Self-similar solution for fractional Laplacian in cones

We construct a self-similar solution of the heat equation for the fractional Laplacian with Dirichlet boundary conditions in every fat cone. As applications, we give the Yaglom limit and entrance law for the corresponding killed isotropic stable L\'{e}vy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone.

The proof of Theorem 1.1 is given in Section 3. In a perspective, the result is the next step in the development of the potential theory of the isotropic α-stable processes after the boundary Harnack principle, Green function, and Dirichlet heat kernel estimates, suggested by the Introduction of Bogdan at al. [11].In view of (1.1) and (1.2), Ψ t (x) may be called a self-similar semigroup solution of the heat equation for the fractional Laplacian with Dirichlet conditions.The property (1.2) also means that Ψ t is an entrance law for p Γ at the origin, see, e.g., Blumenthal [6,p. 104], Haas and Rivero [23] or Bañuelos et al. [2].Furthermore, in Theorems 3.11 and 3.12 below, we prove the existence of the Yaglom limit for Γ.Similar results were obtained in Bogdan et al. [16,Theorems 1.1 and 3.3] for Lipschitz cones.Our approach is different and more versatile than that presented in [16]; we are able to cover more general cones, e.g.Γ = R \ {0} or Γ = R 2 \ ([0, ∞) × {0}), and much more general initial distributions for the Yaglom limit, including distributions with unbounded support.
We next present our second main result.Let 1 q ∞ and L q (Γ) := L q (Γ, dx).For a weight function w > 0, we denote L q (w) := L q (Γ, w(x) dx).For instance, L 1 (M Γ ) = {f /M Γ : f ∈ L 1 }.Then, for 1 q < ∞, we define and, for q = ∞, we let f ∞,M Γ := ess sup Of course, f 1,M Γ = f L 1 (M Γ ) .For a non-negative or integrable function f we let We say that the cone Γ is smooth if its boundary is C 1,1 outside of origin, to wit, there is r > 0 such that at every boundary point of Γ on the unit sphere S d−1 , there exist inner and outer tangent balls for Γ, with radii r.Put differently, for d 2, the spherical cap, Γ ∩ S d−1 , is a C 1,1 subset of S d−1 .For instance, the right-circular cones (see Section 2) are smooth.
Let us comment on our methods and previous developments in the literature.If Γ = R d , then β = 0, M Γ = 1 and p Γ t (x, y) = p t (x, y) is the transition density of the fractional Laplacian on R d (see below).In this case, Ψ t (x) = p t (0, y) and Theorem 1.2 were resolved by Vázquez [36], see also Bogdan et al. [14] when κ = 0 in [14, Eq. (1.1)]; see also Example 4.4 below.For general cones Γ, the behavior of p Γ t is intrinsically connected to properties of M Γ , see, e.g., Bogdan and Grzywny [9], [16], or Kyprianou et al. [28].The identification of the Martin kernel M Γ was accomplished by Bañuelos and Bogdan [3].Its crucial property is the homogeneity of order β ∈ [0, α), which is also reflected in the behavior of the Green function studied by Kulczycki [25] and Michalik [29,Lemma 3.3], at least when Γ is a right-circular cone.As we see in Theorems 1.1 and 1.2, the exponent β determines the self-similarity of the semigroup solution and the asymptotic behavior of the semigroup P Γ t , too.For more information on β we refer the reader to [3] and Bogdan et al. [18].
If Γ is a Lipschitz cone, then Theorem 1.1 follows from [16, Corollary 3.2 and Theorem 3.3].However, the method presented in [16] does not apply to fat cones, in particular, to Γ = R \ {0} or Γ = R 2 \ ([0, ∞) × {0}), which are intrinsically interesting for α ∈ (1, 2).Therefore in this work, we follow the approach suggested by [14], where the authors employ a stationary density of an Ornstein-Uhlenbeck type semigroup corresponding to a homogeneous (self-similar) heat kernel.Another key tool in their analysis is the so-called Doob conditioning using an invariant function or the heat kernel; see also Bogdan et al. [10,Theorem 3.1].In the present paper, we study the semigroup (P Γ t : t 0) of the α-stable Lévy process killed when exiting the cone Γ.Its kernel is the (Dirichlet heat) kernel p Γ t .Although the setting is seemingly different than in [14], due to Theorem 3.1 (below) the Martin kernel M Γ is invariant with respect to P Γ t , which allows for Doob conditioning.Then we form the corresponding Ornstein-Uhlenbeck semigroup and prove existence of a stationary density ϕ in Theorem 3.4 by using the Schauder-Tychonoff fixed-point theorem.As we shall see in the proof of Theorem 1.1, the self-similar semigroup solution Ψ t is directly expressed by ϕ and M Γ .
In Subsection 3.3 we obtain an asymptotic relation between the Martin kernel and the survival probability near the vertex of the cone (see Corollary 3.10).We also obtain a Yaglom limit (quasi-stationary distribution) in Theorem 3.11, which describes the behavior of the stable process starting from a fixed point x ∈ Γ and conditioned to stay in a cone, generalizing Theorem 1.1 of [16].In Theorem 3.12 we extend both results to every initial distribution with finite moment of order α.Note that once the existence and properties of the stationary density ϕ are established, the results of Subsection 3.3 follow by scaling.Notably, our approach applies to rather general self-similar transition densities, at least when they enjoys positive sharp (upper and lower) bounds and invariant function exists.For an approach to entrance laws based on fluctuation theory of Markov additive processes, we refer to [27], see also Chaumont et al. [20].In passing, we also note that Yaglom limit for random walks in cones is discussed by Denisov and Wachtel [27].For a broad survey on quasi-stationary distributions, we refer to van Doorn and Pollet [35].Self-similar solutions for general homogeneous semigroups are discussed in Cholewa and Rodriguez-Bernal [21]; see Patie and Savov [30]

Preliminaries
For x, z ∈ R d , the standard scalar product of is denoted by x • z and |z| is the Euclidean norm.For x ∈ R d and r ∈ (0, ∞), we let B(x, r) = {y ∈ R d : |x − y| < r}, the ball centered at x with radius r, and we write B r := B(0, r).All the considered sets, functions and measures are Borel.For non-negative functions f, g, we write f ≈ g if there is a number c ∈ (0, ∞), i.e., a constant, such that c −1 f g cf , and write f g if there is a constant c such that f cg.
Recall that α ∈ (0, 2) and let where the constant c d,α is such that For t > 0 we let (2.1) By the Lévy-Khintchine formula, p t is a probability density function and We consider the isotropic α-stable Lévy process X = (X t , t 0) in R d , with as transition density.Thus, The Lévy-Khintchine exponent of X is, of course, |ξ| α and ν is the intensity of jumps.By (2.1), for every isometry T on R d .It is well known that (2.4) see, e.g., [7].We then consider the time of the first exit of X from the cone Γ, and we define the Dirichlet heat kernel for Γ, x, y ∈ Γ, t > 0, see [15,22].It immediately follows that p Γ t (x, y) p t (x, y) for all x, y ∈ Γ and t > 0. The Dirichlet heat kernel is non-negative, and symmetric: p Γ t (x, y) = p Γ t (y, x) for x, y ∈ Γ, t > 0. It satisfies the Chapman-Kolmogorov equations: For nonnegative or integrable functions f we define the killed semigroup by In particular, for f ≡ 1 we obtain the survival probability: (2.6) see [12,Remark 1.9].Since t −1/α Γ = Γ, the scaling (2.2) extends to the Dirichlet heat kernel: As a consequence, (2.7) Furthermore, by (2.3), (2.8) The operators P Γ t and the kernel p Γ t (x, y) are the main subject of the paper.In view of (2.8), without loss of generality we may assume that 1 := (0, . . ., 0, 1) ∈ Γ.By [3, Theorem 3.2], there is a unique non-negative function M Γ on R d such that M Γ (1) = 1, M Γ = 0 on Γ c , and for every open bounded set B ⊆ Γ, (2.9) Moreover, M Γ is locally bounded on R d and homogeneous of some order β ∈ [0, α), i.e., (2.10) We call M Γ the Martin kernel of Γ with the pole at infinity.
Throughout the article, we often assume that Γ is fat, i.e., κ ∈ (0, 1) exists such that for all Q ∈ Γ and r ∈ (0, ∞), there is a point The parameter η ∈ (0, π) is usually called the angle of the cone.Of course, every right-circular cone is smooth, and every smooth cone is fat.

Doob conditioning
The Martin kernel M Γ is invariant for the semigroup P Γ t , as follows.Theorem 3.1.For all x ∈ Γ and t > 0, we have . By (2.9) and the strong Markov property, where the last equality follows from the fact that M Γ = 0 outside Γ.We note that ).We consider two scenarios.On {τ Γ = ∞}, for R large enough, we have: On {τ Γ < ∞}, for R large enough we have: too.In both cases, the integrand on the right-hand side of (3.2) converges a.s. to the integrand on the right-hand side of (3.1) as R → ∞.By the local boundedness of M Γ and (2.10), Using [4, Theorem 2.1] and the fact that β ∈ [0, α), we conclude that E x (X * t ) β < ∞.An application of the dominated convergence theorem ends the proof.
3.1.Renormalized kernel.We define the renormalized (Doob-conditioned) kernel Note that ρ is jointly continuous.By Theorem 3.1, and by (2.5), In other words, ρ t is a symmetric transition probability density on Γ with respect to the measure M 2 Γ (y) dy.Furthermore, the following scaling property holds true: for all x, y ∈ Γ and all t > 0, Therefore, By (2.11), for fat cones we have The boundary behavior of (1) For t = 1, (3.9) reads as follows, (3.10) The estimate (3.9) applies to arbitrary cones and arguments t, x, however, it is not optimal.For example, for the right-circular cones, we can confront (3.8) with as provided by [29,Lemma 3.3] and [11,Example 7].
(3) For the right-circular cones, the ratio Clearly, c 2 > 0. We recall the Ikeda-Watanabe formula: . By (3.11), we get (3.10) when x ∈ Γ \ Γ 1 .For arbitrary t > 0, we use (2.7) and (3.10): By the proof of [3, Lemma 4.2], for every R ∈ (0, ∞) there exists a constant c, depending only on α, Γ and R, such that In particular, for fat cones, in view of (2.4) and (3.8), (3.12) with comparability constant depending only on α, Γ and R. Using Lemma 3.2 we also conclude that for every R 1 there is a constant c depending only on R, α and Γ, such that 3.2.Ornstein-Uhlenbeck kernel.Encouraged by [14], we let and, by (3.7), we get the Chapman-Kolmogorov property for ℓ t : Thus, ℓ t is a transition probability density on Γ with respect to M 2 Γ (y) dy.We define the corresponding Ornstein-Uhlenbeck semigroup: We easily see that the operators are bounded on L 1 (M 2 Γ (y) dy).In fact, they preserve densities, i.e., functions f 0 such that Γ f (x)M 2 Γ (x) dx = 1.Before we immerse into details, let us note that the relations (3.12) and (3.13) will be crucial in what follows.Both of them rely on the factorization of the Dirichlet heat kernel (2.11), which is valid for fat sets.For this reason, although it is usually clear from the setting, to avoid unnecessary considerations we assume below in this section that Γ is a fat cone.Theorem 3.4.Assume Γ is a fat cone.Then there is a unique stationary density ϕ for the operators L t , t > 0.
Proof.Fix t > 0 and consider the family F of non-negative functions on Γ that have the form for some sub-probability measure µ concentrated on Γ 1 .By (3.4), F ⊆ L 1 (M 2 Γ (y)).By the scaling (3.7) and the same reasoning as in the proof of [14, Theorem 3.2], L t F ⊆ F .Since L t is continuous, we also have L t F ⊆ F , where F is the closure of F in the norm topology of L 1 (Γ, M 2 Γ (y) dy).Next, we observe that F is convex, therefore by [19,Theorem 3.7], F is equal to the closure of F in the weak topology.In view of (3.12), (3.15) f (y uniformly for f ∈ F .Moreover, (3.4) and (3.12) show that the right-hand side of (3.15) is integrable with respect to M 2 Γ (y) dy.Therefore, the family F is uniformly integrable with respect to M 2 Γ (y) dy.By [8,Theorem 4.7.20],F is weakly pre-compact in L 1 (M 2 Γ (y)), so F is weakly compact.Furthermore, we invoke [19,Theorem 3.10] to conclude that L t is weakly continuous.By the Schauder-Tychonoff fixed point theorem [32,Theorem 5.28], there is a density ϕ ∈ F satisfying L t ϕ = ϕ.It is unique by the strict positivity of the kernel ℓ t , and the same for every t > 0, see the proof of [14,Theorem 3.2].
Let us note that by Theorem 3.4 and [26, Theorem 1 and Remark 2], the following stability result for kernels l t in L 1 (M 2 Γ (y) dy) holds true for every x ∈ Γ: We claim that the convergence in (3.16) is in fact uniform for x in any bounded subset A ⊆ Γ.Indeed, let x, x 0 ∈ A. In view of (3.14) and (3.8) we may write By (3.16), for every z ∈ Γ, Moreover, by the dominated convergence theorem the iterated integral in (3.17) tends to 0 as t → ∞, so the convergence in (3.16) is uniform for all x ∈ A, as claimed.By rewriting (3.16) in terms of ρ, we get that, uniformly for x ∈ A, This leads to the following spacial asymptotics for ρ 1 .
By the continuity of dilations in Thus, by a change of variables in (3.19) and the triangle inequality, we conclude that uniformly for all z ∈ A. To end the proof, we take A = B 1 and x = e t − 1 −1/α z, where t = ln 1 + |x| −α and z = x/|x| ∈ A. .Lemma 3.6.After a modification on set of Lebesgue measure 0, ϕ is continuous on Γ and Proof.By Corollary 3.5 and (3.12), on Γ less a set of Lebesgue measure zero.Theorem 3.4 entails that ϕ = L 1 ϕ a.e., so it suffices to verify that L 1 ϕ is continuous on Γ.To this end we note that ℓ 1 (x, y) is continuous in x, y ∈ Γ. Next, by (3.14) and (3.8), Let R > 1.By (2.4) and (3.9), [11, Remark 3] and (2.7), and the homogeneity (2.10) of M Γ , By the dominated convergence theorem, In what follows, ϕ denotes the continuous modification from Lemma 3.6.
Theorem 3.7.Let Γ be a fat cone.For every t > 0, uniformly in y ∈ Γ we have Proof.If β = 0 then ρ t (x, y) = p t (x, y) and the claim is simply the continuity property of the heat kernel p t .Thus, we assume that β > 0.
We only prove the claim for t = 1; the extension to arbitrary t is a consequence of the scaling (3.6).By (3.7) and the Chapman-Kolmogorov property, for x, y ∈ Γ, We will prove that, uniformly in y ∈ Γ, To this end we first claim that there is c ∈ (0, ∞) dependent only on α and Γ, such that for all x ∈ Γ 1 and y ∈ Γ, (3.21) Indeed, denote ỹ = 2 1/α y.By (3.8), Lemma 3.6 and (3.10), there is c > 0 such that for all z, y ∈ Γ and x ∈ Γ 1 , We split the integral in (3.21) into two integrals.For z ∈ A := B( y, | y|/2) we use the fact that |z| ≈ | y| ≈ |y| and 1 + |z − y| 1, therefore Combining it with (3.22), we arrive at (3.21), as claimed.Let ε > 0. In view of (3.21) and the fact that β > 0, there is R ∈ (0, ∞) depending only on α, β, Γ and ε such that with the implied constant dependent only on α, β, Γ and R, but not otherwise dependent of y.Thus, by Corollary 3.5, for all y ∈ Γ R and x ∈ Γ 1 small enough.Putting (3.24) together with (3.23) we arrive at (3.20).
Using the scaling property (3.7) and Theorem 3.4, The proof is complete.
Note that by the symmetry of ρ t , for x ∈ Γ, Recall also that by (3.12) and (3.4), Thus, by Theorem 3.7 and the dominated convergence theorem, Let us summarize the results of this section in one statement.
We conclude this part by rephrasing (3.25) in terms of Ψ t : 3.3.Yaglom limit.The above results quickly lead to calculation of the Yaglom limit for the stable process (conditioned to stay in a cone).Note that our proof is different than that in [16].We also cover more general cones, including R \ {0} and R 2 \ ([0, ∞) × {0}).First, we obtain the following extension of [16,Theorem 3.1].Corollary 3.10.Let Γ be a fat cone.For every t > 0, Proof.It is enough to prove the claim for t = 1; the general case follows by the scalings (2.7) and (2.10).We have We use (3.12), the dominated convergence theorem, and Theorem 3.7 to get the conclusion.
The first identity below is the Yaglom limit.
Theorem 3.11.Assume Γ is a fat cone and let B be a bounded subset of Γ.Then, uniformly in x ∈ B, lim t→∞ Proof.By (2.6) and the scaling property (2.7), .
Example 3.13.Note that β = 0 if and only if Γ c is a polar set and then M Γ (x) = 1 for all x ∈ Γ, see [3,Theorem 3.2].Consequently, we have p Γ t (x, y) = p t (x, y) and P x (τ Γ > t) = 1 for all x, y ∈ Γ and all t > 0. It follows that ρ t (x, y) = p t (x, y) and a direct calculation using the Chapman-Kolmogorov property entails that ϕ(y) = p 1 (0, y) is the stationary density for the (classical) α-stable Ornstein-Uhlenbeck semigroup, see (3.14) and Theorem 3.4.The statement of Theorem 3.7 thus reduces to the continuity property the heat kernel of the isotropic α-stable Lévy process.Theorems 3.11 and 3.12 trivialize in a similar way.Incidentally, in this case the moment condition on γ in Theorem 3.12 is superfluous.Further examples are given in Section 4.

Asymptotic behavior for the killed semigroup
This section is devoted to examples and applications in Functional Analysis and Partial Differential Equations.Note that in Lemmas 4.1 and 4.2 we do not assume that Γ is fat.
To prove the strong continuity, we fix f ∈ L 1 (M Γ ) and let By the first part of the proof, To this end we let ε > 0 and choose R > 0 such that supp f ∈ B R and Γ\Γ R P Γ t |f |(x)M Γ (x) dx < ε.Then, (4.3) Considering the integrand in (4.3), for all x ∈ Γ R we have (4.4) Since P t f → f uniformly as t → 0 + , for t > 0 small enough we get On the other hand, where K := supp f .We have r := dist(K, Γ c ) > 0, so for t small enough, see, e.g., [31].By (4.3) and (4.4) we get, as required, Recall that The following characterization of hypercontractivity of P Γ t is crucial for the proof of (1.3).
Lemma 4.2.Let q ∈ [1, ∞).We have (4.5) for all t > 0 and all non-negative functions f on R d if and only if Proof.Assume (4.6).Let f 0. With the notation F := f /M Γ we get Let c be the supremum in (4.6).By Minkowski integral inequality, For t > 0, by scaling we get (4.5) as follows: Conversely, assume (4.5).Let y ∈ Γ.Let g n 0, n ∈ N, be functions in C ∞ c (Γ) approximating δ y , the Dirac measure at y, as follows: and lim for every function h continuous near y.For f n := g n /M Γ , f n 1,M Γ = g n 1 = 1 and as n → ∞.By (4.5) and Fatou's lemma, Since y ∈ Γ was arbitrary, we obtain (4.6).
Here is a refinement of Lemma 4.2.
For t > 0 we let By (4.8), Since f has compact support, for sufficiently large t > 0 we have where in the last line we used scaling (3.6) of ρ.By Theorem 3.7, we can make it arbitrary small by choosing small ω, and (4.7) follows in this case.
Step 2. Case q = 1.For t > 0 we let Applying (4.8), we get Since f has compact support, for sufficiently large t.In view of (2.10) and (3.6), by changing variables t −1/α x → x and t −1/α y → y we obtain By Corollary 3.5, we can make it arbitrary small by choosing small ω, so (4.7) is true.
By Hölder inequality we get that, as t → ∞, 1,M Γ → 0, since both factors converge to zero as t → ∞ by Step 1. and Step 2.
Finally, consider arbitrary and f R is compactly supported.Furthermore, due to our assumptions, as R → ∞.Let ε > 0 and choose R > 0 so large that f − f R 1,M Γ < ε.