Heat kernel estimates for the fractional Laplacian with Dirichlet conditions
aa r X i v : . [ m a t h . P R ] N ov The Annals of Probability (cid:13)
Institute of Mathematical Statistics, 2010
HEAT KERNEL ESTIMATES FOR THE FRACTIONALLAPLACIAN WITH DIRICHLET CONDITIONS
By Krzysztof Bogdan , Tomasz Grzywny and Micha l Ryznar Wroc law University of Technology
We give sharp estimates for the heat kernel of the fractionalLaplacian with Dirichlet condition for a general class of domains in-cluding Lipschitz domains.
1. Introduction.
Explicit sharp estimates for the Green function of theLaplacian in C , domains were completed in 1986 by Zhao [43]. Sharpestimates of the Green function of Lipschitz domains were given in 2000by Bogdan [6]. Explicit qualitatively sharp estimates for the classical heatkernel in C , domains were established in 2002 by Zhang [42]. Qualitativelysharp heat kernel estimates in Lipschitz domains were given in 2003 byVaropulous [41]. The development of the boundary potential theory of thefractional Laplacian follows a parallel path. Green function estimates wereobtained in 1997 and 1998 by Kulczycki [29] and Chen and Song [21] for C , domains, and in 2002 by Jakubowski for Lipschitz domains [28]. In2008 Chen, Kim and Song [19] gave sharp explicit estimates for the heatkernel p D ( t, x, y ) of the fractional Laplacian on C , domains D . The maincontribution of the present paper is the following result. Theorem 1. If D is κ -fat, then there is C = C ( α, D ) such that C − P x ( τ D > t ) P y ( τ D > t ) ≤ p D ( t, x, y ) p ( t, x, y ) ≤ CP x ( τ D > t ) P y ( τ D > t )(1) for < t ≤ and x, y ∈ D . Received July 2009; revised January 2010. Supported in part by Grant MNiSW N N201 397137. Supported in part by Grant MNiSW N N201 373136.
AMS 2000 subject classifications.
Primary 60J35, 60J50; secondary 60J75, 31B25.
Key words and phrases.
Fractional Laplacian, Dirichlet problem, heat kernel estimate,Lipschitz domain, boundary Harnack principle.
This is an electronic reprint of the original article published by theInstitute of Mathematical Statistics in
The Annals of Probability ,2010, Vol. 38, No. 5, 1901–1923. This reprint differs from the original inpagination and typographic detail. 1
K. BOGDAN, T. GRZYWNY AND M. RYZNAR
Here p ( t, x, y ) is the heat kernel of the fractional Laplacian on R d , and P x ( τ D > t ) = Z R d p D ( t, x, y ) dy defines the survival probability of the corresponding isotropic α -stable L´evyprocess in D . The result applies also to unbounded domains, in particular, todomains above the graph of a Lipschitz function, where we can take arbitrary t >
0. In fact, (1) holds with C = C ( α, d, κ ) under the mere condition that D is ( κ, t /α )-fat at x and at y ; see Sections 3 and 4 for definitions and results.For exterior domains we have a result free from local geometric assumptions: Corollary 1. If diam( D c ) < ∞ , then (1) holds with C = C ( α, d ) forall t > diam( D c ) α and x, y ∈ D . For exterior domains of class C , a more explicit estimate is given inTheorem 3 below. We also like to note that a useful variant of Theorem 1is given in Theorem 2.Expression (1) is motivated by these applications of the semigroup prop-erty of p D : p D (2 t, x, y ) = Z R d p D ( t, x, z ) p D ( t, z, y ) dz ≤ P x ( τ D > t ) c ( t ) , where c ( t ) = sup z,y ∈ R d p ( t, z, y ) ≥ sup z,y ∈ R d p D ( t, z, y ) [see (12)], and p D (3 t, x, y ) = Z Z p D ( t, x, z ) p D ( t, z, w ) p D ( t, w, y ) dw dz ≤ P x ( τ D > t ) c ( t ) P y ( τ D > t ) . The latter inequality is quite satisfactory for x = y , because c ( t ) = p ( t, x, x ).Off-diagonal ( x, y ) in (1) require, however, a deeper analysis. Our proof of(1) is based on the boundary Harnack principle (BHP) [14] (see also earlier[40]), a version of the Ikeda–Watanabe [27] formula (18), scaling (14) andcomparability of p with its L´evy measure (5); see (28). Counterparts of theseare important in view of possible generalizations.In what follows (1) and analogous sharp estimates will be written as p D ( t, x, y ) C ≈ P x ( τ D > t ) p ( t, x, y ) P y ( τ D > t ) , meaning that either ratio of the sides is bounded by a number C ∈ (0 , ∞ ),and C does not depend on the variables shown (here: t , x , y ). We will skip C from notation if unimportant for our goals.Let δ D ( x ) = dist( x, D c ). As mentioned above, domains D of class C , enjoy the following sharp and explicit estimate of Chen, Kim and Song [19]: p D ( t, x, y ) p ( t, x, y ) ≈ (cid:18) ∧ δ α/ D ( x ) t / (cid:19)(cid:18) ∧ δ α/ D ( y ) t / (cid:19) , < t ≤ , x, y ∈ R d . (2) EAT KERNEL ESTIMATES We note that (2) agrees with (1) because by [10], Corollary 1, P x ( τ D > t ) ≈ ∧ δ α/ D ( x ) t / for 0 < t ≤ , x, y ∈ R d . In fact, starting with (1), we are able to recover and strengthen (2), with asimpler proof; see Example 5 and Proposition 1 below. We note that (1) wasconjectured in [10] based on the cases of C , domains [19] and circular cones[10]. We should also mention that the Gaussian estimates of Varopoulous[41] have a shape similar to (1), in particular, they involve the survivalprobability. Thus, the present paper builds on the evidence accumulated in[19, 41] and [10]. We also note that the upper bound in (2) was proved in2006 by Siudeja for semibounded convex domains ([39], Theorem 1.6), andstated for general convex domains in [39], Remark 1.7. Some of our presenttechniques were inspired by [32], Theorem 4.2, of Kulczycki and Siudeja, [2],Proposition 2.9, of Ba˜nuelos and Kulczycki, and [1], Section 4, of Bogdanand Ba˜nuelos.It is a consequence of Lemma 1 below that we can apply BHP [14, 40] toconveniently estimate P x ( τ D >
1) by some kernel functions of D , namely, bythe Martin kernel with the pole at infinity or the expected survival time [weuse scaling to estimate P x ( τ D > t ) for general t > V : p V ( t, x, y ) p ( t, x, y ) ≈ (1 ∧ δ V ( x ) /t /α ) α/ (1 ∧ | x | /t /α ) α/ − β (1 ∧ δ V ( y ) /t /α ) α/ (1 ∧ | y | /t /α ) α/ − β . (3)Here β ∈ [0 , α ) is a characteristic of the cone, and all t > x, y ∈ R d areallowed. We should add to (1), (2) and (3) that [4, 16] p t ( x ) c ≈ t | x | d + α ∧ t − d/α , t > , x ∈ R d . (4)Here c = c ( α, d ), meaning that c ∈ (0 , ∞ ) may be so chosen to depend onlyon d and α . We like to note that the estimates for general κ -fat domainscannot be as explicit as those for C , domains. In particular, the decay rate β at the vertex of a cone delicately depends on the aperture of the cone;see [1, 10, 35] (see also [6]). Nevertheless, Lipschitz domains offer a naturalsetting for studying the boundary behavior of the Green function and theheat kernel for both the Brownian motion and the isotropic α -stable L´evyprocesses. This is due to the scaling, the rich range of asymptotic behaviorsdepending on the local geometry of the domain’s boundary, connectionsto the boundary Harnack principle and approximate factorization of theGreen function, and applications in the perturbation theory of generators, K. BOGDAN, T. GRZYWNY AND M. RYZNAR in particular, via the 3G Theorem [1, 6, 7, 26, 43] and 3P Theorem [13].The κ -fat sets are a convenient generalization of Lipschitz domains, withsimilar features. It is noteworthy that (1) is an approximate factorization ofthe heat kernel (see also [6, 14] in this connection).We should add that the C , condition specifies the geometry of a domainonly in bounded scales (see Definition 3). This renders the range of time in(2) restricted to 0 < t ≤
1. In what follows we will also study the probabilityof survival for large times (and unbounded domains). This is straightforwardfor special Lipschitz domains (thus for circular cones), but less so for general κ -fat or C , domains. As an interesting case study we consider domainswith bounded complement (i.e., exterior domains) of class C , . These havedistinctive geometries at infinity and at the boundary, resulting in nontrivialcompletion of (2). We remark that exterior C , domains in dimension d > α have been recently studied in [22], too. We also remark that [25], Theorem 4.4bounds the survival probability of the relativistic process in a half-line, and[31] gives an explicit formula for the transition density of the killed Cauchyprocess ( α = 1) on the half-line. Regarding other recent estimates [3, 17, 20,23, 36] for the transition density and potential kernel of jump-type processes,we need to point out that generally these only concern processes withoutkilling. Killing corresponds to the Dirichlet “boundary” condition (analogousto the negative Schr¨odinger perturbation [8, 12]) and it severely influencesthe asymptotics of the transition density and Green function. Needless tosay, the asymptotics are crucial for solving the Dirichlet problem [24, 25].We like to mention possible applications and further directions of research.The estimate (1) fits well into the technique of Schr¨odinger perturbations of[12], which should produce straightforward consequences. Also, the distri-bution of τ D , given by (18) below, can be estimated by using (1). Further,we conjecture that for certain domains D , lim p D ( t, x, y ) /P x ( τ D > t ) existsas x approaches a boundary point of D . This may lead to representationtheorems for nonnegative parabolic functions of the fractional Laplacian(compare [14], Theorems 2 and 3) and construction of excursion laws. Weneed to remark here that our estimates are inconclusive about the (irregular[14]) boundary points of D , but we conjecture that (1) indeed extends to ∂D . Finally, it seems important to understand the behavior of p D ( t, x, y )for domains which are rather small at a boundary point or at infinity. Inthis connection we refer the interested reader to the recent study of intrinsicultracontractivity by Kwa´snicki [33]; see also [10, 19, 30] and the notion of inaccessibility in [14].Our general references to the boundary potential theory of the fractionalLaplacian are [7] and [14]. We also refer the reader to [9] for a broad non-technical overview of the methods and goals of the theory.The paper is composed as follows. In Section 2 we recall basic facts aboutthe killed isotropic α -stable L´evy processes. In Section 3 we prove Theo-rem 1 and Corollary 1. In Section 4 we state and prove Theorem 2 and give EAT KERNEL ESTIMATES applications to specific domains. In particular, we strengthen (2) and partof the results of [19] (see Proposition 1, Theorem 3 and Corollary 2), andwe discuss exterior C , domains in dimension d = 1 < α .
2. Preliminaries.
In what follows, R d denotes the Euclidean space ofdimension d ≥ dy is the Lebesgue measure on R d , and 0 < α <
2. Ourprimary analytic data are as follows: a nonempty open set D ⊂ R d and theL´evy measure given by density function ν ( y ) = 2 α Γ(( d + α ) / π d/ | Γ( − α/ | | y | − d − α . (5)The coefficient in (5) is such that Z R d [1 − cos( ξ · y )] ν ( y ) dy = | ξ | α , ξ ∈ R d . (6)For (smooth compactly supported) φ ∈ C ∞ c ( R d ), the fractional Laplacian is∆ α/ φ ( x ) = lim ε ↓ Z | y | >ε [ φ ( x + y ) − φ ( x )] ν ( y ) dy, x ∈ R d (see [7, 9] for a broader setup). If r > φ r ( x ) = φ ( rx ), then∆ α/ φ r ( x ) = r α ∆ α/ φ ( rx ) , x ∈ R d . (7)We let p t be the smooth real-valued function on R d with Fourier transform, Z R d p t ( x ) e ix · ξ dx = e − t | ξ | α , t > , ξ ∈ R d . (8)In particular, the maximum of p t is p t (0) = 2 − α π − d/ α − Γ( d/α ) / Γ( d/ t − d/α .According to (6) and the L´evy–Khinchine formula, { p t } is a probabilisticconvolution semigroup with L´evy measure ν ( y ) dy ; see [16, 38] or [9]. Wehave the following scaling property, p t ( x ) = t − d/α p ( t − /α x ) , t > , x ∈ R d , (9)which may be considered a consequence of (8). It is noteworthy that by (4)we have p t ( x ) ≈ p t ( x ) , t > , x ∈ R d . (10)We denote p ( t, x, y ) = p t ( y − x ) , and we have Z ∞ s Z R d p ( u − s, x, z )[ ∂ u φ ( u, z ) + ∆ α/ z φ ( u, z )] dz du = − φ ( s, x ) , (11) K. BOGDAN, T. GRZYWNY AND M. RYZNAR where s ∈ R , x ∈ R d , and φ ∈ C ∞ c ( R × R d ); see, for example, [12], (36).We define the isotropic α -stable L´evy process ( X t , P x ) by stipulating tran-sition probability P t ( x, A ) = Z A p ( t, x, y ) dy, t > , x ∈ R d , A ⊂ R d , initial distribution P x ( X (0) = x ) = 1, and c´adl´ag paths. Thus, P x , E x de-note the distribution and expectation for the process starting at x . We definethe time of the first exit from D , or survival time , τ D = inf { t > X t / ∈ D } , and the time of first hitting D , T D = inf { t > X t ∈ D } . We define, as usual, p D ( t, x, y ) = p ( t, x, y ) − E x [ τ D < t ; p ( t − τ D , X τ D , y )] , t > , x, y ∈ R d . We have that 0 ≤ p D ( t, x, y ) = p D ( t, y, x ) ≤ p ( t, x, y ) , (12)hence, Z p D ( t, x, y ) dy = Z p D ( t, x, y ) dx ≤ . (13)If x ∈ D c is regular for the Dirichlet problem on D [14], that is, P x ( τ D = 0) =1, then p D ( t, x, y ) = 0 and (1) is trivially satisfied. By this remark, if all thepoints of ∂D are regular for D , then we can write x, y ∈ R d in Theorem 1,instead of x, y ∈ D . The remark also applies to Examples 1–8 in Section 4.By the strong Markov property, p D is the transition density of the isotropicstable process killed when leaving D , meaning that we have the followingChapman–Kolmogorov equation, Z R d p D ( s, x, z ) p D ( t, z, y ) dz = p D ( s + t, x, y ) , s, t > , x, y ∈ R d , and for nonnegative or bounded (Borel) functions f : R d → R , Z R d f ( y ) p D ( t, x, y ) dy = E x [ τ D < t ; f ( X t )] , t > , x ∈ R d . For s ∈ R , x ∈ R d , and φ ∈ C ∞ c ( R × D ), we have Z ∞ s Z D p D ( u − s, x, z )[ ∂ u φ ( u, z ) + ∆ α/ z φ ( u, z )] dz du = − φ ( s, x ) , EAT KERNEL ESTIMATES which extends (11) and justifies calling p D the heat kernel of the (Dirich-let) fractional Laplacian on D . It is well known that p D is jointly continu-ous and positive for ( t, x, y ) ∈ (0 , ∞ ) × D × D . We have a scaling property, p rD ( r α t, rx, ry ) = r d p D ( t, x, y ), r >
0, or p D ( t, x, y ) = t − d/α p t − /α D (1 , t − /α x, t − /α y ) , t > , x, y ∈ R d , (14)in agreement with (9) and (7). Thus, P rx ( τ rD > r α t ) = P x ( τ D > t ), or P x ( τ D > t ) = Z R d p D ( t, x, y ) dy = P t − /α x ( τ t − /α D > . (15) Remark 1.
For c > ν = cν , the corresponding heat kernels ˜ p ,˜ p D , probability and expectation ˜ P x , ˜ E x . Clearly, ˜ p D ( t, x, y ) = p D ( ct, x, y ).The Green function of D is defined as G D ( x, y ) = Z ∞ p D ( t, x, y ) dt, (16)and scaling of p D yields the following scaling of G D , G rD ( rx, ry ) = r α − d G D ( x, y ) . (17)A result of Ikeda and Watanabe [27] asserts that for x ∈ D the P x -distribution of ( τ D , X τ D − , X τ D ) restricted to X τ D − = X τ D is given by thedensity function ( s, u, z ) p D ( s, x, u ) ν ( z − u ) . (18)For geometrically nice domains, for example, for the ball, P x ( X τ D − = X τ D ) =1 for x ∈ D [14], and then by (16) and (18) the P x -distribution of X τ D hasthe density function given by the Poisson kernel, P D ( x, z ) = Z D G D ( x, u ) ν ( z − u ) du. (19)For x ∈ R d and r > B ( x , r ) = { x ∈ R d : | x − x | < r } and B c ( x , r ) = { x ∈ R d : | x − x | > r } (open complement of a ball).There is a constant C depending only on d , α and p , such that P U ( x , y ) P U ( x , y ) C ≈ P U ( x , y ) P U ( x , y ) , (20)whenever U ⊂ B ( x , r ) ⊂ R d is open, 0 < p < r > x ∈ R d , x , x ∈ U ∩ B ( x , rp ), and y , y ∈ B ( x , r ) c . This boundary Harnack principle (BHP)follows from [14], Lemma 7 and the proof of Theorem 1, and it is essentiallyan approximate factorization of P U . We encourage the interested reader todirectly verify the estimate in the special case of (22) below. K. BOGDAN, T. GRZYWNY AND M. RYZNAR
The Green function and Poisson kernel of B ( x , r ) are known explicitly: G B ( x ,r ) ( x, v ) = B d,α | x − v | α − d Z w s α/ − ( s + 1) d/ ds, (21) P B ( x ,r ) ( x, y ) = C d,α (cid:20) r − | x − x | | y − x | − r (cid:21) α/ | x − y | d , (22)where B d,α = Γ( d/ / (2 α π d/ [Γ( α/ ), C d,α = Γ( d/ π − − d/ sin( πα/ w = ( r − | x − x | )( r − | v − x | ) / | x − v | , | x − x | < r , | v − x | < r , and | y − x | ≥ r ; see [5, 37]. Thus, P x ( | X τ B (0 , | > R ) = Z | y |≥ R P B (0 , ( x, y ) dy ≈ (1 − | x | ) α/ R α , (23)where x ∈ B (0 ,
1) and R ≥
2. Also, for | x − x | ≤ r we have [8] E x τ B ( x ,r ) ( x ) = 2 − α Γ( d/ α Γ(( d + α ) / α/
2) ( r − | x − x | ) α/ . (24)All the sets and functions considered below are Borelian. Positive means strictly positive . Domain means a nonempty open set (connectedness neednot be assumed in this theory).
3. Factorization.
We consider nonempty open set D ⊂ R d . Definition 1.
Let x ∈ D , r > < κ ≤
1. We say that D is ( κ, r )-fat at x if there is a ball B ( A, κr ) ⊂ D ∩ B ( x, r ). If this is true for every x ∈ D , then we say that D is ( κ, r )-fat. We say that D is κ -fat if there is R > D is ( κ, r )-fat for all r ∈ (0 , R ]. Remark 2.
The ball is 1 / Definition 2.
Given B ( A, κ ) ⊂ D ∩ B ( x, U = D ∩ B ( x, | x − A | + κ/ B = B ( A, κ/ ⊂ U and B = B ( A ′ , κ/
6) such that B ( A ′ , κ/ ⊂ B ( A, κ ) \ U ; see the picture: Lemma 1.
There is C = C ( α, d, κ ) such that if D is ( κ, -fat at x , then P x ( τ D > / ≤ CP x ( τ D > . (25) EAT KERNEL ESTIMATES Proof.
Consider x ∈ D and B ( A, κ ) and U as above. For | x − A | < κ/ ≥ P x ( τ D > / ≥ P x ( τ D > ≥ P x ( τ B ( x,κ/ >
3) = P ( τ B (0 ,κ/ > > , and (25) is proved. We will now assume that | x − A | ≥ κ/
2. We note that P x ( τ D > / ≤ P x ( τ U > /
3) + P x ( X τ U ∈ D ) . (26)We have P x ( X τ U ∈ D ) = R D P U ( x, y ) dy . Indeed, if B = B ( x, | x − A | + κ/ U , then P x ( X τ U ∈ ∂U ∩ D ) ≤ P x ( X τ B ∈ ∂B ) = 0; seethe discussion preceding (19) above. Similarly, P x ( X τ U ∈ B ) is an integralof the Poisson kernel P U . We consider BHP for x = x , x = A , p = 1 − κ/ > (1 − κ ) / (1 − κ + κ/ D and B , we obtain P x ( X τ U ∈ D ) P A ( X τ U ∈ D ) ≤ c P x ( X τ U ∈ B ) P A ( X τ U ∈ B ) . We note that (the denominator) P A ( X τ U ∈ B ) ≥ P A ( X τ B ∈ B ) ≥ c > P x ( X τ U ∈ D ) ≤ cP x ( X τ U ∈ B ). We also observe that u R B ν ( y − u ) dy is bounded away from zero and infinity on U . By (19), P x ( X τ U ∈ B ) = Z U G U ( x, u ) Z B ν ( y − u ) dy du ≈ Z U G U ( x, u ) du = E x τ U . Clearly, P x ( τ U > / ≤ E x τ U . By (26), P x ( τ D > / ≤ cE x τ U . By thestrong Markov property, E x τ U ≤ cP x ( X τ U ∈ B ) ≤ cE x [ X τ U ∈ B ; P X τU ( τ B ( X τU ,κ/ > ≤ cP x ( τ D > . (cid:3) K. BOGDAN, T. GRZYWNY AND M. RYZNAR
Remark 3. If D is ( κ, x , then by the above proof we have P x ( τ D > / ≈ P x ( τ D > ≈ P x ( τ D > ≈ P x ( X τ U ∈ D ) ≈ E x τ U . (27)In fact, we can replace 3 by any finite E ≥
1, at the expense of havingthe comparability between each pair of expressions in (27) holding with aconstant C = C ( α, d, κ, E ). Lemma 2.
Consider open D , D ⊂ D such that dist( D , D ) > . Let D = D \ ( D ∪ D ) . If x ∈ D and y ∈ D , then p D (1 , x, y ) ≤ P x ( X τ D ∈ D ) sup s< ,z ∈ D p ( s, z, y ) + E x τ D sup u ∈ D ,z ∈ D ν ( z − u ) and p D (1 , x, y ) ≥ P x ( τ D > P y ( τ D >
1) inf u ∈ D ,z ∈ D ν ( z − u ) . Proof.
By the strong Markov property, p D (1 , x, y ) = E x [ p D (1 − τ D , X τ D , y ) , τ D < , which is E x [ p D (1 − τ D , X τ D , y ) , τ D < , X τ D ∈ D ]+ E x [ p D (1 − τ D , X τ D , y ) , τ D < , X τ D ∈ D ] = I + II . Clearly, I ≤ P x ( X τ D ∈ D ) sup s< ,z ∈ D p ( s, z, y ) . Consider D such that P x ( X τ D ∈ ∂D ∩ D ) = 0, for example, D being anintersection of D with a Lipschitz domain. By (18), the density function of( τ D , X τ D ) at ( s, z ) for z ∈ D equals f x ( s, z ) = Z D p D ( s, x, u ) ν ( z − u ) du. For z ∈ D , f x ( s, z ) = Z D p D ( s, x, u ) ν ( z − u ) du ≤ P x ( τ D > s ) sup u ∈ D ,z ∈ D ν ( z − u ) , hence, by (13), II = Z Z D p D (1 − s, z, y ) f x ( s, z ) dz ds ≤ sup u ∈ D ,z ∈ D ν ( z − u ) Z Z D p D (1 − s, z, y ) P x ( τ D > s ) dz ds ≤ Z P x ( τ D > s ) ds sup u ∈ D ,z ∈ D ν ( z − u ) ≤ E x τ D sup u ∈ D ,z ∈ D ν ( z − u ) . EAT KERNEL ESTIMATES The upper bound follows. The case of general D follows by approximatingfrom below, and continuity of p and ν . The lower bound obtains analogously II ≥ inf u ∈ D ,z ∈ D ν ( z − u ) Z Z D p D (1 − s, z, y ) P x ( τ D > s ) dz ds ≥ P x ( τ D >
1) inf u ∈ D ,z ∈ D ν ( z − u ) Z Z D p D (1 − s, z, y ) dz ds. (cid:3) Remark 4.
Lemma 2 also holds for ˜ ν , ˜ p , ˜ P x and ˜ E x of Remark 1.In what follows we will often use the fact that1 ∧ ν ( z − u ) ≈ p (1 , u, z ) . (28) Lemma 3. If D is ( κ, -fat at x and y , then p D (2 , x, y ) ≤ C ( α, d, κ ) P x ( τ D > P y ( τ D > p (2 , x, y ) . Proof. If | x − y | ≤
8, then p (1 , x, y ) ≈
1, and by the semigroup property,(10) and Lemma 1, p D (1 , x, y ) = Z R d p D (1 / , x, z ) p D (1 / , z, y ) dz ≤ sup z p (1 / , z, y ) P x ( τ D > / ≤ cP x ( τ D > p (1 , x, y ) . Here c = c ( α, d, κ ). If | x − y | >
8, then we will apply Lemma 2 with D = U = D ∩ B ( A, | x − A | + κ/ D = { z ∈ D : | z − x | > | x − y | / } . Since sup s< ,z ∈ D p ( s, z, y ) ≤ cp (1 , x, y ), and sup u ∈ D ,z ∈ D ν ( z − u ) ≤ cp (1 , x, y ) [see (28)], by Remark 3, we obtain p D (1 , x, y ) ≤ cp (1 , x, y )[ P x ( X τ U ∈ D ) + E x τ U ](30) ≤ cP x ( τ D > p (1 , x, y ) , hence, by (29), (30), symmetry, the semigroup property and Lemma 1, p D (2 , x, y ) = Z p D (1 , x, z ) p D (1 , z, y ) dz ≤ cP x ( τ D > P y ( τ D > Z p (1 , x, z ) p (1 , z, y ) dz ≤ cP x ( τ D > P y ( τ D > p (2 , x, y ) . (cid:3) Under the assumptions of Lemma 3, ˜ C = ˜ C ( α, d, κ ) exists such that p D (1 , x, y ) ≤ ˜ CP x ( τ D > P x ( τ D > p (1 , x, y ) . (31) K. BOGDAN, T. GRZYWNY AND M. RYZNAR
Indeed, according to Remark 1, we consider ˜ ν = ν and the corresponding˜ p , ˜ p D , ˜ P x , obtaining p D (1 , x, y ) = ˜ p D (2 , x, y ) ≤ ˜ C ˜ P x ( τ D >
2) ˜ P x ( τ D > p (2 , x, y )= ˜ CP x ( τ D > P x ( τ D > p (1 , x, y ) . Lemma 4. If r > , then there is a constant C = C ( α, d, r ) such that p B ( u,r ) ∪ B ( v,r ) (1 , u, v ) ≥ Cp (1 , u, v ) , u, v ∈ R d . Proof.
For | u − v | ≥ r/ D = B ( u, r ) ∪ B ( v, r ), D = B ( u, r/
8) and D = B ( v, r/ p B ( u,r ) ∪ B ( v,r ) (1 , u, v ) ≥ P u ( τ D > P v ( τ D >
1) inf u ∈ D ,z ∈ D ν ( z − u ) ≥ c [ P ( τ B (0 ,r/ > p (1 , u, v ) . For | u − v | ≤ r/
2, by (4), we simply have p B ( u,r ) ∪ B ( v,r ) (1 , u, v ) ≥ inf | z |
Consider U x , B x , and U y , B y , selected according to Definition 2for x and y , correspondingly. By the semigroup property, Lemma 4 with r = κ/
6, and (4), p D (3 , x, y ) ≥ Z B y Z B x p D (1 , x, u ) p D (1 , u, v ) p D (1 , v, y ) du dv ≥ cp (1 , x, y ) Z B x p D (1 , x, u ) du Z B y p D (1 , v, y ) dv. For u ∈ B x = B ( A ′ , κ/ D = U x = U and D = B ( A ′ , κ/ p D (1 , x, u ) ≥ P x ( τ U > P ( τ B (0 ,κ/ >
1) inf w ∈ U,z ∈ D ν ( z − w ) ≥ cP x ( τ U > ≥ cP x ( τ D > . Similarly, p D (1 , v, y ) ≥ cP y ( τ D > p D (3 , x, y ) ≥ cP y ( τ D > p (1 , x, y ) P x ( τ D > ≥ cP y ( τ D > p (3 , x, y ) P x ( τ D > . (cid:3) EAT KERNEL ESTIMATES Under the assumptions of Lemma 5 we also have that p D (1 , x, y ) ≥ ˜ C ( α, d, κ ) P x ( τ D > P y ( τ D > p (1 , x, y ) . (32)This is proved analogously to (31). Proof of Theorem 1.
Assume that R ≥ D is ( κ, r )-fat for0 < r ≤ R . If t /α ∈ (0 , R ], then t − /α D is ( κ, C = C ( α, d, κ )in (1). If R <
1, then we argue as in the case of (31) C = C ( α, d, κ, R ) or,alternatively, we use Remark 6 below. (cid:3) Proof of Corollary 1.
Note that D is (1 / , r )-fat for r ≥ D c ),and so we obtain (1) for t ≥ α diam( D c ) with the same constant C . If weconsider ˜ ν = 2 − α ν and argue like in the case of (31), then we obtain thewider range of t , as in the statement of Corollary 1. (cid:3) Remark 5.
Since the κ -fatness condition is more restrictive when κ isbigger, the above constants C = C ( α, d, κ ) may be chosen decreasing withrespect to κ . Also, if D has a tangent inner ball of radius 1 at every boundarypoint, then the constants in Lemmas 3 and 5 depend only on α and d . Remark 6. If D is ( κ, r )-fat at x and 1 ≤ K < ∞ , then D is ( κ/K, rK )-fat at x . This observation together with scaling allows to easily increasetime , compare (31) or (32), at the expense of enlarging the constants ofcomparability. The argument, however, does not allow to decrease time.Remark 1 is more flexible in this respect.
4. Applications.
We let s D ( x ) = E x τ D = R G D ( x, y ) dv if this expecta-tion is finite for x ∈ D , otherwise we let s D ( x ) = M D ( x ), the Martin kernelwith the pole at infinity for D , M D ( x ) = lim D ∋ y, | y |→∞ G D ( x, y ) G D ( x , y ) . We should note that this (alternative) definition of s D is natural in view of[14], Theorem 2. The choice of x ∈ D is merely a normalization, M D ( x ) =1, and will not be reflected in the notation. By the scaling of the Greenfunction (17), we obtain s rD ( rx ) s rD ( ry ) = s D ( x ) s D ( y ) , x, y ∈ D, r > . (33)We denote by A r ( x ) or A r ( x, κ, D ) every point A such that B ( A, κr ) ⊂ D ∩ B ( x, r ), as in Definition 1. It is noteworthy that A r ( x ) approximatelydominates x in terms of the distance to ∂D : δ D ( A r ( x )) ≈ r ∨ δ D ( x ) . (34) K. BOGDAN, T. GRZYWNY AND M. RYZNAR If D is ( κ, x , then rD is ( κ, r )-fat at rx , and (every) rA ( x, κ, D )may serve as A r ( rx, κ, rD ). Theorem 2. If D is ( κ, t /α )-fat at x and y , then P x ( τ D > t ) C ≈ s D ( x ) s D ( A t /α ( x )) , (35) where C = C ( d, α, κ ) and, furthermore, p D ( t, x, y ) C ≈ s D ( x ) s D ( A t /α ( x )) p ( t, x, y ) s D ( y ) s D ( A t /α ( y )) . (36) Proof.
To verify (35), we first let t = 1 and assume that D is ( κ, x . Let A = A ( x ). If E x τ D < ∞ , then we consider the set U ⊂ D ofDefinition 2, and we obtain E x τ D = E x τ U + E x s D ( X τ U ) . By Remark 3, E x τ U ≈ P x ( τ D > E A τ U ≈
1, we trivially have E x τ U E A τ U ≈ P x ( τ D > . Similarly, P A ( X τ U ∈ D ) ≈
1. By BHP and Remark 3, we obtain E x s D ( X τ U ) E A s D ( X τ U ) ≈ P x ( X τ U ∈ D ) P A ( X τ U ∈ D ) ≈ P x ( X τ U ∈ D ) ≈ P x ( τ D > . (37)This yields (35) in the considered case. If E x τ D = ∞ , then s D is harmonicand we have s D ( x ) = E x s D ( X τ U ) (see [14], Theorem 2 and (77)) and weproceed directly via (37). The case of general t in (35) is obtained by thescaling of (33) and (15). Finally, (36) follows from (35) and Theorem 1. Theresulting comparability constants depend only on α , d and κ . (cid:3) Remark 7.
Assume that D is κ -fat, so that there is R > D is ( κ, r )-fat for every r ≤ R . Then (35) and (36) hold with C = C ( d, α, κ ) forall x, y ∈ D and t ≤ R α .Below we give a number of applications. Example 1.
We let
R > D = B (0 , R ) ⊂ R d . By (24), the ex-pected survival time is s D ( x ) C ≈ δ α/ D ( x ) R α/ , where C = C ( d, α ). By (34), s D ( A t /α ( x )) C ≈ ( t /α ∨ δ D ( x )) α/ R α/ , therefore, for all t ≤ R α and x, y ∈ R d , P x ( τ D > t ) C ≈ δ α/ D ( x )( t /α ∨ δ D ( x )) α/ = (cid:18) ∧ δ D ( x ) t /α (cid:19) α/ (38) EAT KERNEL ESTIMATES and p D ( t, x, y ) C ≈ (cid:18) ∧ δ α/ D ( x ) t / (cid:19) p ( t, x, y ) (cid:18) ∧ δ α/ D ( y ) t / (cid:19) . (39)To be explicit, δ B (0 ,R ) ( x ) = ( R − | x | ) ∨
0, and δ B (0 ,R ) c ( x ) = ( | x | − R ) ∨ D c follow because all x ∈ D c are regular for D . Example 2.
Let D ⊂ R d be a half-space. The Martin kernel with thepole at infinity for D is s D ( x ) = δ α/ D ( x ) [1]. We see that (38) and (39) holdwith C = C ( d, α ) for all t ∈ (0 , ∞ ) and x, y ∈ R d . Example 3.
Let D = B c (0 , ⊂ R d and d ≥ α . By the Kelvin transform([18] or [14]) and (21), M D ( x ) = lim y →∞ | x | α − d | y | α − d G B ( x/ | x | , y/ | y | ) | x | α − d | y | α − d G B ( x / | x | , y/ | y | ) = | x | α − d G B ( x/ | x | , | x | α − d G B ( x / | x | , , where G B ( z,
0) = B d,α | z | α − d Z | z | − − s α/ − ( s + 1) d/ ds, < | z | < . Thus, there is c = c ( x , d, α ) such that M D ( x ) = c Z | x | − s α/ − ( s + 1) d/ ds, | x | ≥ . (40)If d > α , then s D ( x ) ≈ ∧ δ α/ D ( x ), s D ( A t /α ( x )) ≈ ∧ ( t /α ∨ δ D ( x )) α/ , thus, P x ( τ D > t ) C ≈ ∧ δ α/ D ( x )1 ∧ ( t /α ∨ δ D ( x )) α/ = 1 ∧ δ α/ D ( x )(1 ∧ t /α ) α/ (41)and p D ( t, x, y ) C ≈ (cid:18) ∧ δ α/ D ( x )1 ∧ t / (cid:19) p ( t, x, y ) (cid:18) ∧ δ α/ D ( y )1 ∧ t / (cid:19) for all 0 < t < ∞ and x, y ∈ R d . Here C = C ( d, α ).For α = d = 1, (40) yields s D ( x ) ≈ log(1 + δ / D ( x )), s D ( A t /α ( x )) ≈ log(1 +( t ∨ δ D ( x )) / ) , thus, for all 0 < t < ∞ and x, y ∈ R d we have P x ( τ D > t ) ≈ log(1 + δ / D ( x ))log(1 + ( t ∨ δ D ( x )) / ) = 1 ∧ log(1 + δ / D ( x ))log(1 + t / )(42)and p D ( t, x, y ) p ( t, x, y ) ≈ (cid:18) ∧ log(1 + δ / D ( x ))log(1 + t / ) (cid:19)(cid:18) ∧ log(1 + δ / ( y ))log(1 + t / ) (cid:19) . Sharp explicit estimates for p B c (0 ,R ) with arbitrary R > K. BOGDAN, T. GRZYWNY AND M. RYZNAR
Example 4.
Let D = B c (0 , ⊂ R d and 1 = d < α . We have that G { } c ( x, y ) = G D ( x, y ) + E x G { } c ( X T B , y ) . Let c α = [ − α ) cos( πα/ − . By [18], Lemma 4, for x, y ∈ R , G { } c ( x, y ) = c α ( | y | α − + | x | α − − | y − x | α − ) . If follows that G D ( x, y ) = c α ( | x | α − − | x − y | α − − E x ( | X τ D | α − − | X τ D − y | α − )) . Since | X τ D | ≤ y →∞ ( −| x − y | α − + E x | X τ D − y | α − ) = 0, for every x ∈ R . If | x | ≥
2, then we can find c = c ( α, x ) such that M D ( x ) = | x | α − − E x | X τ D | α − | x | α − − E x | X τ D | α − = c ( | x | α − − E x | X τ D | α − ) ≈ | x | α − ≈ δ D ( x ) α − . On the other hand, by BHP, M D ( x ) ≈ δ α/ D ( x ) if δ ( x ) ≤ s D ( x ) ≈ δ α − D ( x ) ∧ δ α/ D ( x ), s D ( A t /α ( x )) ≈ ( t /α ∨ δ D ( x )) α − ∧ ( t /α ∨ δ D ( x )) α/ , and for all 0 < t < ∞ , x, y ∈ R d , we obtain P x ( τ D > t ) C ≈ δ α − D ( x ) ∧ δ α/ D ( x )( t /α ∨ δ D ( x )) α − ∧ ( t /α ∨ δ D ( x )) α/ , (43)hence, p D ( t, x, y ) C ≈ (cid:18) ∧ δ α − D ( x ) ∧ δ α/ D ( x ) t − /α ∧ t / (cid:19) p ( t, x, y ) (cid:18) ∧ δ α − D ( y ) ∧ δ α/ D ( y ) t − /α ∧ t / (cid:19) . Here C = C ( α ). To estimate p B c (0 ,R ) with arbitrary R >
0, we use scaling.
Definition 3.
We say that (open) D is of class C , at scale r > Q ∈ ∂D there exist balls B ( x ′ , r ) ⊂ D and B ( x ′′ , r ) ⊂ D c tangentat Q . If D is C , at some (unspecified) positive scale (hence also at smallerscales), then we simply say D is C , . C , domains may be equivalently defined using local coordinates [34]. Remark 8. If D is C , at scale r , then it is (1 / , p )-fat for all p ∈ (0 , r ]. Remark 9.
Let D be C , at scale r . Let x ∈ D , and let Q ∈ ∂D besuch that δ D ( x ) = | x − Q | . Consider the above balls B ( x ′ , r ) and B ( x ′′ , r ).If δ D ( x ) < r , then let B x = B ( x ′ , r ), otherwise B x = B ( x, δ D ( x )). Thus, δ B x ( x ) = δ D ( x ), and the radius of B x is r ∨ δ D ( x ). EAT KERNEL ESTIMATES Example 5.
We will verify (2) for C , domains D . For the proof weinitially assume that D = R d is C , at scale r = 1. Let x ∈ D . We adopt thenotation of Remark 9 and consider (the ball) B x and (the open complementof a ball) B c ( x ′′ ,
1) tangent at Q ∈ ∂D . Since B x ⊂ D ⊂ B c ( x ′′ , P x ( τ B x > ≤ P x ( τ D > ≤ P x ( τ B c ( x ′′ , > . Clearly, δ B x ( x ) = δ D ( x ) = | Q − x | = δ B c ( x ′′ , ( x ). By (38) and (41)–(43), P x ( τ D > t ) ≈ (cid:18) ∧ δ D ( x ) t /α (cid:19) α/ , t ≤ . By Remark 8 and Theorem 1, there is C = C ( d, α ) such that, for all x, y ∈ R d , p D ( t, x, y ) C ≈ (cid:18) ∧ δ D ( x ) α/ t / (cid:19)(cid:18) t | x | d + α ∧ t − d/α (cid:19)(cid:18) ∧ δ D ( y ) α/ t / (cid:19) , t ≤ . If D is C , at a scale r <
1, then r − D is C , at scale 1. This yields (2)in time range 0 < t ≤ r α . Remark 3 allows for an extension to all t ∈ (0 , d , α and r . The case of D = R d is trivial.Further estimates for C , domains will be given in Proposition 1, Theo-rem 3 and Corollary 2. Example 6.
Let d ≥
2. For x = ( x , . . . , x d − , x d ) ∈ R d we denote ˜ x =( x , . . . , x d − ), so that x = (˜ x, x d ). Let λ < ∞ . We consider a Lipschitz func-tion γ : R d − → R , that is, | γ (˜ x ) − γ (˜ y ) | ≤ λ | ˜ x − ˜ y | . We define a specialLipschitz domain D = { x = (˜ x, x d ) ∈ R d : x d > γ (˜ x ) } . For such D the geo-metric notions of Theorem 2 become more explicit as we will see below.We note that D is ((2 √ λ ) − , r )-fat for all r > x = (˜ x, x d ) ∈ D and r > x ( r ) = (˜ x, γ (˜ x ) + r ). If x is close to ∂D ,then x (1) dominates x in the direction of the last coordinate. We note that P x (1) ( τ D > ≥ c >
0. Here c = c ( d, α, λ ). By Remark 3 and BHP, P x ( τ D > C ≈ ∧ M D ( x ) M D ( x (1) ) , x ∈ D, (44)where C = C ( α, d, λ ). By scaling, the Martin kernel with the pole at infinityfor rD is a constant multiple of M D ( x/r ). By (44), we obtain P x ( τ D > t ) = P t − /α x ( τ t − /α D > C ≈ ∧ M D ( x ) M D ( x ( t /α ) ) , x ∈ D. (45)We note in passing that (45) agrees with (35) because r M D ( x ( r ) ) isincreasing [15]. Or, in our previous notation we can take A r ( x, κ, D ) = K. BOGDAN, T. GRZYWNY AND M. RYZNAR x ( r ∨ ( x d − γ (˜ x )) . We substitute (45) into (1) so that for all 0 < t < ∞ and x, y ∈ D (in fact, by regularity, for x, y ∈ R d ) we have p D ( t, x, y ) C ≈ (cid:18) ∧ M D ( x ) M D ( x ( t /α ) ) (cid:19) p ( t, x, y ) (cid:18) ∧ M D ( y ) M D ( y ( t /α ) ) (cid:19) . Example 7.
For circular cones V [10] we have M V ( x ) = | x | β M V ( x/ | x | ) , x = 0 , (46)where 0 ≤ β < α is a characteristic of the cone; see [1]. By [35], Lemma 3.3, M V ( x ) ≈ δ V ( x ) α/ | x | β − α/ , x ∈ R d ;see also [10] and [35]. Considering (44), by simple manipulations, we obtain1 ∧ δ V ( x ) α/ | x | β − α/ (1 ∨ | ˜ x | ) β − α/ C ≈ (1 ∧ δ V ( x ) α/ )(1 ∧ | x | ) β − α/ , (47)where C = C ( λ ). By (1) and scaling, we get (3).The interested reader may find more references on stable processes andBrownian motion in cones in [10]. Note that (46) holds for generalized opencones, that is, open sets ∅ = V ⊂ R d such that kV = V for all k > Example 8.
Let d = 1 , , . . . and V = R d \ { x d = 0 } . This generalizedcone is non-Lipschitz but it is (1 / , r )-fat for every r >
0. Let 1 < α <
2. From[1], Example 3.3, we have M V ( x ) = | x d | α − (the decay near a hyperplane isslower than near a half-space). We consider t = 1 in (36). We let A ( x ) =(˜ x, x d + 1 /
2) if x d > A ( x ) = (˜ x, x d − /
2) otherwise. Thus, M V ( x ) M V ( A ( x )) = | x d | α − ( | x d | + 1 / α − ≈ (1 ∧ | x d | ) α − . By (1) and scaling, we obtain the following analogue of (3): p V ( t, x, y ) p ( t, x, y ) ≈ (cid:18) ∧ δ V ( x ) t /α (cid:19) α − (cid:18) ∧ δ V ( y ) t /α (cid:19) α − , t > , x, y ∈ R d . (48)We note that V is the complement of a point if d = 1.If D is bounded and κ > D is not ( κ, r ) at large scales r ,and the asymptotics of the probability of survival are exponential. Indeed,for the fractional Laplacian with Dirichlet condition on D c we let λ > φ > L ( D, dx )]; see [30]. The following approximation results from the intrinsicultracontractivity of every bounded domain [30]: p D ( t, x, y ) ≈ φ ( x ) φ ( y ) e − λ t , t ≥ , x, y ∈ R d . EAT KERNEL ESTIMATES Here comparability constants depend on D and α (see also Proposition 1below). Given that infinity is inaccessible [14] from bounded D , it is ofconsiderable interest to understand the behavior of the heat kernel relatedto accessible and inaccessible points of D (see also [33] in this connection).In the remainder of the paper we will study C , domains in more de-tail. We focus on unbounded domains, large times and dependence of thecomparability constants on global geometry of the domains.Example 1 and intrinsic ultracontractivity yield the following result. Lemma 6.
There exist λ = λ ( α, d ) > and C = C ( α, d ) such that forall r > , t > and x ∈ R d we have P x ( τ B (0 ,r ) > t ) C ≈ (cid:20) ∧ (cid:18) δ B (0 ,r ) ( x ) r ∧ t /α (cid:19) α/ (cid:21) e − λ t/r α . Lemma 7.
Let d > α , < r < R , W = B (0 , r ) ∪ B c (0 , R ) . There is c = c ( α, d ) such that for all t > and x ∈ R d we have P x ( τ W > t ) ≥ c (cid:18) rR (cid:19) α (cid:20) ∧ (cid:18) δ B (0 ,r ) ( x ) r ∧ t /α (cid:19) α/ (cid:21) . Proof.
By scaling, we only need to consider r = 1 < R . By [5], we obtain P x ( T B (0 , = ∞ ) = Γ( d/ d − α ) / α/ Z | x | − u α/ − ( u + 1) d/ du ≈ ∧ δ α/ B c (0 , ( x )[compare (40)]. Thus, there is c = c ( d, α ) such that P y ( T B (0 ,R ) = ∞ ) ≥ c > , | y | > R. Let x ∈ B (0 , t ≥ P x ( τ W > t ) ≥ P x ( τ W = ∞ ) ≥ E x {| X τ B (0 , | ≥ R ; P X τB (0 , ( T B (0 ,R ) = ∞ ) }≥ cP x ( | X τ B (0 , | ≥ R ) ≥ c R α δ α/ B (0 , ( x ) . By (38), for t ≤ P x ( τ W > t ) ≥ P x ( τ B (0 , > t ) ≈ ∧ (cid:18) δ B (0 , ( x )1 ∧ t /α (cid:19) α/ . (cid:3) The C , condition at a given scale fails to determine the fatness of D at larger scales and, consequently, the exact asymptotics of the survivalprobability. The following is a substitute. K. BOGDAN, T. GRZYWNY AND M. RYZNAR
Proposition 1. If D is C , at some scale r > , then C − e − λ t/ ( r ∨ δ D ( x )) α (cid:20) ∧ (cid:18) δ D ( x ) r ∧ t /α (cid:19) α/ (cid:21) (49) ≤ P x ( τ D > t ) ≤ C (cid:20) ∧ (cid:18) δ D ( x ) r ∧ t /α (cid:19) α/ (cid:21) for all t > and x ∈ R d . Here C = C ( α, d ) and λ = λ ( α, d ) .If also d > α and diam( D c ) < ∞ , then for all t > and x ∈ R d , P x ( τ D > t ) ≥ C − (cid:18) r diam( D c ) (cid:19) α (cid:20) ∧ (cid:18) δ D ( x ) r ∧ t /α (cid:19) α/ (cid:21) . (50) Proof.
Consider x ∈ D , B x ⊂ D and B ( x ′′ , r ) ⊂ D c of Remark 9. Clearly, τ B x ≤ τ D ≤ T B ( x ′′ ,r ) , thus, P x ( τ B x > t ) ≤ P x ( τ D > t ) ≤ P x ( T B ( x ′′ ,r ) > t ) . Lemma 6 yields the estimate C − e − λ t/ ( r ∨ δ D ( x )) α (cid:20) ∧ (cid:18) δ D ( x )( r ∨ δ D ( x )) ∧ t /α (cid:19) α/ (cid:21) ≤ P x ( τ D > t )and P x ( τ D > t ) ≤ C (cid:20) ∧ (cid:18) δ D ( x ) r ∧ t /α (cid:19) α/ (cid:21) , which simplifies to (49) as δ D ( x ) > r yields δ D ( x ) / [( r ∨ δ D ( x )) ∧ t /α ] ≥ ρ = diam( D c ) ≥ r , the center, say, x , of B x ,and W := B x ∪ B c ( x , ρ + r ∨ δ D ( x )) ⊂ D . By Lemma 7 and Remark 9, P x ( τ D > t ) ≥ P x ( τ W > t ) ≥ c (cid:18) r ∨ δ D ( x ) ρ + r ∨ δ D ( x ) (cid:19) α (cid:20) ∧ (cid:18) δ D ( x )( r ∨ δ D ( x )) ∧ t /α (cid:19) α/ (cid:21) ≥ c (cid:18) rρ (cid:19) α (cid:20) ∧ (cid:18) δ D ( x ) r ∧ t /α (cid:19) α/ (cid:21) . (cid:3) In view of Theorem 1, (49) mildly strengthens [19], Theorem 1.1(i) [i.e.,(2) above]. We also get the following result.
Theorem 3.
Let d > α . If D is C , at scale r and diam( D c ) < ∞ , then C − (cid:18) r diam( D c ) (cid:19) α EAT KERNEL ESTIMATES ≤ p D ( t, x, y )[1 ∧ ( δ D ( x ) / ( r ∧ t /α )) α/ ] p ( t, x, y )[1 ∧ ( δ D ( y ) / ( r ∧ t /α )) α/ ] ≤ C for all t > and x, y ∈ R d . Here C = C ( α, d ) . Proof.
The result follows from (50) and Corollary 1. (cid:3)
A similar result (with less control of the constants) is given in [22]. Remark 10.
We consider the recurrent case α ≥ d = 1. If D ⊂ R is thecomplement of a finite union of bounded closed intervals, then P x ( τ D > t ) C ≈ ∧ δ D ( x ) α − ∧ δ D ( x ) α/ t − /α ∧ t / , t > , x ∈ R d , if α > P x ( τ D > t ) C ≈ ∧ log(1 + δ D ( x ) / )log(1 + t / ) , t > , x ∈ R d , if α = 1,where C = C ( D, α ). The estimates follow easily from Examples 2 and 3.
Corollary 2. If D ⊂ R is the complement to a finite union of boundedclosed intervals, then C = C ( D, α ) exists such that for all t > and x, y ∈ R , p D ( t, x, y ) p ( t, x, y ) C ≈ (cid:20) ∧ δ D ( x ) α − ∧ δ D ( x ) α/ t − /α ∧ t / (cid:21)(cid:20) ∧ δ D ( y ) α − ∧ δ D ( y ) α/ t − /α ∧ t / (cid:21) for α > , while for α = 1 we have p D ( t, x, y ) p ( t, x, y ) C ≈ (cid:20) ∧ log(1 + δ D ( x ) / )log(1 + t / ) (cid:21)(cid:20) ∧ log(1 + δ D ( y ) / )log(1 + t / ) (cid:21) . Acknowledgments.
Results of the paper were reported at the workshop
Schr¨odinger operators and stochastic processes , Wroc law, 14–15 May 2009,and at
The Third International Conference on Stochastic Analysis and ItsApplications , Beijing, 13–17 July 2009. The authors are grateful to the or-ganizers for this opportunity. We also thank an anonymous referee for aquestion which led to our conjecture in the Introduction about (1) for irreg-ular boundary points of D . Paper [22] appeared on arXiv after the first draft [11] of the present paper. K. BOGDAN, T. GRZYWNY AND M. RYZNAR
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