodic of period 1 in each component. Assumption H2. The functions b, c, e are of class Cbβ with exponent β satisfying 1−
α < β < 1. 2
The functions g and u0 are both continuous. Assumption H3. The function σ : Rd × Rd → Rd satisfies the following conditions. (1). Regularity. For every x ∈ Rd , the function y → σ(x, y) is of class C 2 . There exists a constant C > 0, such that for any x1 , x2 , y ∈ Rd , |σ(x1 , y) − σ(x2 , y)| ≤ C|x1 − x2 ||y|. (2). Oddness. For all x, y ∈ Rd , σ(x, −y) = −σ(x, y). (3). Bounded inverse Jacobian. The Jacobian matrix with respect to the second variable ∇y σ(x, y) is non-degenerate for all x, y ∈ Rd , and there exists a constant C > 0 such that |(∇y σ(x, y))−1 | ≤ C for all x, y ∈ Rd , where | · | is the operator norm on L (Rd , Rd ). (4). Growth condition. There exists a positive bounded function φ : Rd → R+ , such that for all x, y ∈ Rd , φ(x)−1 |y| ≤ |σ(x, y)| ≤ φ(x)|y|. Remark 2.1. Some comments on our assumptions will be helpful: (1). As mentioned in the end of the introduction, b, c, e, g, u0 and the function x → σ(x, y), for every y ∈ Rd , can be regarded as functions on Td , and we have b, c, e ∈ C β (Td ), g, u0 ∈ C(Td ), under Assumptions H1 and H2. (2). Both the oddness and the growth condition in Assumption H3 imply that σ(·, 0) ≡ 0. 4
(3). The bounded inverse Jacobian condition implies that |∇y σ| ≥ C −1 . Since by Hadamard’s inequality (see, for instance, [30]), |(∇y σ)−1 | ≤ C ⇒ | det((∇y σ)−1 )| ≤ C n ⇔ | det(∇y σ) ≥ C −n | ⇒ |∇y σ| ≥ C −1 . (4). The growth condition implies that for any γ > α, we have Z sup |σ(x, y)|γ ν α (dy) < ∞. x∈Rd
(2.4)
B\{0}
˜ tx, ), f (Xtx, )), for any f ∈ ˜ tx ) (or f (X This ensures that we can apply Itˆo’s formula to f (X Cbγ (Rd ) with γ > α (cf. [26, Lemma 4.2]). (5). By virtue of the oddness condition in Assumption H3 and the symmetry of the jump intensity measure ν α , for any x ∈ Rd , Z Z i α P.V. σ (x, y)ν (dy) = P.V. σ i (x, y)ν α (dy) = 0. (2.5) σ(x,·)−1 B\B
B\σ(x,·)−1 B α
Consequently we can rewrite the operator Aσ,ν in (1.2) as Z σ,ν α A w(x) = w(x + z) − w(x) − z i ∂i w(x)1B (z) ν σ,α (x, dz),
(2.6)
Rd \{0}
where the kernel {ν σ,α (x, ·)|x ∈ Rd } is given by Z σ,α ν (x, A) := 1A (σ(x, y))ν α (dy),
A ∈ B(Rd \ {0}).
(2.7)
Rd \{0}
Moreover, for any γ > α, the growth condition in Assumption H3 implies that Z Z γ σ,α sup (|z| ∧ 1)ν (x, dz) = sup (|σ(x, y)|γ ∧ 1)ν α (dy) d d d d x∈R x∈R R \{0} R \{0} Z Z α γ α ≤ sup (φ(x)|y|) ν (dy) + ν (dy) x∈Rd
≤
(2.8)
|y|≥φ(x)−1
|y|≤φ(x)
1 1 kφk2γ−α kφkαL∞ < ∞. L∞ + γ−α α
Remark 2.2. The special case σ(x, y) = σ0 (x)y with certain σ0 is considered in [10], where the author assumed the function σ0 : Rd → GL(Rd ) is periodic and of class C 3 . In our context, Assumption H3 amounts to saying that σ0 : Rd → GL(Rd ) is periodic and Lipschitz.
(2.9)
Since these imply the regularity condition immediately, the bounded inverse Jacobian and growth conditions are fulfilled by continuity and periodicity, together with the observation supx∈Rd kσ0 (x)k ∨ kσ0 (x)−1 k < ∞. The oddness condition is trivial in this case. 5
We will need some regularities for the ’partial’ inverse of σ. For a function F : Rd × Rd 3 1− d d d (x, y) → F (x, y) ∈ R, we say F ∈ L∞ 2 (R ; C1 (R ; R )), if there exists a constant C > 0 such that for all x, y ∈ Rd , |F (x, y)| ≤ C, and for all x1 , x2 , y ∈ Rd , |F (x1 , y) − F (x2 , y)| ≤ C|x1 − x2 |. Then the regularity and growth conditions in Assumption H3 imply that the 1− d d d function (x, y) → σ(x, y)/|y| is of class L∞ 2 (R ; C1 (R ; R )). Lemma 2.3. Under Assumption H3, for every x ∈ Rd , the function y → σ(x, y) is a C 2 diffeomorphism. Denote the inverse by τ (x, z) := σ(x, ·)−1 (z), then for every z ∈ Rd , the function x → τ (x, z) is periodic of period one. Moreover, the function (x, z) → τ (x, z)/|z| is 1− d d d of class L∞ 2 (R ; C1 (R ; R )). Proof. Fix x ∈ Rd . Since the function y → σ(x, y) is of class C 2 , by the bounded inverse Jacobian condition in Assumption H3, together with Hadamard’s global inverse function theorem (see [18, Theorem 6.2.4]), σ(x, ·) is a C 2 -diffeomorphism. The periodicity is obvious. Now using the bounded inverse Jacobian condition, the Jacobian matrix of τ (x, z) with respect to z satisfies |∇z τ (x, z)| ≤ C, for all x, z ∈ Rd . Then by the growth condition and regularity condition, the second assertion follows from the following derivation, sup z
|τ (x, z)| ≤ φ(x), |z|
|τ (x1 , σ(x1 , y)) − τ (x2 , σ(x1 , y))| |τ (x1 , z) − τ (x2 , z)| = sup |z| |σ(x1 , y)| y z |τ (x2 , σ(x2 , y)) − τ (x2 , σ(x1 , y))| |σ(x2 , y) − σ(x1 , y)| = sup ≤ kφkL∞ |∇z τ | sup |σ(x1 , y)| |y| y y sup
≤ CkφkL∞ |∇z τ ||x1 − x2 |. 1− d d Assumption H4. det(∇z τ ) ∈ L∞ 2 (R ; C1 (R ; R)).
Remark 2.4. In the case σ(x, y) = σ0 (x)y, the Jacobian of τ (x, z) with respect to z is ∇z τ (x, z) ≡ σ0 (x)−1 . Then Assumption H4 reduce to that the function det(σ0 )−1 : Rd → R is Lipschitz, which is a direct consequence of (2.9). Now by (2.7), ν σ,α (x, dz) =
dz dz dτ (x, z) = | det ∇ τ (x, z)| . z dz |τ (x, z)|d+α |τ (x, z)|d+α
If we let h(x, z) =
| det ∇z τ (x, z)| |τ (x,z)|d+α |z|d+α
,
(2.10)
then ν σ,α (x, dz) = h(x, z) |z|dz d+α . Using the growth condition, we also find that for all x, z ∈ Rd , |τ (x, z)| kφk−1 ≤ kφkL∞ . (2.11) L∞ ≤ |z| Combining (2.10), (2.11), Lemma 2.3 and Assumption H4, together with [8, Proposition 1.2.6], we conclude that 6
1− d d d Proposition 2.5. Under Assumptions H3 and H4, h ∈ L∞ 2 (R ; C1 (R ; R )). Moreover, d there exists a positive constant C such that h(x, z) ≥ C for all x, z ∈ R .
In particular, the kernel ν σ,α is comparable to the jump intensity measure of an isotropic α-stable process. Remark 2.6. Thanks to Proposition 2.5, the general assumptions in [3, 16] are satisfied. Thus, the regularity results and heat kernel estimates therein are available in our context. γ d d d Actually, these two papers only need that h ∈ L∞ 2 (R ; C1 (R ; R )) for some 0 < γ < 1, this is the case by virtue of the natural embedding C 1− ⊂ C γ . Note that [3] also needs α + β not to be an integer, this can be fulfilled by choosing an appropriate β.
3
Nonlocal Poisson equation with zeroth-order term
As mentioned in the introduction, we will apply Zvonkin’s transform to study the homogenization of SDEs and nonlocal PDEs. Before that, we shall investigate the strong well-posedness of the SDEs presented in the previous section, and Zvonkin’s transform will also play an important role in this step (see next section). The key is to consider the following nonlocal Poisson equation with zeroth-order term, κu − Lα u = f,
(3.1)
where κ > 0, and Lα is the linear integro-partial differential operator given by α
Lα := Aσ,ν + b · ∇,
(3.2)
˜ of (2.3) once which may be regarded as the infinitesimal generator of the solution process X we prove its well-posedness in the next section.
3.1
Well-posedness of nonlocal Possoin equation
We first revisit the maximum principle and the solvability of Poisson equations with zeroth-order term studied in [26]. In this subsection we always assume that Assumptions H2, H3 and H4 are in force. Proposition 3.1. If u ∈ Cb1+γ (Rd ), 1 + γ > α, is a solution to κu − Lα u = f with κ > 0 and f ∈ Cb (Rd ), then κkuk0 ≤ kf k0 . α
Proof. Note that the nonlocal operator Aσ,ν can be rewritten in the form (2.6). For u ∈ Cb1+γ (Rd ), we have Z |u(x + z) − u(x) − z · ∇u(x)| ≤ |z|
1
|∇u(x + rz) − ∇u(x)|dr ≤ 0
7
[∇u]γ 1+γ |z| . 1+γ
(3.3)
Then by (2.8), there exists a constant C > 0 such that Z α |L u(x)| ≤ |u(x + z) − u(x) − z · ∇u(x)|ν σ,α (x, dz) B\{0} Z |u(x + z) − u(x)|ν σ,α (x, dz) + |b(x) · ∇u(x)| + c B Z 1+γ σ,α ≤ 2kuk1+γ (|z| ∧ 1)ν (x, dz) + kbk0 Rd \{0}
≤ Ckuk1+γ . Based on this estimate, the rest of the proof is exactly the same as that of [26, Proposition 3.2], even though it is set up with σ(·, y) ≡ y there. Now we investigate the solvability of the Poisson equation with a zeroth-order term involved. The results generalize the Schauder estimates in [26] to the anisotropic nonlocal case. Theorem 3.2. For any κ > 0 and f ∈ Cbβ (Rd ), the nonlocal Poisson equation (3.1) has a unique solution u = uκ ∈ Cbα+β (Rd ). In addition, there exists a positive constant C = C(κ, kbkβ ) such that kuκ kα+β ≤ C(kuκ k0 + kf kβ ). (3.4) Proof. The a priori estimate (3.4) is from [3, Theorem 7.1, Theorem 7.2]. We thus need to show that the equation (3.1) has a unique solution uκ ∈ Cbα+β (Rd ). Now we prove the existence and uniqueness of solution in Cbα+β (Rd ). It is shown in [26, Theorem 3.4] that when σ(·, y) ≡ Idy (y) := y, the existence and uniqueness hold in Cbα+β (Rd ). For the general σ, we apply the method of continuity (see [11, Section 5.2]). α α Define a family of linear operators by Lθ := θAσ,ν + (1 − θ)AIdy ,ν + b · ∇. We consider the family of equations: κu − Lθ u = f. (3.5) We can also rewrite the nonlocal term in Lθ into the form (2.6), with the kernel given by νθ := θν σ,α + (1 − θ)ν α . Then the a priori estimate (3.4) also holds for uθ (cf. Remark 2.6). As a result, the operator Lθ can be considered as a bounded linear operator from the Banach space Cbα+β (Rd ) into the Banach space Cbβ (Rd ). α Note that L0 = AIdy ,ν + b · ∇, which is the case considered in [26], and L1 = Lα . The solvability of the equation (3.1) for any f ∈ Cbβ (Rd ) is then equivalent to the invertibility of the operator Lθ . We can see from the proof of Proposition 3.1 that kuθ k0 ≤ Ckf k0 . Then together with the estimate (3.4) for uθ , we have the bound kuθ kα+β ≤ Ckf kβ , with the constant C being independent of θ. Since, as discussed in [26], the operator L0 = α AIdy ,ν + b · ∇ maps Cbα+β (Rd ) onto Cbβ (Rd ), the method of continuity is applicable and the result follows. Remark 3.3. If we take the periodicity assumption H1 into account, then we can slightly strengthen the conclusions in Theorem 3.2. That is, if f ∈ C β (Td ), then the unique solution of (3.1) is of class C α+β (Td ). 8
3.2
Feller property
In this subsection, we will study further the operator Lα . It turns out that it is the generator of a Feller semigroup. As a corollary, the solution of equation (3.1) can be represented in terms of a semigroup, and satisfies a finer estimate. All these results will be used in the next section. Theorem 3.4. The linear operator (Lα , D(Lα )), D(Lα ) = C α+β (Td ), defined on the Banach space (C(Td ), k · k0 ), is closable and dissipative, its closure generates a Feller semigroup {Pt }t≥0 on C(Td ). Proof. Similar to (3.3), one can find that for u ∈ C α+β (Td ), u(x + σ(x, y)) − u(x) − σ(x, y) · ∇u(x) ≤ [∇u]α+β−1 |σ(x, y)|α+β . α+β Combining this with (2.4), a straightforward application of the dominated convergence theorem yields that limy→x Lα u(y) = Lα u(x) for any u ∈ C α+β (Td ) and x ∈ Td . This amounts to saying that Lα (C α+β (Td )) ⊂ C(Td ). Therefore, the operator Lα : C(Td ) ⊃ C α+β (Td ) → C(Td ) is a densely defined unbounded operator on C(Td ). Now Proposition 3.1 implies that for any κ > 0 and u ∈ C α+β (Td ), k(κ−Lα )uk0 ≥ κkuk0 , that is, Lα is dissipative. By Theorem 3.2, we have C β (Td ) ⊂ (κ − Lα )(C α+β (Td )) for any κ > 0, which yields that the operator κ − Lα has dense range in C(Td ). In addition, Lα α satisfies the positive maximum principle, due to the equivalent form (2.6) of Aσ,ν and Courr`ege’s theorem (see [14, Corollary 4.5.14]). Now the final assertion follows form the celebrate Hille-Yosida-Ray Theorem (see, for instance, [7, Theorem 4.2.2]). Let us recall the notion of martingale problem (see [7, Section 4.3]). First recall that D = D(R+ ; Td ) is the space of all Td -valued c`adl`ag functions on R+ , equipped with the Skorokhod topology. Let wt (ω) = ω(t), ω ∈ D, be the coordinate process on (D, B(D)), and {Ftw }t≥0 := σ(ws : 0 ≤ s ≤ t) be the canonical filtration. Given a probability measure ν on Td , we say that a probability measure Pν on (D, B(D)) is a solution of the martingale problem for (Lα , ν), if Pν ◦ w0−1 = ν and the process Z t f M (t) := f (wt ) − f (w0 ) − Lα f (ws )ds 0
is a (D, B(D), {Ftw }t≥0 , Pν )-martingale, for any f ∈ D(Lα ) = C α+β (Td ). Theorem 3.5. For every x ∈ Td , the martingale problem for (Lα , δx ) has a unique solution Px . Moreover, the coordinate process {wt }t≥0 is a Feller process with generator being the closure of (Lα , C α+β (TRd )), and has a jointly continuous transition probability density p(t; x, y), i.e., Px (wt ∈ A) = A p(t; x, y)dy, A ∈ B(Td ), which satisfies the following estimates X X d d t t −α −α −1 ∧t ≤ p(t; x, y) ≤ C1 ∧t , C1 |x − y + j|d+α |x − y + j|d+α d d j∈Z
j∈Z
9
1 −α
|∇x p(t; x, y)| ≤ C2 t
X j∈Zd
d t ∧ t− α d+α |x − y + j|
,
where C1 > 1, C2 > 0 are two constants. Proof. The existence of solution of the martingale problem is in [20, Proposition 3]. Taking Theorem 3.4 into account, the uniqueness and the Feller property follow from [7, Theorem 4.4.1]. The existence of transition density and the two estimates can be found in [16, Theorem 1.4]. Remark 3.6. Combining Theorem 3.4 and Theorem 3.5, we see that the Feller semigroup {Pt }t≥0 generated by the closure of Lα has the representation Z f (y)p(t; x, y)dy, f ∈ C(Td ), Pt f (x) = Td
and the following gradient estimate holds Z
X
d t |∇Pt f (x)| ≤ C2 kf k0 t ∧ t− α d+α |y + j| Td j∈Zd Z 1 d t −α −α = C2 kf k0 t ∧t dy |y|d+α Rd 1 1 kf k0 t− α . ≤ C2 1 + α 1 −α
dy (3.6)
Corollary 3.7. For any κ > 0 and f ∈ C β (Td ), the unique solution uκ of equation (3.1) admits the representation Z ∞ uκ (x) = e−κt Pt f (x)dt, (3.7) 0
where {Pt }t≥0 is the Feller semigroup generated by the closure of Lα , and the integral on the right hand side converges. Moreover, there exists a constant C > 0 independent of u, f, b, κ such that α+β−1 (3.8) κkuκ k0 + κ α k∇uκ k0 + [∇uκ ]α+β−1 ≤ Ckf kβ . Proof. Theorem 3.2 tells that the interval (0, +∞) is contained in the resolvent set of Lα . Then by the integral representation of the resolvent (see [6, Theorem II.1.10.(ii)]), we arrive at Z t uκ = (κ − Lα )−1 f = lim
t→∞
e−κs Ps f ds,
0
where the limit is taken in (C(Td ), k · k0 ). The representation (3.7) then follows. Now thanks to the gradient estimate (3.6) and representation (3.7), the estimate (3.8) is then obtained by the same argument as the proof of [26, Theorem 3.3, Part I].
10
In the next section, we will remove the large jumps from the SDEs and study their well-posedness by Zvonkin’s transform. Thus we consider the following operator Z α,[ L w(x) = w(x + σ(x, y)) − w(x) − σ i (x, y)∂i w(x) ν α (dy) − bi (x)∂i w(x). (3.9) B\{0}
We have the following regularity result for Lα,[ . Corollary 3.8. There exists a constant κ∗ > 0 such that for any κ > κ∗ and f ∈ C β (Td ), there exists a unique solution u = u[κ ∈ C α+β (Td ) to the equation κu − Lα,[ u = f.
(3.10)
In addition, there exists a constant C > 0 independent of u, f, b, κ, such that for any κ > κ∗ , (κ − κ∗ )ku[κ k0 + κ
α+β−1 α
k∇u[κ k0 + [∇u[κ ]α+β−1 ≤ Ckf kβ .
(3.11)
Proof. To obtain the a priori estimate (3.11), we rewrite the equation (3.10) in the form Z α [u(x + σ(x, y)) − u(x)]ν α (dy). κu − L u = f − Bc
The estimate (3.8) implies that κku[κ k0 + κ
α+β−1 α
k∇u[κ k0 + [∇u[κ ]α+β−1 ≤ C(kf kβ + 2ν α (B c )kuk0 ).
Then (3.11) follows by choosing κ∗ = 2Cν α (B c ). Now define a family of operators by Z [ α,[ Lθ = L + θ [u(x + σ(x, y)) − u(x)]ν α (dy). Bc
Then L[1 = Lα , L[0 = Lα,[ . The well-posedness of equation (3.10) follows from the method of continuity and the a priori estimate (3.11), just as in the proof of Theorem 3.2.
4
SDEs with multiplicative stable L´ evy noise
The goal of this section is to study the strong well-posedness of SDEs (2.2) and (2.3), as ˜ x, for each > 0. As corollaries, we well as the ergodic properties of the solution processes X also obtain the Feynman-Kac formula and the well-posedness of nonlocal Poisson equation without zeroth-order term, which will be used to study homogenization in the next two sections.
11
4.1
Strong well-posedness of SDEs
We only consider the strong well-posedness for SDE (2.3) since (2.2) has the same form. The key is to reduce the SDE (2.3), whose coefficients have low regularity, to an SDE with Lipschitz coefficients by using Zvonkin’s transform. For κ > κ∗ , let ˆbκ ∈ C α+β (Td ) be the solution of κˆbκ − Lα,[ˆbκ = b, where Lα,[ is the operator in (3.9). The existence and uniqueness of solution ˆbκ is ensured by Corollary 3.8. Define a map Φκ : Rd → Rd by Φκ (x) = x + ˆbκ (x). Then Φκ is of class C α+β . Moreover, we have Lemma 4.1. For κ > 0 large enough, the map Φκ : Rd → Rd is a C 1 -diffeomorphism and α+β its inverse Φ−1 . κ is also of class C Proof. By the estimate in Corollary 3.8, we have κ
α+β−1 α
k∇ˆbκ k0 ≤ Ckbkβ ,
Now by choosing κ > κ∗ ∨ (2Ckbkβ )
α α+β−1
κ > κ∗ .
, we get that k∇ˆbκ k0 ≤ 21 . Thus
3 1 |x1 − x2 | ≤ Φκ (x1 ) − Φκ (x2 ) ≤ |x1 − x2 |, 2 2 i.e., Φκ is bi-Lipschitz. In particular, Φκ is a C 1 -diffeomorphism. Moreover, −1 ∇(Φ−1 κ ) = Inv ◦ ∇Φκ ◦ Φκ ,
where the matrix inverse map Inv : GL(Rd ) → GL(Rd ) is of class C ∞ . We arrive at the second conclusion of the lemma by applying [8, Proposition 1.2.7]. To solve SDE (2.3), by a standard interlacing technique (cf. [1, Section 6.5]), it suffices to solve the following SDE with no jumps greater than one: Z t Z tZ x,[ x,[ x,[ ˜ (dy, ds). ˜ ˜ ˜ s− Xt = x + b(Xs− )ds + σ(X , y)N 0
0
B
Now fix κ > 0 large enough such that the conclusions in Lemma 4.1 hold. We introduce Zvonkin’s transform ˜ t(κ) = Φκ (X ˜ tx,[ ). X Then by applying Itˆo’s formula, we have Z t Z tZ (κ) (κ) (κ) (κ) ˜ t = Φκ (x) + ˜ s− )ds + ˜ s− ˜ (dy, ds), X b (X σ (κ) (X , y)N 0
0
B
where b(κ) (x) = κˆbκ (Φ−1 κ (x)), −1 −1 −1 σ (κ) (x, y) = ˆbκ (Φ−1 κ (x) + σ(Φκ (x), y)) − bκ (Φκ (x)) + σ(Φκ (x), y).
12
(4.1)
˜ x = {X ˜ tx }t≥0 to SDE Theorem 4.2. For each x ∈ Rd , there is a unique strong solution X (2.3). Proof. By the above argument, we only need to show that SDE (4.1) admits a unique strong solution. Note that for γ ∈ (0, 1), f ∈ Cb1+γ (Rd ), x, u, v ∈ Rd , there exists a constant C > 0 such that |f (u + x) − f (u) − f (v + x) − f (v)| ≤ Ckf k1+γ |u − v||x|γ , the proof can be found in [3, Theorem 5.1.(c)]. Then for any x1 , x2 ∈ Rd , (κ) σ (x1 , y) − σ (κ) (x2 , y) −1 −1 ≤ ˆbκ (Φ−1 κ (x1 ) + σ(Φκ (x1 ), y)) − bκ (Φκ (x1 )) −1 −1 − ˆbκ (Φ−1 κ (x2 ) + σ(Φκ (x1 ), y)) + bκ (Φκ (x2 )) −1 −1 −1 ˆ + ˆbκ (Φ−1 κ (x2 ) + σ(Φκ (x1 ), y)) − bκ (Φκ (x2 ) + σ(Φκ (x2 ), y)) −1 + σ(Φ−1 κ (x1 ), y) − σ(Φκ (x2 ), y) −1 α+β−1 −1 ≤ Ckˆbκ kα+β Φκ (x1 ) − Φ−1 κ (x2 ) σ(Φκ (x1 ), y) −1 + (k∇ˆbκ k0 + 1) σ(Φ−1 κ (x1 ), y) − σ(Φκ (x2 ), y) α+β−1 ≤ Ckˆbκ kα+β k∇(Φ−1 |x1 − x2 ||y|α+β−1 κ )k0 kφkL∞ + C(k∇ˆbκ k0 + 1)k∇(Φ−1 )k0 |x1 − x2 ||y|, κ
where we have used the regularity condition for σ in Assumption H3, and φ is the positive bounded function in the growth condition in that assumption. Noting that 2(α + β − 1) > α by Assumption H2, we arrive at Z (κ) σ (x1 , y) − σ (κ) (x2 , y) 2 να (dy) ≤ C|x1 − x2 |2 . B
In addition, it is obvious that (κ) b (x1 ) − b(κ) (x1 ) ≤ C|x1 − x2 |. Now the strong well-posedness of SDE (4.1) follows from the classical result [13, Theorem 4.9.1]. The proof is complete. ˜ x is a Feller process with generator being the closure Corollary 4.3. The solution process X ˜ x is a strong Markov process. of (Lα , C α+β (Td )). In particular, X Proof. By applying Itˆo’s formula, it is easy to see that for any f ∈ D(Lα ) = C α+β (Td ), the following process is a (Ω, F, P, {Ft }t≥0 )-martingale Z t f x x ˜ (t) := f (X ˜ t ) − f (X ˜0 ) − ˜ sx )ds. Lα f (X M 0
˜ x has c`adl`ag paths almost surely. Let P ˜ x := P ◦ X ˜ x be the It is easy to see that X X ˜ x on (D, B(D)), then P ˜ x is a solution of martingale pushforward probability measure of X X α problem for (L , δx ). By Theorem 3.5, we find that PX˜ x = Px , the Feller property follows. The strong Markov property follows from [27, Theorem III.3.1]. 13
Remark 4.4. The Feller semigroup {Pt }t≥0 in Theorem 3.4 is the semigroup associated with ˜ x , that is, the solution process X ˜ x )), Pt f (x) = E(f (X t
f ∈ C(Td ).
As a consequence of the Feller property, we can obtain the well-posedness of the parabolic nonlocal PDE and the corresponding Feynman-Kac representation. See [23] for the classical version for second order PDE. Theorem 4.5. The parabolic nonlocal PDE ( ∂u (t, x) = Lα u(t, x) + gu(t, x), t > 0, x ∈ Rd , ∂t u(0, x) = u0 (x), x ∈ Rd , admits a unique mild solution in the sense that u(t) = u0 + L
α
Rt 0
u(s)ds ∈ D(Lα ) for all t ≥ 0 and
t
Z
Z u(s)ds + g
0
t
u(s)ds. 0
Moreover, the unique solution has the following Feynman-Kac representation Z t x x ˜ ˜ u(t, x) = E u0 (Xt ) exp g(Xs )ds . 0
Proof. Choose G > 0 large enough such that kgk0 < G. Define Z t g x x ˜ )ds − Gt , ˜ ) exp Pt f (x) = E f (X g(X s t
f ∈ C(Td ).
0
Then by an argument similar to that used in [1, Section 6.7.2], one can show that {Ptg }t≥0 is a Feller semigroup with generator being the closure of (Lα + g − G, C α+β (Td )). This yields that {eGt Ptg }t≥0 is a C0 -semigroup on C(Td ) with generator being the closure of (Lα + g, C α+β (Td )). Note that u0 ∈ C(Td ). Now applying the classic result [6, Proposition II.6.4] in the theory of C0 -semigroups, we conclude that the parabolic nonlocal PDE admits a unique mild solution, which can be given by the orbit map u(t) = eGt Ptg u0 . The desired conclusions follow immediately.
4.2
Ergodicity
Now we deal with the ergodicity of SDEs. By the above discussion, for every > 0, ˜ x, which is a Td -valued Feller process. SDE (2.2) also admits a unique strong solution X ˜ x, , by {Pt }t≥0 the associated Denote by p (t; x, y) the transition probability density of X Feller semigroup.
14
˜ x, possesses a unique invariant distribution Proposition 4.6. For each ≥ 0, the process X d µ on T . Moreover, there exist positive constants C and ρ such that for any periodic bounded Borel function f on Rn (i.e., f is Borel bounded on Td ), Z sup Pt f (x) − f (y)µ (dy) ≤ Ckf k0 e−ρt x∈Td
Td
for every t ≥ 0. Proof. Thanks to the Doeblin-type result presented in [5, Theorem 3.3.1], it is enough to ensure that the transition probability density p (t; x, y) is bounded from below by a positive constant (cf. [10, Proposition 1]), which follows immediately from the density estimates in Theorem 3.5. ˜ x in (2.3). Denote by µ = µ0 the invariant probability measure for the limit process X t d d Then from the fact that Pt f → Pt f in C(T ) as → 0 for any f ∈ C(T ) and t ≥ 0, we can prove the following lemma, as in [12, Lemma 2.4]. Lemma 4.7. As → 0, we have µ → µ. Now we combine Proposition 4.6 and Lemma 4.7 to get the following ergodic theorem. Theorem 4.8. Let f be a bounded Borel function on Td . Then for any t > 0, Z t x, Z X s ds → 0 f − f (x)µ(dx) Td 0
(4.2)
in probability, as → 0. Proof. For > 0, 0 ≤ s < t, let f¯ be a bounded measurable function on Td satisfying R f¯(x)µ (dx) = 0. Then by Proposition 4.6, Td h h i Z i x, ˜ x, x, ¯ ¯ ˜ ˜ |f (y)| p (t − s, Xs , y)dy − µ (dy) ≤ Ckf¯k0 e−ρ(t−s) . E |f (Xt )| Xs = Td
The rest of the proof is similar to that of [21, Proposition 2.4] by applying Lemma 4.7. For every γ > 0, denote by Cµγ (Td ) the class of allR f ∈ C γ (Td ) which are centered with respect to the invariant measure µ in the sense that Td f (x)µ(dx) = 0. It is easy to check that Cµγ (Td ) is closed, and hence a sub-Banach space of C γ (Td ) under the norm k · kγ . Thanks to Theorem 3.2 and Proposition 4.6, we can use the Fredholm alternative to obtain the solvability of the following Poisson equation without zeroth-order term in the smaller space Cµα+β (Td ), Lα u + f = 0, (4.3) for f ∈ Cµβ (Td ). Before that, we need some lemmas. Lemma 4.9. The restrictions {Ptµ := Pt |Cµ (Td ) }t≥0 form a C0 -semigroup on the Banach space (Cµ (Td ), k · k0 ), with generator given by Lαµ f := Lα f , D(Lαµ ) := Cµα+β (Td ). 15
Proof. Since µ is invariant with respect to {Pt }t≥0 , for any f ∈ Cµ (Td ) and t ≥ 0, we have Z Z Pt f (x)µ(dx) = f (x)µ(dx) = 0. Td
Td
That is, Cµ (Td ) is {Pt }t≥0 -invariant, in the sense that Pt (Cµ (Td )) ⊂ Cµ (Td ) for all t ≥ 0. The lemma then follows from the corollary in [6, Subsection II.2.3]. Lemma 4.10. If f ∈ Cµβ (Td ), then the unique solution uκ of (3.1) is of class Cµα+β (Td ), for any κ > 0. Proof. Since f is centered with respect to µ, by Proposition 4.6 we have kPt f k0 ≤ Ckf k0 e−ρt .
(4.4)
Note the fact that µ is invariant with respect to {Pt }t≥0 . Then combining (4.4) and the representation (3.7), a straightforward application of Fibini’s theorem implies that Z Z ∞ Z Z ∞ Z −κt −κt e Pt f (x)µ(dx) dt e Pt f (x)dtµ(dx) = uκ (x)µ(dx) = 0 Td Td 0 Td Z Z ∞ −κt f (x)µ(dx) dt = 0. e = 0
Td
That is, uκ is also centered with respect to µ. The following theorem will solve the well-posedness of equation (4.3), which is more general than the results in [10, Proposition 3]. We formulate it as follows, referring to [24, Theorem 1] for the classical version for second order partial differential operators. Theorem 4.11. For any f ∈ Cµβ (Td ), there exists a unique solution in Cµα+β (Td ) to the equation (4.3), which satisfies the estimate kukα+β ≤ C(kuk0 + kf kβ ),
(4.5)
where C = C(kbkβ ) is a positive constant. Moreover, the unique solution admits the representation Z ∞ u(x) = Pt f (x)dt. (4.6) 0
Proof. The a priori estimate (4.5) is also from [3, Theorem 7.1]. First, we show that if the equation has a solution u ∈ Cµ (Td ) for f ∈ Cµβ (Td ), then u must have the representation (4.6), this also implies the uniqueness. By the exponential ergodicity result in Proposition 4.6, we have kPtµ f k0 ≤ Ckf k0 e−ρt for any f ∈ Cµ (Td ) and t ≥ 0. This yields that, using [6, Theorem II.1.10.(ii)] as in the proof of Corollary 3.7, the set {z ∈ C|Rez > −ρ} is contained in the resolvent set of Lαµ . Noting that u = (0 − Lαµ )−1 f , the representation and uniqueness follow. Now we prove the existence. Let κ0 be a fixed positive constant. Thanks to Lemma 4.10, the linear map κ0 − Lα : Cµα+β (Td ) → Cµβ (Td ) is invertible. Furthermore, by virtue of Proposition 3.1 and the energy estimate (3.4), together with the compact embedding 16
Cµα+β (Td ) ⊂ Cµβ (Td ) (see, for instance, [11, Lemma 6.36]), the resolvent Rκ0 := (κ0 − Lα )−1 is compact from Cµβ (Td ) to Cµβ (Td ). Consider then the equation f ∈ Cµβ (Td ),
u − κ0 Rκ0 u = Rκ0 f,
(4.7)
Then the Fredholm alternative (see [11, Section 5.3]) implies that the equation (4.7) always has a unique solution u ∈ Cµβ (Td ) provided the homogeneous equation u − κ0 Rκ0 u = 0 has only the trivial solution u = 0. To rephase these statements in terms of the Poisson equation (4.3), we observe first that since Rκ0 maps Cµβ (Td ) onto Cµα+β (Td ), any solution u ∈ Cµβ (Td ) of (4.7) must also belong to Cµα+β (Td ). Hence, operating on (4.7) with κ0 − Lα we obtain −Lα u = (κ0 − Lα )(u − κ0 Rκ0 u) = f. Thus, the solutions of (4.7) are in one-to-one correspondence with the solutions of the Poisson equation (4.3). Consequently, (4.7) has a unique solution in Cµα+β (Td ) if we can show that the homogeneous equation Lα u = 0 has only the zero solution, while the latter follows from the representation (4.6). Remark 4.12. The assumption that f is centered with respect to µ in Theorem 4.11 is necessary. To see this informally, let’s recall the Riesz-Schauder theory for compact operators (cf. [33, Theorem X.5.3]). The equation (4.7) admits a solution u ∈ C(Td ) if and only if Rκ f ∈ Ker(I ∗ − κR∗κ )⊥ , where the superscript ∗ denotes the adjoint of operators. This is equivalent to say that the equation (4.3) admits a solution u ∈ C(Td ) if and only if f ∈ Ker(Lα,∗ )⊥ . On the other hand, we have µ ∈ Ker(Lα,∗ ) since µ is the invariant measure with respect to {Pt }t≥0 . Thus a necessary condition for the existence of (4.3) is hµ, f i = 0, regarding µ as an element in the dual space of C(Td ).
5
Homogenization of SDEs
The aim of this section is to show the homogenization result of the solutions X x, of SDEs 1 in (2.1). For this purpose, (2.1). It is quite natural to get rid of the drift term involving α−1 we again use Zvonkin’s transform, x, x X x, x, t ˆ t := Xt + ˆb X − ˆb , (5.1) where ˆb is the solution of the Poisson equation Lαˆb + b = 0,
(5.2)
with the linear operator Lα given by (3.2). Note that the transform here is slightly different from that used in Section 4. Due to Theorem 4.11, ˆb ∈ Cµα+β (Td ) is uniquely determined under the following assumption. Assumption H5. The functions b and e satisfy the centering condition, Z Z b(x)µ(dx) = 0, e(x)µ(dx) = 0. Td
Td
17
Note that this Assumption is quite natural in the homogenization problems and the reader can also find it in [5, 12, 21]. To prove the homogenization results of SDEs and nonlocal PDEs, we need the following scaling condition for the coefficient σ. Assumption H6. The function σ satisfies that Z y α+β 1 ν α (dy) → 0 as → 0. sup σ (x, y) − σ x, d x∈R B\{0} In the case σ(x, y) = σ0 (x)y, Assumption H6 holds automatically. We will let Assumptions H5 and H6 hold true in this and next section. We also need an elementary lemma. The proof is very similar to (3.3) and shall be omitted. Lemma 5.1. Let 0 < γ ≤ 1 and f ∈ Cb1+γ (Rd ). For any x, u, v ∈ Rd , it holds that |f (x + u) − f (x + v) − (u − v) · ∇f (x)| ≤
[∇f ]γ |u − v|1+γ . 1+γ
Now we are in a position to study the homogenization of SDEs with multiplicative stable noise. Theorem 5.2. In the sense of weak convergence on the space D, we have that, X x, ⇒ X x ,
where Xtx := x + Bt + Lt ,
as → 0. The homogenized coefficient B is given by Z (I + ∇ˆb)c(x)µ(dx), B=
(5.3)
(5.4)
Td
and {Lt }t≥0 is a symmetric α-stable L´evy processes with jump intensity measure Z Z 1A (σ(x, y))µ(dx)ν α (dy), A ∈ B(Rd \ {0}). Π(A) = Rd \{0}
(5.5)
Td
ˆ x, ⇒ X x , as → 0 (cf. Proof. Since ˆb is bounded, the theorem will follow if we prove that X [32, Theorem 2.7.(iv)]). By applying Itˆo’s formula, and note that ˆb ∈ C α+β (Td ) is the solution of Poisson equation (5.2), x, x, Z t Z t Z t Xs− 1 α x, x, σ,ν α ˆ Xs− ˆ ˆ Xt = x + (I + ∇b)c ds − A b ds + Aσ ,ν ˆb (Xs− ) ds α−1 0 0 0 Z tZ i h x, x, x, ˆ ˆ ˜ α (dy, ds) + b (Xs− + σ (Xs− , y)) − b (Xs− ) N 0
Rd \{0}
Z tZ
x, ˜ α (dy, ds) σ (Xs− , y) N
+ 0
=: x +
B\{0}
Λ1 (c)t
−
Z tZ 0
α Λ2 (ˆb, Aσ,ν )t
+
x, σ (Xs− , y) N α (dy, ds)
+
α Λ3 (ˆb, Aσ ,ν )t
18
Bc
˜ α )t + Λ (σ, N ˜ α )t + Λ (σ, N α )t . + Λ4 (ˆb, N 5 6
For the last three stochastic integral terms, we figure out the characteristics of them as semimartingales (cf. [15, Proposition IX.5.3]). Choose the truncation function h1 (x) = x, x, x, x1B (x). Denote by Ξ (s, y) := [ˆb (Xs− + σ (Xs− , y)) − ˆb (Xs− )]. Note that Ξ (·, 0) ≡ 0 by ˜ α) virtue of σ(·, 0) ≡ 0 as mentioned in Remark 2.1 (2). Then the characteristics of Λ4 (ˆb, N associated with h1 is given by Z tZ Ξ (s, y)1B c (Ξ (s, y))ν α (dy)ds, B4 (t) = − d 0 R \{0} C4 ≡ 0, Z tZ 1A (Ξ (s, y))ν α (dy)ds, A ∈ B(Rd \ {0}). ν4 (A × [0, t]) = 0
Rd \{0}
˜ α ) + Λ6 (σ, N α ) is given by The characteristics of Λ5 (σ, N Z tZ x, x, B5+6 (t) = σ (Xs− , y) [1B (σ (Xs− , y)) − 1B (y)] ν α (dy)ds, d \{0} 0 R C5+6 ≡ 0, Z tZ x, 1A (σ (Xs− , y)) ν α (dy)ds, A ∈ B(Rd \ {0}). ν5+6 (A × [0, t]) = d 0
R \{0}
By the same argument as in (2.5), we have B5+6 ≡ 0. Then the theorem is a consequence of the functional central limit theorem in [15, Theorem VIII.2.17] and the following lemma.
Lemma 5.3. For any t ∈ R+ , and any bounded continuous function f : Rd → R which vanishes in a neighbourhood of the origin, the following convergences hold in probability P when → 0: (i) sup0≤s≤t |Λ1 (c)s − Bs| → 0; ˆ σ ,ν α ˆ σ,ν α )s − Λ2 (b, A )s → 0; (ii) sup0≤s≤t Λ3 (b, A (iii) sup0≤s≤t |B4 (s)| → 0; RtR (iv) 0 Rd \{0} f (x)ν4 (dx, ds) → 0; (v)
RtR 0
Rd \{0}
f (x)ν5+6 (dx, ds) → t
R Rd \{0}
f (x)Π(dx);
where B and Π are defined in (5.4) and (5.5), respectively. Proof. (i). By Theorem 4.8, the convergence in probability of the first integral is immediate, x, Z t X s− (I + ∇ˆb)c ds → 0, → 0. sup |Λ1 (c)s − Bs| ≤ − B 0≤s≤t 0
19
(ii). Note that ν α (A) = −α ν α (A), A ∈ B(Rd \ {0}). By the oddness condition in Assumption H3, 1 σ,ν α ˆ x A b Zα−1 h x x y x x y x i = ˆb +σ , − ˆb − σi , ∂iˆb 1B (y) ν α (dy). Rd \{0} Then we have x σ ,ν α 1 α σ,ν ˆb (x) − A A ˆb α−1 Z x x x y x 1 ˆ ˆ = −b +σ , b + σ , y B\{0} x 1 i x i x y ˆ ν α (dy) − σ ,y − σ , ∂i b Z h x y x i α ˆ x + +σ , − ˆb ν (dy) b Bc =: I1 + I2 , where I1 and I2 are the two integral terms in the second equality. By the boundness of ˆb and the bounded convergence theorem, I2 go to zero as → 0. It follows from Lemma 5.1 and the scaling condition in Assumption H3 that Z x y α+β 1 1 x σ ,y − σ , ν α (dy) → 0. I1 ≤ α + β B\{0} Thus, α α sup Λ3 (ˆb, Aσ ,ν )s − Λ2 (ˆb, Aσ,ν )s 0≤s≤t x, Z t σ ,ν α 1 α σ,ν ˆb Xs− ds A ˆb Xs− − A ≤ α−1 0 → 0, → 0. ˜ α is definable, the (iii) and (iv). Since the stochastic Rintegral of Ξ with respect to N tR 2 third characteristic of Λ4 satisfies that 0 Rd \{0} (|x| ∧ 1)ν4 (dx, ds) < ∞ for each > 0 and t ∈ R+ (cf. [15, Proposition II.2.9]). By the hypothesis, there exist ρ > 0 and M > 0 such that |f | ≤ M on Bρc and f = 0 on Bρ . Then for any t ∈ R+ , Z tZ 0
Rd \{0}
f (x)ν4 (dx, ds)
Z tZ ≤M 0
Rd \{0}
1Bρc (x)ν4 (dx, ds),
which goes to zero almost surely as → 0 by the boundness of ˆb and the dominated convergence theorem, and (iv) follows. 20
For B4 , we have the estimate sup 0≤s≤t
|B4 (s)|
Z t Z
|x|2 ν4 (dx, ds)
≤ Rd \{0}
0
=:
p
J1 ·
p
21 Z t Z 0
21
Rd \{0}
1B c (x)ν4 (dx, ds)
J2 .
By (iv) and a usual approximation procedure, J2 goes to zero surely as → 0. For J1 , Z tZ h i 2 ˆ x, x, x, α ˆ b (X + σ (X , y)) − b (X ) J1 = s− s− s− ν (dy)ds 0 Rd \{0} Z t Z Z = + | · · · |2 ν α (dy)ds 0
Bc
B \{0}
2 Z Z t 4tkˆbk20 λ(Sd−1 ) 2−α Xs− 2 ˆ ≤ + kbk1 , y dsν α (dy). σ α B \{0} 0 By the growth condition in Assumption H3, 2 2 Z Z t d−1 2−α Z t σ Xs− , y dsν α (dy) ≤ tλ(S ) φ Xs− ds. 2−α B \{0} 0 0 Then (iii) follows from these estimates and Theorem 4.8. (v). It follows from Theorem 4.8 that, Z tZ Z Z t x, Xs− 5+6 f (y)ν (dy, ds) = f σ ,y dsν α (dy) d 0 Rd \{0} ZR \{0} Z0 f (σ(x, y))µ(dx)ν α (dy) →t d d ZR \{0} T f (y)Π(dy), → 0, =t Rd \{0}
where the convergence is in probability.
6
Homogenization of linear nonlocal PDEs Define Yt
Z t := 0
1 α−1
e
Xsx,
+g
Xsx,
ds.
(6.1)
Thanks to Theorem 4.5, the nonlocal PDE (1.1) has a unique mild solution, which is given by the Feynman-Kac formula, u (t, x) = E [u0 (Xtx, ) exp(Yt )] . ˆ x, , we define Similar to X Yˆt
:=
Yt
x Yt + eˆ − eˆ . 21
(6.2)
Here eˆ ∈ Cµα+β (Td ), thanks to Theorem 4.11 and Assumption H5, is the unique solution of the Poisson equation Lα eˆ + e = 0, with Lα given by (3.2). Again using Itˆo’s formula, x, x, Z t Z t Z t Xs− Xs− 1 σ ,ν α σ,ν α ˆ (g + ∇ˆ ec) Yt = ds − ds + A e ˆ X A e ˆ s− ds α−1 0 0 0 Z tZ x, x, x, ˜ α (dy, ds) + [ˆ e (Xs− + σ (Xs− , y)) − eˆ (Xs− )] N Rd \{0}
0
α α ˜ α )t . =: Λ1 (c, g)t − Λ2 (ˆ e, Aσ,ν )t + Λ3 (ˆ e, Aσ ,ν )t + Λ4 (ˆ e, N
Then in the same way as the proof of Theorem 5.2, we have the convergence of Y . Lemma 6.1. In the sense of weak convergence on the space D, both Y and Yˆ converge in distribution to a deterministic path y(t) = Et as → 0, where the homogenized coefficient E is given by Z (g + ∇ˆ ec)(x)µ(dx).
E :=
(6.3)
Td
Now we are in the position to prove the main result of this section. Since ˆb and eˆ are bounded on Rd , u has the same limit behavior with uˆ (t, x) := E[u0 (Xtx, ) exp(Yˆt )]
(6.4)
as → 0. Theorem 6.2. For each t ≥ 0, x ∈ Rd , u (t, x) → u(t, x),
→ 0,
(6.5)
where u satisfies the limit nonlocal PDE, ( ∂u (t, x) = AIdy ,Π u(t, x) + B · ∇u(t, x) + Eu(t, x), t > 0, x ∈ Rd , ∂t u(0, x) = u0 (x), x ∈ Rd , where Idy ,Π
A
Z
w(x + y) − w(x) − y i ∂i w(x)1B (y) Π(dy),
w(x) := Rd \{0}
and the constant coefficients B, E and measure Π are given by (5.4), (6.3) and (5.5) respectively. The solution is given by u(t, x) = E[u0 (x + Bt + Lt )]eEt . Proof. We only need to show uˆ (t, x) → u(t, x), → 0 for any t ≥ 0, x ∈ Rd . For the α α ˜ α )t as convenience of notation, we shall write Λ1 (c, g)t , Λ2 (ˆ e, Aσ,ν )t , Λ3 (ˆ e, Aσ ,ν )t , Λ4 (ˆ e, N Λ1 (t), Λ2 (t), Λ3 (t), Λ4 (t), respectively. We fix a t ∈ R+ . 22
Firstly, we prove the uniform integrability of the set {eΛ4 (t) |0 < ≤ 1} for each t ∈ R+ . This follows by proving that it is uniformly bounded in L2 (Ω, P). Denoting the integrand ˜ α ) by in Λ4 (ˆ e, N x, x, x, Γ (s, y) := [ˆ e (Xs− + σ (Xs− , y)) − eˆ (Xs− )] . Then by Itˆo’s formula, e
2Λ4 (t)
Z tZ = 1− Z t0 Z + 0
2e2Λ4 (s−) Γ (s, y)ν α (dy)ds
Bc (s,y)
e2Λ4 (s−) e2Γ
˜ (dy, ds) −1 N
B\{0}
Z tZ Z0 t ZB
− 1 N (dy, ds)
c
e2Λ4 (s−) e2Γ (s,y) − 1 − 2Γ (s, y) ν α (dy)ds.
+ 0
(s,y)
e2Λ4 (s−) e2Γ
+
B\{0}
Since eˆ is bounded, Γ has a uniform bound for all > 0. Then there exists a large constant C > 0 such that for each > 0 and t ∈ R+ , Z tZ Z t 2Λ (s−) 2Γ (s,y) 2 α α c 4 E e e − 1 ν (dy)ds ≤ Cν (B )E e2Λ4 (s−) ds < ∞. 0
Bc
0
Hence combining these, there exists θ ∈ (0, 1) such that Z tZ 2Λ4 (t) Ee =1+E e2Λ4 (s−) e2Γ (s,y) − 1 − 2Γ (s, y) ν α (dy)ds 0
Z ≤1+E
B\{0} t
e
2Λ4 (s−)
Z
(s,y)
2e2θΓ
|Γ (s, y)|2 ν α (dy)ds
B\{0}
0
Z ≤ 1 + C(kˆ ek0 )E
t
e
2Λ4 (s−)
0
Z
|Γ (s, y)|2 ν α (dy)ds.
B\{0}
As shown in the proof of part (iii) and (iv) in Lemma 5.3, Z Z λ(Sd−1 ) 2 2 2 α 2 2 |σ (Xs− , y)|2 ν α (dy) ≤ |Γ (s, y)| ν (dy) ≤ kˆ ek1 kˆ ek1 |ψ (Xs− )|2 . 2 − α B\{0} B\{0} Thus, Ee
2Λ4 (t)
2
d−1
≤ 1 + C(α, λ(S
Z ), kˆ ek1 , kψkL∞ )
t
Ee2Λ4 (s−) ds.
0 Λ4 (t)
By Gr¨onwall’s inequality, the uniform boundness of {e |0 < ≤ 1} in L2 (Ω, P) follows. Secondly, the proof of Lemma 5.3 shows that the set {Λ3 (t) − Λ2 (t)|0 < ≤ 1} is bounded. The set {Λ1 (t)|0 < ≤ 1} is bounded by virtue of the boundness of c, g and eˆ. Also since u0 is periodic and continuous, {u0 (Xt )|0 < ≤ 1} is bounded. Thus, the set {u0 (Xtx, ) exp(Yˆt )|0 < ≤ 1} is uniformly integrable. 23
ˆ
Finally, we pass to the limit. It is easy to see that eYt → ey(t) in probability as → 0 (by, for instance, [32, Theorem 2.7.(iii)]). Then for any subsequence {n } → 0, there exists ˆ nk a subsubsequence {nk } → 0 such that eYt → ey(t) almost uniformly (cf. [17, Lemma 4.2]). That is, for any ρ > 0, there exists a set N ∈ F with P(N ) ≤ ρ, such that
n
Yˆt k y(t) −e → 0, k → ∞. (6.6)
e L∞ (N c ,P)
By the boundness of u0 , we know the set {u0 (Xtx, )[exp(Yˆt ) − exp(y(t))]|0 < ≤ 1} is also uniformly integrable. Then for any δ > 0, there exist ρ0 > 0 and N0 ∈ F with P(N0 ) ≤ ρ0 , such that ˆ (6.7) E u0 (Xt ) eYt − ey(t) 1N0 < δ. Now along the sequence {nk }, we combining (6.6) with (6.7) to get nk nk Yˆt y(t) −e E u0 (Xt ) e ≤ E |· · · 1N0 | + E · · · 1N0c
n
ˆ k ≤ δ + ku0 kL∞ P(N0c ) eYt − ey(t)
L∞ (N c ,P)
≤ 2δ. To summarize these together, for any subsequence {n } → 0, there exists a subsubsequence {nk } → 0 such that nk n ˆ E u0 (Xt k ) eYt − ey(t) → 0, k → ∞, which implies that the convergence holds on the whole line 0 < ≤ 1. On the other hand, by Theorem 5.2, we know that E|u0 (Xt ) − u0 (Xt )| → 0 as → 0. The result (6.5) follows immediately. Remark 6.3. We close this section by some comments for the proof of Theorem 6.2. In [21], the author applied Girsanov’s transform to get rid of the stochastic integral term involved in Yˆ , since this term may not possess the uniformly integrability. While in our case, since the stochastic integral term in Yt has an infinitesimal integrand Γ (s, y), the uniform integrability of {exp(Yˆt )|0 < ≤ 1} is easier to treat. Acknowledgements. The research of J. Duan was partly supported by the NSF grant 1620449. The research of Q. Huang was partly supported by China Scholarship Council (CSC), and NSFC grants 11531006 and 11771449. The research of R. Song is supported in part by a grant from the Simons Foundation (# 429343, Renming Song).
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