Principal eigenvalue of the fractional Laplacian with a large incompressible drift

We study the principal Dirichlet eigenvalue of the operator LA=Δα/2+Ab(x)·∇\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L_A=\Delta^{\alpha/2}+Ab(x)\cdot\nabla}$$\end{document}, on a bounded C1,1 regular domain D. Here α∈(1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha\in(1,2)}$$\end{document}, Δα/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{\alpha/2}}$$\end{document} is the fractional Laplacian, A∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${A\in\mathbb{R}}$$\end{document}, and b is a bounded d-dimensional divergence-free vector field in the Sobolev space W1,2d/(d+α)(D). We prove that the eigenvalue remains bounded, as A→ + ∞, if and only if b has non-trivial first integrals in the domain of the quadratic form of Δα/2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\Delta^{\alpha/2}}$$\end{document} for the Dirichlet condition.


Introduction
This article is motivated by the following result of Berestycki et al. [4] for the Laplacian perturbed by a divergence-free drift in dimensions d ≥ 2. Let D ⊂ R d be a bounded C 2 regular open set and let b(x) = (b 1 (x), . . . , b d (x)) : R d → R d be a bounded d-dimensional vector field such that div b = 0 on D in the sense of distributions (distr.), i.e.
The result can be interpreted intuitively in the following way: functions w satisfying (1.2) are constant along the flow of the vector field Ab(x) (see Sect. 5.2), and the existence of (non-trivial) first integrals allows for flow lines that are contained in D. On the other hand, if no such w exist, then the flow leaves D with speed proportional to A. Adding the Laplacian Δ to b · ∇, or equivalently the Brownian motion to the flow, results in a stochastic process whose trajectories gradually depart from the integral curves of b, but the general picture is similar: if nontrivial first integrals exist, then the trajectories may remain in D with positive probability during a finite time interval, even as A → +∞. In this case we are lead to a nontrivial limiting transition mechanism between the flow lines. The result described in the foregoing enjoys many extensions and has proved quite useful in various applications describing the influence of a fluid flow on a diffusion, see for example [5,16,20,39]. In the context of a compact, connected Riemannian manifold a sufficient and necessary condition for λ A to remain bounded, as A → +∞, expressed in terms of the eigenspaces of the advection operator b(x) · ∇, has been given in [21,Theorem 1].
The purpose of the present paper is to verify that a similar property of the principal eigenvalue holds when the classical Laplacian is replaced by the fractional Laplacian Δ α/2 with α ∈ (1, 2). We consider I α 0 defined as the set of all the nonzero first integrals in the Sobolev space H The main difficulty in our development stems from roughness of general elements of H α/2 0 (D) and non-locality of Δ α/2 , which prevent us from a direct application of the differential calculus in the way it has been done in [4]. Instead, we use conditioning suggested by a paper of Bogdan and Dyda [9], approximation techniques for flows given by DiPerna and Lions in [17], and the properties of the Green function and heat kernel of gradient perturbations of Δ α/2 obtained by Bogdan and Jakubowski in [11] for α ∈ (1, 2) and bounded C 1,1 -regular open sets D. These properties allow to define and study, via the classical Krein-Rutman theorem and compactness arguments, the principal eigen-pair (λ A , φ A ) for L A = Δ α/2 + Ab · ∇ and α ∈ (1, 2). Our main result can be stated as follows.
paper will be Borel. We denote by the ball of center x ∈ R d and radius r > 0. We will consider nonempty, bounded open set D ⊂ R d , whose boundary is of class C 1,1 . The latter means that r > 0 exists such that for every Q ∈ ∂D there are balls B(x , r) ⊂ D and B(x , r) ⊂ R d \D, which are tangent at Q (the inner and outer tangent ball, respectively). We will refer to such sets D as to C 1,1 domains, without requiring connectivity. For an alternative analytic description and localization of C 1,1 domains we refer to [10,Lemma 1]. We note that each connected component of D contains a ball of radius r, and the same is true for D c . Therefore D and D c have a finite number of components, which will play a role in a later discussion of extensions of the vector field to a neighborhood of D. The distance of a given x ∈ R d to D c will be denoted by Constants mean positive numbers, that do not depend on the considered arguments of the functions being compared. Accordingly, notation f ( We will employ the function space L 2 (D), consisting of all square integrable real valued functions, with the usual scalar product Generally, given 1 ≤ p ≤ ∞, the norms in L p (D) shall be denoted by · p . Customarily, C ∞ c (D) denotes the space of smooth functions on R d with compact support in D. Also, In the last equality we have used Plancherel theorem. The Fourier transform of f is given byf

Isotropic α-stable Lévy process on R d
The discussion in Sects. 2.2 and 2.3 is valid for any α ∈ (0, 2). Let The coefficient is chosen in such a way that Vol. 21 (2014) Principal eigenvalue of the fractional Laplacian 545 We define the fractional Laplacian as the L 2 (R d )-closure of the operator Its Fourier symbol is given by Δ α/2 φ(ξ) = −|ξ| αφ (ξ), cf (2.2). The fractional Laplacian is the generator of the semigroup of the isotropic α-stable Lévy process (Y t , P x ) on R d . Here P x and E x are the law and expectation for the process starting at x ∈ R d . These are defined on the Borel σ-algebra of the canonical cádlág path space D([0, +∞); R d ) via transition probability densities as follows. We let (Y t ) be the canonical process, i.e.
and define time-homogeneous transition density p(t, x, y) : According to (2.3) and the Lévy-Khinchin formula, {p t } is a probabilistic convolution semigroup of functions with the Lévy measure ν(y)dy, see e.g. [8].

Stable process killed off D
Let τ D = inf{t > 0 : Y t / ∈ D} be the time of the first exit of the (canonical) process from D. For each (t, x) ∈ (0, +∞) × D, the measure is absolutely continuous with respect to the Lebesgue measure on D. Its density p D (t, x, y) is continuous in (t, x, y) ∈ (0, +∞) × D 2 and satisfies G. Hunt's formula (see [10,30]), (2.7) In addition, the kernel is symmetric (see [6,10] for discussion and references): and defines a strongly continuous semigroup on L 2 (D), We shall denote P t = P R d t . The Green function of Δ α/2 for D is defined as We have ( [11]), and (2.11) The following estimate has been proved by Kulczycki [30] and Chen and Song [14] (see also [24,Theorem 21]), In particular, From (2.12) it also follows that (2.14) We define H α/2 = H α/2 (R d ) as the subspace of L 2 (R d ) made of those elements for which Here (the Dirichlet form) E α is given as follows (cf [22]): If f ∈ H α/2 0 (D) and g belongs to the domain of the fractional Laplacian, then

Gradient perturbations of Δ α/2 on R d
Throughout the remainder of the paper we always assume that 1 < α < 2, and for each n ≥ 1 It follows from [26, Theorem 2 and Example 2] that series (2.21) converges uniformly on compact subsets of (0, +∞) × (R d ) 2 . From the results of [10], we know thatp(t, x, y) is a transition probability density function, i.e. it is non-negative and R dp In addition,p is continuous on (0, +∞) × (R d ) 2 , and where c T → 1 if T → 0. In fact, this holds under much weaker, Kato-type condition on b, see [10, Theorems 1 and 2]. We denote byP x andẼ x the law and expectation on D([0, +∞); R d ) for the (canonical) Markov process Y starting at x and defined by the transition probability densityp, 548 K. Bogdan and T. Komorowski NoDEA Remark 1.P x may also be defined by solving stochastic differential equation where the isotropic α-stable Lévy process is now denoted by Y (0) . Such equations have been studied in dimension 1 in [37] under the assumptions of boundedness and continuity of the vector field; also for α = 1. We refer the reader to [34, formula (13)], for a closer description of a connection to (2.21) and (2.20).

Remark 2.
We may also define the perturbation series for −b(x). In what follows, objects pertaining to −b will be marked with the superscript hash ( # ), e.g.p # (t, x, y) = ∞ n=0 (−1) n p n (t, x, y).

Antisymmetry of the perturbation
The last equality extends to arbitrary f, g ∈ H 1 0 (D).

Remark 3.
A strengthening of Proposition 2.1 will be given in Corollary 3.11 below.

Gradient perturbations with Dirichlet conditions
We recall that D is a bounded C 1,1 domain in R d . Hunt's formula may be used to define the transition probability density of the (first) perturbed and (then) killed process [11]. Thus, for t > 0, x, y ∈ D 2 we let We havẽ By [11, formula (40)], there exist constants c, C > 0 such that We define the Green function of D for L: The main result of [11] asserts that andG D (x, y) is continuous for x = y (for estimates ofp D see [13]). We consider the integral operators In light of (2.32) and (2.
It has been shown in [11] that the following crucial recursive formula holds and for all ϕ ∈ C ∞ c (D) and Later on we shall also consider the operator

Comparison of the domains of generators
The following pointwise version of (2.39) is proved in [11,Lemma 12], We define After a series of auxiliary estimates, we will prove H to be compact on L 2 (D).
Vol. 21 (2014) Principal eigenvalue of the fractional Laplacian 551 Proof. We only need to examine points close to ∂D. Given such a point we consider the integral over its neighborhood O. The neighborhood can be chosen in such a way that, after a bi-Lipschitz change of variables (see [11, formula (75)], [12, formula (11)]), we can reduce our consideration to the case when

Remark 5. The result is valid for all bounded open
Lipschitz sets [12] in all dimensions d ∈ N.
The above result shall be used to establish that operator K (thus also H) is L 2 bounded via Schur's test, see [23,Theorem 5.2]. Note that we have pK ≤ cq, for some constant Summarizing, we will require the following conditions: , or equivalently when |y − x| < (C/N ) 1/(d+1−α) , and then there are constants C, C 1 > 0 such that This estimate for H(x, y) − H N (x, y) actually holds for all x, y ∈ D, hence (3.11) yields Since H is a norm limit of compact operators, it is compact, too.
In view of Proposition 3.5 we may regard the gradient operator b · ∇ as a small perturbation of Δ α/2 when α ∈ (1, 2). In fact b · ∇ is relatively compact with respect to Δ α/2 with Dirichlet conditions in the sense of [27, IV.1.3]. For future reference we remark that I − H * is invertible, too.

Proof. By [11, Lemma 10],
for any bounded function g. Invoking the argument used in Proposition 3.5, we conclude that there is a number c independent of g for which (3.14) By approximation, (3.13) and (3.14) extend to all g ∈ L 2 (D). Furthermore, if {g j } ⊂ C ∞ c (D) and lim j→+∞ g j = g in L 2 (D), then lim j→+∞ |G D g−G D g j | 1 = 0, cf. (2.1). Each G D g j is continuous on D and G D g j = 0 on ∂D, hence G D g j ∈ H 1 0 (D) (see [1,Theorem 5.37]). Therefore, G D g ∈ H 1 0 (D).
We shall also observe the following localization principle for our perturbation problem.
D as operators on L 2 (D), hence also as functions defined in Sect. 2.6. A similar conclusion forP D t follows from the fact thatG −1 D is the generator of the semigroup. We will identify the adjoint operator of H on L 2 (D). Proof. If f, g ∈ L 2 (D), then by Corollary 3.7 and (2.25), Functions {G D g, g ∈ L 2 (D)} form a dense set in L 2 (D), which ends the proof. When definingP D t , via (2.31), we may encounter the situation when b is given only on D. We may extend the field outside D by letting e.g. b = 0 on D c . Of course such an extension needs not satisfy divb = 0, even though the condition may hold on D. However, we still have a local analogue of Proposition 2.1. Recall that the transition probability densities and Green function corresponding to the vector field −b are marked with a hash ( # ). Corollary 3.11. If div b = 0 on D, then for all t > 0 and x, y ∈ D, Proof. We shall first prove (3.17), or, equivalently, that Applying the operator G D from the left to both sides of the equality we obtaiñ By Lemma 3.10, Taking adjoints of both sides of (3.12) we also obtain, We have already noted in the proof of Theorem 3.6 that I − H * is a linear automorphism of L 2 (D). Therefore (3.20) and (3.21) give (3.18). Furthermore, let L * denote the adjoint of L on L 2 (D). We have where the last equality follows from [18, Lemma XII.1.6]. Let (P D * t ) be the semigroup adjoint to (P D t ). By [33,Corollary 4.3.7], the generator of (P D * t ) is L * . Since L # = L * , the semigroups are equal. The corresponding kernels are defined pointwise, therefore they satisfy (3.16).

Lemma 4.2.G D is compact on L 2 (D).
Proof. Let N > 0, andG D (x, y) is compact. Indeed, it has a finite Hilbert-Schmidt norm: on L 2 (D) may be directly estimated as follows, see, e.g., Theorem 3, p. 176 of [32]. We let N → ∞. The functionsG D (x, ·) (and G D (·, y)) are uniformly integrable on D by (2.35). SinceG D is approximated in the norm topology by compact operators, it is compact.
In the special case when b(x) ≡ 0, (P D t ) equals (P D t ), a symmetric contraction semigroup on L 2 (D), whence the Green operator G D is symmetric, compact and positive definite. The spectral theorem yields the following.
Lemma 4.4. If (4.2) holds for some φ ∈ L 1 (D) and λ = 0, then φ ∈ C(D) and there is C = C(α, b, D, λ) such that Proof. Starting from (4.2), for an arbitrary integer n ≥ 1 we obtain D (x, y) :=G D (x, y) and To estimateG (n) D , we use basic properties of the Bessel potentials, which can be found in [2, Ch.II. §4]. Recall that for α > 0 the Bessel potential kernel G α is the unique, extended-continuous, probability density function on R d , whose Fourier transform isĜ again. Considering such n we conclude that φ is bounded. The boundary decay of φ follows from (4.2), (2.36) and (2.13). The continuity of φ is a consequence of the continuity ofG D (x, y) for y = x, and the uniform integrability of the kernel, which stems from (2.35).

Proof of Theorem 1.1
We say that w is a first integral of b if (D) and w is not equal to 0 a.e. Recall that for A ∈ R, the operator L A = Δ α/2 + Ab · ∇ is considered with the Dirichlet exterior condition on D, i.e. it acts on G D (L 2 (D)), see Theorem 3.6. The Green operator and Krein-Rutman eigen-pair of L A shall be denoted byG A and (λ A , φ A ), respectively. We also recall that φ A ∈ D(L A ) and The proof of (1.3) shall be obtained by demonstration of lower and upper bounds for λ A (as A → ∞).

Proof of the lower bound (1.4) Proposition If f ∈ D(L
Proof. Let f ∈ D(L A ). According to Proposition 3.8, f belongs to D(Δ α/2 ) and H 1 0 (D). In addition, L A f = Δ α/2 f + Ab · ∇f . Taking the scalar product of both sides of the equality against f and using (2.25) we get the result, because the second term vanishes.
According to Proposition 5.1, 3) Suppose A n → +∞, as n → ∞, but λ An stay bounded. By (5.3) and Corollary 4.3, the sequence φ An = G Suppose that w is a weak limit of φ An in H α/2 0 (D), thus a strong limit in L 2 (D). We have w 2 = 1, and for ψ ∈ C ∞ c (D), Dividing both sides by A n and passing to the limit, we obtain that

The proof of the upper bound (1.5)
The proof of (1.5) uses "conditioning" of truncations of w 2 by the principal eigenfunction inspired by [4] and [9] [see (5.21) below]. Here w is a first integral in H α/2 0 (D). An important part of the procedure is to prove that the truncation of w 2 is also a first integral in H α/2 0 (D). This is true if w ∈ H 1 0 (D): suppose that f : R → R and f are bounded and f (0) = 0. Let div b = 0 and that div(wb) = b · ∇w = 0 a.e. Then, a.e. we have   [17,Theorem III.1], there exists a unique a.e. defined jointly Borelian family of mappings X(·, x) : R → R d (flow generated by b) with the following properties: first, for a.e. x ∈ R d , the function R t → b (X(t, x)) is continuous and (X(s, x))ds, t ∈ R, (5.5) so that, in particular, second, for all t ∈ R and Borel measurable sets We let p ∈ [1, +∞], u 0 ∈ L p (R d ), and define Note that, (t, u 0 (·)) → u(t, ·), t ∈ R, defines a group of isometries on L p (R d ).
If X(s, x))ds + t 0 div(vb)(X(s, x))ds, for a.e. x ∈ R d and all t ∈ R, because div b = 0 implies b · ∇v = div(vb). Then for all t ∈ R, Let g ∈ C ∞ c (B(0, 1)), g ≥ 0 and R d g dx = 1 (a mollifier). Define The function is an approximate solution of the transport equation ∂u ∂t − b · ∇u = 0, (5.13) in the following sense: if  X(t, ·). A word of explanation may be helpful. Suppose that b ∈ W 1,q loc (R d ), div b = 0 a.e., and w is an L p -integrable first integral, i.e. b · ∇w = 0, as distributions on space-time R × R d , see [17, (13)]. By [17,Corollary II.1], there is a unique solution to the transport equation (5.13) with the initial condition u(0, ·) = w. The equation is understood in the sense of distributions on space-time, too. By [17,Theorem III.1], the solution has the form (t, x) → w (X(t, x)). However, since w is a first integral, the mapping (t, x) → w(x) defines another solution. Thus, by uniqueness, w(X(t, x)) = w(x) a.e. In our case this argument needs to be slightly modified since the first integral is defined only on D and not on the entire R d . In particular, the identity w(X(t, x)) = w(x) is bound to hold only for small times t.
As an immediate consequence of Lemma 5.2 we obtain the following.  Proposition 5.5. Suppose that A ∈ R, ε > 0, w ∈ I α 0 and w is bounded. Then, However, the second term on the right hand side vanishes, because − D w 2 (z)b(z) · ∇ log[φ A (z) + ε]dz = 0, and w 2 = w 2 N , for a sufficiently large N , is a first integral by virtue of Corollary 5.4. Thus (5.19) follows from (2.19).

The upper bound when the drift is defined only on D.
Suppose that b ∈ W 1,q (D), with q = 2d/(d + α), and div b(x) ≡ 0 on D. Here, as usual, D is a bounded domain with the C 1,1 class boundary. By the discussion in Sect. 2.1, R d \D has finitely many, say, N + 1 connected components. Denote them by O 0 , . . . , O N , and assume that ∞ belongs to the compactification of O 0 . We start with the following extension result.