Large-time behavior of two families of operators related to the fractional Laplacian on certain Riemannian manifolds

This note is concerned with two families of operators related to the fractional Laplacian, the first arising from the Caffarelli-Silvestre extension problem and the second from the fractional heat equation. They both include the Poisson semigroup. We show that on a complete, connected, and non-compact Riemannian manifold of non-negative Ricci curvature, in both cases, the solution with $L^1$ initial data behaves asymptotically as the mass times the fundamental solution. Similar long-time convergence results remain valid on more general manifolds satisfying the Li-Yau two-sided estimate of the heat kernel. The situation changes drastically on hyperbolic space, and more generally on rank one non-compact symmetric spaces: we show that for the Poisson semigroup, the convergence to the Poisson kernel fails -but remains true under the additional assumption of radial initial data.

Let M be a complete, non-compact Riemannian manifold and ∆ be its Laplace-Beltrami operator.It is well understood that the long time behavior of solutions to the heat equation is strongly related to the global geometry of M. This applies also to the heat kernel h t x, y , that is, the minimal positive fundamental solution of the heat equation or, equivalently, the integral kernel of the heat semigroup exp (t∆) (see for instance [23]).
The connection between the long time behavior of the solution u(t, x) of (1) for initial data f ∈ L 1 (M) with respect to the Riemannian measure µ on M and that of the heat kernel h t (x, y) has recently been the subject of extensive studies, see for example [5,25,39] or see [1,2,5,29] for other Laplacians or settings.Denote by M = M f (x) dµ(x) the mass of the initial data.In the case when M = R n with the euclidean metric, the heat kernel is given by and the solution to (1) satisfies as t → ∞ (2) u(t, .) − M h t ( ., x 0 ) L 1 (R n ) −→ 0 and (3) By interpolation, a similar convergence holds with respect to any L p norm when 1 < p < ∞: where p ′ is the Hölder conjugate of p.
Note that (2) holds for any choice of x 0 , which means that in the long run the solution u (t, x) and the heat kernel h t (x, x 0 ) "forget" about the initial function f , resp.initial point x 0 .We refer to a recent survey [37] for more details about this property in the euclidean setting.It is worth mentioning that this longtime asymptotic convergence result corresponds to the Central Limit Theorem of probability in the PDE setting.
On manifolds of non-negative Ricci curvature, the results (2) and (3) were generalized in [25].The situation is drastically different in hyperbolic spaces.It was shown by Vázquez [39] that (2) fails for general absolutely integrable initial data f but is still true if f is spherically symmetric around x 0 .Similar results were obtained in [5] in a more general setting of symmetric spaces of non-compact type by using tools of harmonic analysis.Note that these spaces have nonpositive sectional curvature.Recall that in hyperbolic spaces Brownian motion X t tends to escape to ∞ along geodesics, which means that it "remembers" at least the direction of the starting point x 0 .In [25], it was also shown that (3) fails on connected sums R n #R n , n ≥ 3.
The fractional Laplacian is the operator (−∆) σ , σ ∈ (0, 1), defined as the spectral σ-th power of the Laplace-Beltrami operator, with Dom(−∆) ⊂ Dom((−∆) σ ).It is connected to anomalous diffusion, which accounts for much of the interest in modeling with fractional equations (quasi-geostrophic flows, turbulence and water waves, molecular dynamics, and relativistic quantum mechanics of stars).It also has various applications in probability and finance.On certain "good" non-compact Riemannian manifolds M (e.g.Cartan-Hadamard manifolds or manifolds with non-negative Ricci curvature, see [7,Proposition 3.3]) one can obtain the fractional Laplacian through a Dirichlet-to-Neumann map extension problem introduced by Caffarelli and Silvestre [12], as well as a Poisson formula and a fundamental solution, see the work of Stinga and Torrea [35].More precisely, let H σ (M) denote the usual Sobolev space on M. Then for any given f ∈ H σ (M) there exists a unique solution of the extension problem with v(0, x) = f (x) and the fractional Laplacian can be recovered through Notice that equation ( 4) gives rise to the first family of operators this note is concerned about.The second family of operators we consider arises from the fractional heat equation These two families of operators have drawn much attention, see for instance [2,7,9,10,11,14,15,33,38] and the references therein.It is worth mentioning that both families of operators include the Poisson semigroup (for σ = 1/2 and α = 1 respectively).
The aim of this paper is to study the long time behavior of these two families of operators on certain Riemannian manifolds, for absolutely integrable initial data.More precisely, we treat the case of non-negative Ricci curvature and generalizations of these, that is, doubling volume, complete, non-compact manifolds with double-sided heat kernel estimates of the Li-Yau type, and show that the results are essentially euclidean, in the sense of convergence proved in [38].This is no longer the case in negatively curved manifolds.More precisely, we consider the Poisson semigroup on real hyperbolic space, and more generally on rank one symmetric spaces of non-compact type and show that in this case, the long time results are vastly different: the aforementioned euclidean-type results fail even for compactly supported initial data.
Notice that for all manifolds considered here, these two families of operators admit integral kernels which are actually probability measures.
Our main result is the following.

Remark. By interpolation between
The situation changes drastically in real hyperbolic space, and generally, in rank one non-compact symmetric spaces.More precisely, the Poisson semigroup fails to satisfy these convergences.Theorem 2. Let X be a rank one non-compact symmetric space.Assume that f ∈ L 1 (X) and let M = X f denote its mass.If e −t √ −∆ is the Poisson semigroup and p t the Poisson kernel, then, in general

LARGE-TIME BEHAVIOR OF OPERATORS RELATED TO THE FRACTIONAL LAPLACIAN 4
However, the convergence holds if f is radial.
To the best of our knowledge, even though the fractional Laplacian and related diffusion equations have been the center of many studies, the above asymptotics have been established only on euclidean space, [38].We use different methods to prove our results.For the case of manifolds with non-negative Ricci curvature, we use subordination formulas to use information for the heat kernel, such as double-sided bounds and a quantitative Hölder continuity estimate, which is a key component to our proof.Notice that one could have pursued gradient estimates, but the approach used here is more general.For the case of rank one non-compact symmetric spaces, we rely on tools of harmonic analysis available in this setting and large-time asymptotics of the Poisson kernel.An essential idea of the proof in the rank one case is to describe the critical region of the Poisson kernel, since this kernel is a probability measure.

P
From now on, M denotes a complete, connected, non-compact Riemannian manifold of dimension n ≥ 2. Let µ be the Riemannian measure on M. Let d(x, y) be the geodesic distance between two points x, y ∈ M, and V(x, r) = µ (B (x, r)) be the Riemannian volume of the geodesic ball B(x, r) of radius r centered at x ∈ M.
Throughout the paper we follow the convention that C, C 1 , c, c 1 ... denote positive constants.These constants may depend on M but do not depend on the variables x, y, t.Moreover, the notation A B between two positive expressions means that A ≤ CB, and A ≍ B means cB ≤ A ≤ CB.Also, A(t) ∼ B(t) means that A(t)/B(t) → 1 as t → +∞.
We say that M satisfies the volume doubling property if, for all x ∈ M and r > 0, we have (8) V(x, 2r) ≤ C V(x, r).
It follows from ( 8) that there exist some positive constants ν, ν ′ > 0 such that ν for all x ∈ M and 0 < r ≤ R (see for instance [23,Section 15.6]).Moreover, (9) implies that, for all x, y ∈ M and r > 0, Notice that hyperbolic spaces as well as all non-compact symmetric spaces fail to be doubling (they are locally doubling, though).More precisely, in the rank one case, if n = dim X and ρ 2 > 0 is the bottom of the spectrum, it holds The integral kernel h t x, y of the heat semigroup exp(t∆) is the smallest positive fundamental solution to the heat equation (1).It is known that h t x, y is smooth in t, x, y , symmetric in x, y, and satisfies the semigroup identity (see for instance [23], [36]).Besides, for all y ∈ M and t > 0 The manifold M is called stochastically complete if for all y ∈ M and t > 0 It is known that if M is geodesically complete and, for some x 0 ∈ M and all large enough r, V (x 0 , r) ≤ e Cr 2 , then M is stochastically complete.In particular, the volume doubling property (9) and the volume bounds (10) for rank one symmetric spaces imply that all manifolds considered in this paper are stochastically complete.
When the Ricci curvature of M is non-negative, the following two-sided estimates of the heat kernel were proved by Li and Yau [27]: Apart from manifolds with non-negative Ricci curvature, the above-described manifolds cover many other examples.Let us recall that, on a complete Riemannian manifold, the following three properties are equivalent: • The two-sided estimate (11) of the heat kernel; • The uniform parabolic Harnack inequality: u(t, x), (12) where u(t, x) is a non-negative solution of the heat equation ∂ t u = ∆u in a cylinder (0, T) × B(x, r) with x ∈ M, r > 0 and T = r 2 .
• The conjunction of the volume doubling property (8) and the Poincaré inequality: for all x ∈ M, t > 0, and bounded Lipschitz functions f in B(x, r).Here, f B is the mean of f over B(x, r).
See, for instance, [19,22,30,31] for more details.Manifolds satisfying these equivalent conditions include complete manifolds with non-negative Ricci curvature, connected Lie groups with polynomial volume growth, co-compact covering manifolds whose deck transformation has polynomial growth, and many others.We refer to [32, pp.417-418] for a list of examples.

Corollary 3. Let M be a geodesically complete non-compact manifold that satisfies one of the following equivalent conditions:
• the two-sided estimate (11) of the heat kernel; • the uniform parabolic Harnack inequality (12); • the conjunction of the volume doubling property (8) and the Poincaré inequality (13).
Then the conclusions of Theorem 1 are true.
Let us now recall a consequence of the two-sided estimate (11), which will be essential for this note, see for instance [31,Theorem 5.4.12]:there exists 0 < θ ≤ 1 such that for all t > 0, x, y, z ∈ M, and d(y, z) ≤ √ t, Notice that a pointwise gradient estimate of the form is the limit case θ = 1 of ( 14), but requires more structure on the manifold, such as non-negative Ricci curvature, see for instance [27].In this sense, the Hölder continuity estimate ( 14) is more general.
Last, recall that on spaces of essentially negative curvature the above estimates of the heat kernel typically fail: for example, these are hyperbolic spaces [17], non-compact symmetric spaces [3,4], asymptotically hyperbolic manifolds [13], and fractal-like manifolds [6].
We finally recall the definition of real hyperbolic space and more generally, of rank one symmetric spaces, as well as some indispensable tools from Fourier analysis on these spaces.

Rank one non-compact symmetric spaces.
Our main reference for this subsection is [26].Let G be a connected, noncompact semisimple Lie group with finite center.Let K be a maximal compact subgroup of G and X = G/K be the corresponding symmetric space.We consider a Cartan decomposition g = k ⊕ p of the Lie algebra of G. Fix a maximal abelian subspace a of p and consider the decomposition g = n ⊕ a ⊕ k.If a R, then we say that the symmetric space X has rank one.
From now on, we assume that rankX = 1.In this case, after fixing some order on the non-zero restricted roots, there are at most two roots which are positive with respect to this order, which we denote by α and 2α.Let m α and m 2α be the multiplicities of these roots, and define the number ρ by ρ := (m α + 2m 2α )/2.A rank one non-compact symmetric space is one of the following: the real, the complex, the quaternionic hyperbolic space and the octonionic hyperbolic plane.We have n = dim X = m α + m 2α + 1 and ρ equal to (n − 1)/2, n/2, n/2 + 1 and 11, respectively.
The group G admits the following decompositions, Let H 0 be the unique element of a with the property that α, H 0 = 1 and normalize the Killing form on g such that |H 0 | = 1.Denote by τ(g) the real number such that To simplify the notation, we often identify the Lie subgroup A = exp a with the real line R using the map τ → exp(τ H 0 ).Notice that we may also identify In the Cartan decomposition, the Haar measure on G writes with density (16) δ(r) = (sinh r) m α (sinh 2r) m 2α e 2ρ r .
Here K is equipped with its normalized Haar measure and "const" is a positive normalizing constant, so that for right-K invariant functions, we have Notice that viewed on G/K, r is the distance of gK to the origin o = {K}.
Finally, we describe Fourier analysis on rank one non-compact symmetric spaces.For continuous compactly supported functions, the Helgason-Fourier transform is defined by Here, M denotes the centralizer of exp a in K. Let us also define the spherical transform of continuous compactly supported and radial functions by where ϕ λ is the elementary spherical function of index λ ∈ C. The functions ϕ λ are normalized eigenfunctions of ∆, that is, They are also radial and have the property that ϕ λ = ϕ −λ .Finally, let us recall that in the case of radial Schwartz functions, we have

F L
This section deals with two families of operators related to the fractional Laplacian but both subordinated via integral representations to the heat semigroup.Interestingly, the Poisson semigroup belongs to both, for special values of their parameters.
In recent years there has been intensive research on various kinds of fractional order operators.Being nonlocal objects, local PDE techniques to treat nonlinear problems for the fractional operators do not apply.To overcome this difficulty, in the euclidean case, Caffarelli and Silvestre [12] studied the extension problem associated with the Laplacian and realized the fractional power as the map taking Dirichlet data to Neumann data.In [35] Stinga and Torrea related the extension problem for the fractional Laplacian to the heat semigroup, providing a subordination formula and conditions for the existence of an integral kernel.On certain classes of non-compact manifolds, which include symmetric spaces of non-compact type, the extension problem has been studied by Banica, González and Sáez [7].Interestingly, in the non-compact setting one needs to have a precise control of the behavior of the metric at infinity and geometry plays a crucial role.
To begin with, using the spectral theorem, one can define fractional powers of the Laplacian via the heat semigroup, see [40, (5), p.260].Then, the relation between the fractional Laplacian and the extension problem (4) is the following.
Theorem 4. [35].Let σ ∈ (0, 1).Then for f ∈ Dom((−∆) σ ), a solution to the extension problem is given by Moreover, the fractional Laplacian on M can be recovered through From a probabilistic point of view, the extension problem corresponds to the property that all symmetric stable processes can be obtained as traces of degenerate Bessel diffusion processes, see [34].
However, despite the subordination of {T σ t } t>0 to the heat semigroup, passing from the heat kernel to a Poisson kernel is a non-trivial issue in the case of non-compact manifolds since one needs to control the behavior at infinity.By [7,35] and under the description of the heat semigroup given in the present paper, one needs to check whether, given x 0 , there exists a constant C x 0 and ǫ > 0 such that the heat kernel on the manifold M satisfies (20) h t ( . Thus the problem of an integral kernel for the operator T σ t reduces to obtaining suitable upper bounds for the heat kernel (from where one may derive information for its time derivatives as well, see for instance [24]).Inequality (20) is true on real hyperbolic space as well as on non-compact symmetric spaces of arbitrary rank.It is also true on manifolds satisfying a volume doubling condition and the local Poincaré inequality [7, Proposition 3.3], such as manifolds of non-negative Ricci curvature, or more generally the manifolds considered in Corollary 3.Then, the function T σ t f in ( 19) is given by where the integral kernels are given by ( 21) We now pass to the fractional heat equation, again for the manifolds mentioned above.Let α ∈ (0, 2) and take η α t to be the inverse Laplace transform of the function exp{−t (.) α/2 }.The fractional Laplacian (−∆) α/2 is the infinitesimal generator of a standard isotropic α-stable Lévy motion X α t .This process is a Lévy process, which can be viewed as the long-time scaling limit of a random walk with power law jumps ([28, Theorem 6.17]).Via subordination to the heat semigroup, we may write [40, (7), p.260].Then, for the manifolds considered in this note (those of Corollary 3 and rank one non-compact symmetric spaces), for any reasonable f we have that where the kernels are given by ( 22) The family {W α t } t>0 is a C 0 -semigroup, [40].For every L 1 (M) (for an optimal class of initial data on euclidean space, see [8]), the function w(t, .) = W α t f solves the initial value problem In the case of hyperbolic spaces, the α-stable process with transition densities P α t (x, y) was first defined by Getoor [20] (see also [21] for general non-compact symmetric spaces).
Finally, observe that owing to the subordination formulas and the properties of the heat kernel, both kernels Q σ t , P α t are non-negative and symmetric.They are also probability measures: to see this, recall first that the manifolds considered here are stochastically complete.Then, by a Fubini argument, the claim follows for Q σ t by the definition of Gamma function, while for P α t from the fact that ∞ 0 η α t (u) du = 1, [40, Eq (14), p.262].

A L : -R
Throughout this section, we consider M to be a manifold of non-negative Ricci curvature -or more generally, we consider the Riemannian manifolds of Corollary 3-and study the long-time properties of the families of operators considered in Section 3.An essential idea of the proof is the use of the Hölder continuity of the heat kernel, owing to subordination formulas of their kernels.

Remark. By convexity between (24) and (25), we obtain for any
Let us stress that the conditions (P1)-(P4) are not necessarily optimal.The class P γ and the above conditions will simply allow us to avoid repeating several steps when proving convergence to the fundamental solution for the two families of operators {T σ t } t>0 , σ ∈ (0, 1) and {W α t } t>0 , α ∈ (0, 2).To prove Theorem 5, it is sufficient to consider the action of the operator K γ t on continuous compactly supported functions f .Then, owing to the properties (P1)-(P4), one can show that the desired convergence remains valid for the whole class of L 1 (M) initial data by using a density argument, see for instance the arguments in [5, pp.17-18] or [25, pp.11-12].Therefore, it suffices to prove the following result, which also gives a rate of convergence for continuous compactly supported initial data.Proposition 6. Fix γ > 0 and a basepoint x 0 ∈ M. Let f ∈ C c (B(x 0 , ξ)) for some ξ > 0 and set M = M f dµ.Then, for all t > t 0 (ξ, γ), it holds Proof.First of all, observe that the operator K γ t is bounded on L 1 (M), due to (P1) and (P2).Write Therefore, by ( 26), (P1), (P4) and by integrating in x over M, we obtain for t large enough such that d(x 0 , y) ≤ ξ ≤ t 1/γ , where in the last step we used (P2).This proves the desired L 1 (M) result.
We now turn to the proof of the sup norm asymptotics.For t large enough so that d(x 0 , y) ≤ ξ ≤ t 1/γ , by (P3) we get ψ γ t (x, x 0 ) ≍ ψ γ t (x, y) for all x ∈ M. Therefore, by ( 26), (P1) and (P4) we get where in the last step we used (P2).The claim follows.

Asymptotics for solutions to the Caffarelli-Silvestre extension problem.
In this subsection, we study the large-time asymptotic behavior of the family of operators {T σ t } t>0 .To this end, we first give some indispensable estimates concerning the kernels Q σ t and use them to prove that they belong to the class P 1 for all σ ∈ (0, 1).Lemma 7.For all x, y ∈ M, all t > 0, and all constants c > 0 and κ > 1, it holds , t + d(x, y)) where the implied constants depend only on c, κ and M.
In addition, for fixed r > 0, we have where the implied constant depends in addition on r, N. Proof.Set Then, for C := min{c, 1/4}, we have where for the last two inequalities we used the volume doubling property (9).The lower bound follows similarly.
Last, as far as asymptotics are concerned, observe that for all C > 0, we have The additional exponential term on the right-hand side allows for the fast decay in t as t → +∞, while for the remaining integral we may conclude as before.
Corollary 8.There is a constant C ≥ 1 such that if d(y, z) ≤ t, then Proof.Observe first that for any γ > 0, if d(y, z) ≤ t 1/γ , then the triangle inequality implies Recall next the subordination formula (21) for Q σ t and the double-sided heat kernel estimates (11).Then, the claim is a simple consequence of Lemma 7 for κ = σ + 1, which yields ( 28) the doubling volume condition (9) and the inequality (27) for γ = 1.
We are now in a position to prove that (Q σ t ) ∈ P 1 for all σ ∈ (0, 1).Indeed, as already mentioned, due to the subordination formula (21), the kernel Q σ t satisfies (P1) as well as the first assertion of (P2).The second assertion of (P2) follows immediately from (28).(P3) follows by Corollary 8.

Write
where Let us first start with I 2 .Since u ≥ ξ 2 > d 2 (x 0 , y), we can use the Hölder estimate (14) for the heat kernel.Therefore, applying Lemma 7 for κ = 1 + σ + θ 2 , we get , where in the last step we used the bounds (28).It remains to treat I 1 .For this, observe that by the second part of Lemma 7 for κ = 1 + σ and by (28), we have for all x, y ∈ M and for all t large enough Therefore, ). Taking t large enough so that d(x 0 , y) < ξ < t and making use of Corollary 8 we finally estimate for all x, y, x 0 ∈ M such that d(x 0 , y) < ξ and t large enough.This proves (P4) for θ 1 = θ, where θ is the constant from the heat kernel Hölder inequality (14).

Asymptotics for solutions to the fractional heat equation.
As in the previous section, we start by proving upper and lower bounds for the kernel P α t , by using the subordination formula (22) and the double-sided estimates of the heat kernel (11).
It is well-known that the subordinator η α t cannot be written explicitly, except for the case α = 1.Let us recall, however, that where [21].From these, we get that (31) Remark.For κ = 0, the estimates of Lemma 9 amount to bounds for the kernel of the fractional heat semigroup (see the next corollary), which were already known in the literature: we refer to [33,Eq (3.5)]where such bounds are actually established on a more general setting.It is mostly the case κ > 0 that will be of interest to us.Corollary 10.There is a constant C ≥ 1 such that if d(y, z) ≤ t 1/α , then Proof.Recall first the subordination formula (22) and the double-sided heat kernel estimates (11).Then, by Lemma 9 for κ = 0 we get for all x, y ∈ M and all t > 0 that (34) Then the claim follows as in Corollary 8.
Finally, we prove (P4).Fix a basepoint x 0 ∈ M and consider y ∈ M such that d(x 0 , y) < ξ.Write where Let us first start with I 2 .Since u ≥ ξ 2 > d 2 (x 0 , y), we can use the Hölder estimate (14) for the heat kernel.Therefore, applying Lemma 9 for κ = θ/2, we get , where in the last step we used the bounds (34).It remains to treat I 1 .For this, observe that by the second part of Lemma 9 for κ = 0 and by (34), we have for all x, y ∈ M and for all t large enough, ξ 2 0 h u (x, y) η α t (u) du t −N P α t (x, y), for all N > 0.
Therefore, I 1 t −N (P α t (x, y) + P α t (x, x 0 )).Taking t large enough so that d(x 0 , y) < ξ < t 1/α and making use of Corollary 10 we finally estimate Altogether, we get for all t large enough and all x, y, x 0 ∈ M such that d(x 0 , y) < ξ.This proves (P4) for θ α = θ/α, where θ is the constant from the heat kernel Hölder inequality (14).

4.3.1.
Final remarks on the rate of convergence.The rate of convergence for continuous and compactly supported initial data is optimal, both for the extension problem (O(t −θ ) from ( 29)) and for the fractional heat equation −θ/α ) from ( 35)), in the following sense: on euclidean space, the heat kernel Hölder inequality (14) holds for θ = 1, and it is known that, concerning the fractional heat equation, the optimal rate for convergence in the L 1 norm for compactly supported initial data is O(t −1/α ), [38,Theorem 3.2].
Moreover, instead of using (P4) for P α t , one could pursue using a Hölder continuity estimate for P α t , that is, that there is a constant Θ > 0 such that when d(y, z) ≤ t 1/α .For a proof, see [14,Theorem 4.14] for stable-like processes on a rather general setting (alternatively, one can modify for all α ∈ (0, 2) the result of [18,Theorem 4] for the Poisson operator, using the semigroup property, the fact that the fractional heat kernel is a probability measure and Corollary 10).However, for compactly supported initial data, this would imply L 1 (M) convergence at speed O(t −Θ/α ).In this sense, our approach gives more information on the rate of convergence for this class of data.Notice moreover that as far as the extension problem is concerned, for σ 1/2, the operators {T σ t } t>0 do not form a semigroup.Indeed, observe that where K σ (•) is the modified Bessel function of the second kind and index σ (recall that for σ = 1/2, we have K 1/2 (x) = π 2x e −x , so the right hand side above becomes e −tλ ).
Finally, following some ideas from the euclidean setting in [37,38], let us show how one can prescribe any rate of convergence to solutions of the fractional heat equation by choosing appropriate initial data (the proof works also for T σ t , with obvious modifications).More precisely, we shall show that given any decreasing and positive function φ(t) such that φ(t) → 0 as t → +∞, there is a solution w with mass M = 1 satisfying (36) w(t, .) − P α t ( ., x 0 ) V( ., t 1/α ) L ∞ (M) kφ(t k ), for a sequence of times t k → +∞ that can be chosen.
To prove (36), fix a basepoint x 0 ∈ M. Let (m k ) k≥1 be a nonnegative summable sequence with ∞ k=1 m k = ǫ < 1, and consider initial data Observe that the total mass is 1.Also, the points x k ∈ M, k ≥ 1, where the weighted Dirac measures are located are such that r k := d(x 0 , x k ) → +∞.In this case, the action of W α t yields the following solution of the fractional heat equation, Therefore at x = x 0 we have for some constant c 1 > 0 due to (34).Now, again due to (34), there is a constant for some C > 1 owing to (9).Consider now φ(t) ց 0 as t → +∞, and choose iteratively t k and x k as follows: given choices for the steps 1, 2, ..., k − 1, pick t k to be much larger than t k−1 and such that φ(t k ) ≤ m k /(2k).This is possible since φ(t) decreases to zero.Choose now d(x k , x 0 ) = r k so large that 1 + ), which proves the claim.

T P -
This section deals with the Poisson semigroup convergence on rank one noncompact symmetric spaces.Our aim is to show that in this case, non-euclidean phenomena occur, namely, the convergence results of the previous sections fail.More precisely, our aim is to prove Theorem 2.
The Poisson semigroup e −t √ −∆ has been studied in various settings, including hyperbolic space (and more generally, non-compact symmetric spaces), see for instance [3,16] and the references therein.Information for its kernel p t can be deduced by its subordination to the heat kernel.Therefore, in the rank one case, by well-known properties of the heat kernel, the Poisson kernel p t is a radial positive function, and for f ∈ L 1 (X), we may write Lemma 13.Let y = y 0 K be in a bounded region of X.Then, for every x = gK in the critical region Ω t , Here, k is the left component of g in the Cartan decomposition and exp(τ(k −1 y 0 )H 0 ) is the middle component of k −1 y 0 in the Iwasawa decomposition.
Proof.The arguments follow closely those of [5,Lemma 3.8], but we include the proof for the sake of completeness.Write x = gK, where g = k (exp rH 0 ) k ′ in the Cartan decomposition.Then d(gK, o) = r.Consider the Iwasawa decomposition k The next lemma is crucial for our proof.
Proof.Write r = d(x, o), s = d(x, y) and let d(y, o) < ξ, for some ξ > 0. By the triangle inequality, we have |r − s| ≤ ξ and since x is in the critical region, we have t 2−ǫ ≤ r ≤ t 2+ǫ .In addition, for t large enough, we have having proven the desired convergence in the L 1 norm for radial C c (X) functions, one may conclude for the whole class of radial L 1 (X) initial data by a density argument, see [5].
To begin with, let x Ω t .Let also ξ > 0 be a constant such that the compact support of f is contained in B(o, ξ).Then we have Notice that x ∈ X Ω t and y ∈ B(o, ξ) imply x ∈ X Ω t,y , where working as in Proposition 12 for X Ω t,y p t (d(x, y)) dµ(x).Thus, This proves the desired convergence outside the critical region for all f ∈ C c (X).
We now turn to x ∈ Ω t .By Lemma 14, the right-K-invariance of τ(k −1 .)and f , and the definition (17)  Notice that f (± iρ, kM) = H f (± iρ) = M when f is radial, see Subsection 2.1.Therefore in this case, we deduce the desired convergence by integrating (41) over the critical region.
On the other hand, using the Cartan decomposition (15) we have f (y 0 ) e 2ρ τ(k −1 y 0 ) − 1 dy 0 dk as t → +∞.The last integral is not constantly zero when f is not radial.For example, consider f to be a Dirac measure supported on some point y = y 0 K other than the origin, thus for y 0 K.In other words, the solution now coincides with p t ( ., y) and the mass is equal to 1.In this case, however, the last integral is equal to K e 2ρ τ(k −1 y 0 ) − 1 dk, thus does not vanish identically.