Two-term trace estimates for relativistic stable processes
aa r X i v : . [ m a t h . P R ] S e p TWO-TERM TRACE ESTIMATES FOR RELATIVISTIC STABLEPROCESSES
RODRIGO BA ˜NUELOS, JEBESSA MIJENA, AND ERKAN NANE
Abstract.
We prove trace estimates for the relativistic α -stable process extendingthe result of Ba˜nuelos and Kulczycki (2008) in the stable case. Introduction and statement of main results
For m > , an R d -valued process with independent, stationary increments havingthe following characteristic function E e iξ · X α,mt = e − t { ( m /α + | ξ | ) α/ − m } , ξ ∈ R d , is called relativistic α -stable process with mass m . We assume that sample paths of X α,mt are right continuous and have left-hand limits a.s. If we put m = 0 we obtainthe symmetric rotation invariant α − stable process with the characteristic function e − t | ξ | α , ξ ∈ R d . We refer to this process as isotropic α − stable L´evy process. For therest of the paper we keep α, m and d ≥ α, m in the notation, whenit does not lead to confusion. Hence from now on the relativistic α -stable process isdenoted by X t and its counterpart isotropic α − stable L´evy process by ˜ X t . We keepthis notational convention consistently throughout the paper, e.g., if p t ( x − y ) is thetransition density of X t , then ˜ p t ( x − y ) is the transition density of ˜ X t .In Ryznar [13] Green function estimates of the Sch¨odinger operator with the freeHamiltonian of the form ( − ∆ + m /α ) α/ − m, were investigated, where m > L ( R d ).Using the estimates in Lemma 2.6 below and proof in Ba˜nuelos and Kulczycki (2008)we provide an extension of the asymptotics in [3] to the relativistic α stable processesfor any 0 < α < E e iξ · B t = e − t | ξ | , ξ ∈ R d . Let β = α/
2. Ryznar showed that X t can be represented as a time-changed Brownianmotion. Let T β ( t ) , t >
0, denote the strictly β -stable subordinator with the followingLaplace transform(1.1) E e − λT β ( t ) = e − tλ β , λ > . Key words and phrases.
Relativistic stable process, trace, asymptotics. et θ β ( t, u ) , u >
0, denote the density function of T β ( t ). Then the process B T β ( t ) isthe standard symmetric α -stable process.Ryznar [13, Lemma 1] showed that we can obtain X t = B T β ( t,m ) , where a subor-dinator T β ( t, m ) is a positive infinitely divisible process with stationary incrementswith probability density function θ β ( t, u, m ) = e − m /β u + mt θ β ( t, u ) , u > . Transition density of T β ( t, m ) is given by θ β ( t, u − v, m ). Hence the transitiondensity of X t is p ( t, x, y ) = p ( t, x − y ) given by(1.2) p ( t, x ) = e mt Z ∞ πu ) d/ e −| x | u e − m /β u θ β ( t, u ) du. Then p ( t, x, x ) = p ( t,
0) = e mt Z ∞ πu ) d/ e − m /β u θ β ( t, u ) du. The function p ( t, x ) is a radially symmetric decreasing and that(1.3) p ( t, x ) ≤ p ( t, ≤ e mt Z ∞ πu ) d/ θ β ( t, u ) du = e mt t − d/α ω d Γ( d/α )(2 π ) d α , where ω d = π d/ Γ( d/ is the surface area of the unit sphere in R d . For an open set D in R d we define the first exit time from D by τ D = inf { t ≥ X t / ∈ D } .We set(1.4) r D ( t, x, y ) = E x [ p ( t − τ D , X τ D , y ); τ D < t ] , and(1.5) p D ( t, x, y ) = p ( t, x, y ) − r D ( t, x, y ) , for any x, y ∈ R d , t >
0. For a nonnegative Borel function f and t >
0, let P Dt f ( x ) = E x [ f ( X t ) : t < τ D ] = Z D p D ( t, x, y ) f ( y ) dy, be the semigroup of the killed process acting on L ( D ), see, Ryznar [13, Theorem 1].Let D be a bounded domain (or of finite volume). Then the operator P Dt maps L ( D ) into L ∞ ( D ) for every t >
0. This follows from (1.3), (1.4), and the gen-eral theory of heat semigroups as described in [10]. It follows that there exists anorthonormal basis of eigenfunctions { ϕ n : n = 1 , , , · · · } for L ( D ) and corre-sponding eigenvalues { λ n : n = 1 , , , · · · } of the generator of the semigroup P Dt satisfying λ < λ ≤ λ ≤ · · · , with λ n → ∞ as n → ∞ . By definition, the pair { ϕ n , λ n } satisfies P Dt ϕ n ( x ) = e − λ n t ϕ n ( x ) , x ∈ D, t > . nder such assumptions we have(1.6) p D ( t, x, y ) = ∞ X n =1 e − λ n t ϕ n ( x ) ϕ n ( y ) . In this paper we are interested in the behavior of the trace of this semigroup(1.7) Z D ( t ) = Z D p D ( t, x, x ) dx. Because of (1.6) we can write (1.7) as(1.8) Z D ( t ) = ∞ X n =1 e − λ n t Z D ϕ n ( x ) dx = ∞ X n =1 e − λ n t . We denote d -dimensional volume of D by | D | .Our first result is the Weyl’s asymtotic for the eigenvalues of the relativistic Lapla-cian Proposition 1.1. (1.9) lim t → t d/α e − mt Z D ( t ) = C | D | , where C = ω d Γ( d/α )(2 π ) d α . Let N ( λ ) be the number of eigenvalues { λ j } which do not exceed λ . It follows from(1.9) and the classical Tauberian theorem (see for example [11], p.445 Theorem 2)where L ( t ) = C | D | e m/t is our slowly varying function at infinity that(1.10) lim λ →∞ λ − d/α e − m/λ N ( λ ) = C | D | Γ(1 + d/α ) . This is the analogue for relativistic stable process of the celebrated Weyl’s asymp-totic formula for the eigenvalues of the Laplacian.
Remark . The first author presented (1.10) at a conference in Vienna at theScrh¨odingier Institute in 2009 (see [1]) and at the 34th conference in stochastic pro-cesses and their applications in Osaka in 2010 (see [2]). Thus this result has beenknown to the authors, and perhaps to others, for number of years.Our goal in this paper is to obtain the second term in the asymptotics of Z D ( t )under some additional assumptions on the smoothness of D . Our result is inspiredby result for trace estimates for stable processes by Ba˜nuelos and Kulczycki [3].To state our main result we need the following property of the domain D . Definition 1.3.
The boundary, ∂D , of an open set D in R d is said to be R − smoothif for each point x ∈ ∂D there are two open balls B and B with radii R such that B ⊂ D, B ⊂ R d \ ( D ∪ ∂D ) and ∂B ∩ ∂B = x . heorem 1.4. Let D ⊂ R d , d ≥ , be an open bounded set with R − smooth boundary.Let | D | denote the volume ( d − dimensional Lebesgue measure) of D and | ∂D | denoteits surface area ( ( d − − dimensional Lebesgue measure) of its boundary. Suppose α ∈ (0 , . Then (1.11) (cid:12)(cid:12)(cid:12)(cid:12) Z D ( t ) − C ( t ) e mt | D | t d/α + C ( t ) | ∂D | (cid:12)(cid:12)(cid:12)(cid:12) ≤ C e mt | D | t /α R t d/α , t > , where C ( t ) = 1(4 π ) d/ Z ∞ z − d/ e − ( mt ) /β z θ β (1 , z ) dz → C = ω d Γ( d/α )(2 π ) d α , as t → ,C ( t ) = Z ∞ r H ( t, ( x , , · · · , , ( x , , · · · , dx ≤ C e mt t /α t d/α , t > C = Z ∞ ˜ r H (1 , ( x , , · · · , , ( x , , · · · , dx ,C = C ( d, α ) , H = { ( x , · · · , x d ) ∈ R d : x > } and r H is given by (1.4) .Remark . When m = 0, 0 < α ≤ C ( t ) = C t /α /t d/α . Then the result inTheorem 1.4 becomes for bounded domains with R − smooth boundary(1.12) (cid:12)(cid:12)(cid:12)(cid:12) Z D ( t ) − C | D | t d/α + C | ∂D | t /α t d/α (cid:12)(cid:12)(cid:12)(cid:12) ≤ C | D | t /α R t d/α , where C , C are as in Theorem 1.4. This was established by Ba˜nuelos and Kulczycki[3] recently.The asymptotic for the trace of the heat kernel when α = 2 (the case of theLaplacian with Dirichlet boundary condition in a domain of R d ), have been extensivelystudied by many authors. For Brownian motion Van den Berg [5], proved that underthe R − smoothness condition(1.13) (cid:12)(cid:12)(cid:12)(cid:12) Z D ( t ) − (4 πt ) − d/ (cid:18) | D | − √ πt | ∂D | (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C d | D | t − d/ R , t > . For domains with C boundaries the result(1.14) Z D ( t ) = (4 πt ) − d/ (cid:18) | D | − √ πt | ∂D | + o ( t / ) (cid:19) , as t → , was proved by Brossard and Carmona [8], for Brownian motion. . Preliminaries
Let the ball in R d with center at x and radius r, { y : | y − x | < r } , be denotedby B ( x, r ) . We will use δ D ( x ) to denote the Euclidean distance between x and theboundary, ∂D , of D . That is, δ D ( x ) = dist( x, ∂D ). Define ψ ( θ ) = Z ∞ e − v v p − / ( θ + v/ p − / dv, θ ≥ , where p = ( d + α ) / . We put R ( α, d ) = A ( − α, d ) /ψ (0) , where A ( v, d ) = (Γ(( d − v ) / / ( π d/ v | Γ( v/ | ) . Let ν ( x ) , ˜ ν ( x ) be the densities of the L´evy measures of therelativistic α − stable process and the standard α − stable process, respectively. Thesedensities are given by(2.1) ν ( x ) = R ( α, d ) | x | d + α e − m /α | x | ψ ( m /α | x | ) , and(2.2) ˜ v ( x ) = A ( − α, d ) | x | d + α . We need the following estimate of the transition probabilities of the process X t which is given in ([14], Lemma 2.2): For any x, y ∈ R d and t > c > c > , (2.3) p ( t, x, y ) ≤ c e mt min (cid:26) t | x − y | d + α e − c | x − y | , t − d/α (cid:27) . We will also use the fact([7], Lemma 6) that if D ⊂ R d is an open bounded setsatisfying a uniform outer cone condition, then P x ( X ( τ D ) ∈ ∂D ) = 0 for all x ∈ D. For the open bounded set D we will denoted by G D ( x, y ) the Green function for theset D equal to, G D ( x, y ) = Z ∞ p D ( t, x, y ) dt, x, y ∈ R d . For any such D the expectation of the exit time of the processes X t from D is givenby the integral of the Green function over the domain. That is: E x ( τ D ) = Z D G D ( x, y ) dy. Lemma 2.1.
Let D ⊂ R d be an open set. For any x, y ∈ D we have r D ( t, x, y ) ≤ c e mt (cid:18) tδ d + αD ( x ) e − c δ D ( x ) ∧ t − d/α (cid:19) . roof. Using (1.4) and (2.3) we have r D ( t, x, y ) = E y ( p ( t − τ D , X ( τ D ) , x ); τ D < t ) ≤ c e mt E y (cid:18) t | x − X ( τ D ) | d + α e − c | x − X ( τ D ) | ∧ t − d/α (cid:19) ≤ c e mt (cid:18) tδ d + αD ( x ) e − c δ D ( x ) ∧ t − d/α (cid:19) . (cid:3) We need the following result for the proof of Proposition 1.1.
Lemma 2.2. (2.4) lim t → p ( t, e − mt t d/α = C , where C = (4 π ) d/ Z ∞ u − d/ θ β (1 , u ) du = ω d Γ( d/α )(2 π ) d α . Proof.
By (1.2) we have p ( t, x, x ) = p ( t,
0) = e mt Z ∞ πu ) d/ e − m /β u θ β ( t, u ) du. Now using the scaling of stable subordinator θ β ( t, u ) = t − /β θ β (1 , ut − /β ) and achange of variables we get p ( t,
0) = e mt (4 π ) d/ t d/α Z ∞ z − d/ e − m /β t /β z θ β (1 , z ) dz = C ( t ) e mt t d/α , then by dominated convergence theorem we obtainlim t → p ( t, e − mt t d/α = 1(4 π ) d/ Z ∞ z − d/ θ β (1 , z ) dz, and this last integral is equal to the density of α -stable process at time 1 and x = 0which was calculated in [3] to be ω d Γ( d/α )(2 π ) d α . (cid:3) We next give the proof of Proposition 1.1.
Proof of Proposition 1.1.
By (1.4) we see that(2.5) p D ( t, x, x ) C e mt t − d/α = p ( t, C e mt t − d/α − r D ( t, x, x ) C e mt t − d/α . ince the first term tend to 1 as t → t d/α C e mt Z D r D ( t, x, x ) dx → , as t → . For q ≥
0, we define D q = { x ∈ D : δ D ( x ) ≥ q } . Then for 0 < t < , considerthe subdomain D t / α = { x ∈ D : δ D ( x ) ≥ t / α } and its complement D Ct / α = { x ∈ D : δ D ( x ) < t / α } . Recalling that | D | < ∞ , by Lebesgue dominated convergencetheorem we get | D Ct / α | → , as t →
0. Since p D ( t, x, x ) ≤ p ( t, x, x ), by (1.3) we seethat r D ( t, x, x ) C e mt t − d/α ≤ , for all x ∈ D . It follows that(2.7) t d/α C e mt Z D Ct / α r D ( t, x, x ) dx → , as t → . On the other hand, by Lemma 2.2 in [14] we obtain r D ( t, x, x ) C e mt t − d/α = E x [ p ( t − τ D , X τ D , x ); t ≥ τ D ] C e mt t − d/α ≤ c E y min (cid:26) t d/α | x − X ( τ D ) | d + α e − c | x − X ( τ D ) | , (cid:27) ≤ c min (cid:26) t d/α δ D ( x ) d + α e − c δ D ( x ) , (cid:27) . (2.8)For x ∈ D t / α and 0 < t <
1, the right hand side of (2.8) is bounded above by ct d/ α +1 / and hence(2.9) t d/α C e mt Z D / αt r D ( t, x, x ) dx ≤ ct d/ α +1 / | D | , and this last quantity goes to 0 as t → (cid:3) For an open set D ⊂ R d and x ∈ R d , the distribution P x ( τ D < ∞ , X ( τ D ) ∈ · ) willbe called the relativistic α − harmonic measure for D. The following Ikeda-Watanabeformula recovers the relativistic α − harmonic measure for the set D from the Greenfunction. Proposition 2.3 ([14]) . Assume that D is an open, nonempty, bounded subset of R d , and A is a Borel set such that dist ( D, A ) > . Then (2.10) P x ( X ( τ D ) ∈ A, τ D < ∞ ) = Z D G D ( x, y ) Z A v ( y − z ) dzdy, x ∈ D. Here we need the following generalization already stated and used in [3]. roposition 2.4. [14, Proposition 2.5] Assume that D is an open, nonempty, boundedsubset of R d , and A is a Borel set such that A ⊂ D c \ ∂D and ≤ t < t < ∞ , x ∈ D. Then we have P x ( X ( τ D ) ∈ A, t < τ D < t ) = Z D Z t t p D ( s, x, y ) ds Z A v ( y − z ) dzdy. The following propostition holds for a large class of L´evy processes
Proposition 2.5. [3, Proposition 2.3]
Let D and F be open sets in R d such that D ⊂ F. Then for any x, y ∈ R d we have p F ( t, x, y ) − p D ( t, x, y ) = E x ( τ D < t, X ( τ D ) ∈ F \ D ; p F ( t − τ D , X ( τ D ) , y )) . Lemma 2.6. [13, Lemma 5]
Let D ⊂ R d be an open set. For any x, y ∈ D and t > the following estimates hold; p D ( t, x, y ) ≤ e mt ˜ p D ( t, x, y ) r D ( t, x, y ) ≤ e mt ˜ r D ( t, x, y ) . (2.11)We need the following lemma given by Van den Berg in [5]. Lemma 2.7. [5, Lemma 5]
Let D be an open bounded set in R d with R-smoothboundary ∂D and for ≤ q < R denote the area of boundary of ∂D q by | ∂D q | . Then (2.12) (cid:18) R − qR (cid:19) d − | ∂D | ≤ | ∂D q | ≤ (cid:18) RR − q (cid:19) d − | ∂D | , ≤ q < R. Corollary 2.8. ( [3] , Corollary 2.14) Let D be an open bounded set in R d with R-smooth boundary. For any < q ≤ R we have(i) − d +1 | ∂D | ≤ | ∂D q | ≤ d − | ∂D | , (ii) | ∂D | ≤ d | D | R , (iii) (cid:12)(cid:12)(cid:12)(cid:12) | ∂D q | − | ∂D | (cid:12)(cid:12)(cid:12)(cid:12) ≤ d dq | ∂D | R ≤ d dq | D | R . Proof of main result
Proof of Theorem 1.4.
For the case t /α > R/ Z D ( t ) ≤ Z D p ( t, x, x ) dx ≤ c e mt | D | t d/α ≤ c e mt | D | t /α R t d/α . y Corollary 2.8 and Lemma 2.6 we also have C ( t ) | ∂D | ≤ C e mt | ∂D | t /α t d/α ≤ d C e mt | D | t /α Rt d/α ≤ d +1 C e mt | D | t /α R t d/α C ( t ) e mt | D | t d/α ≤ C e mt | D | t /α R t d/α . Therefore for t /α > R/ t /α ≤ R/
2. From (1.5) and the fact that p ( t, x, x ) = C ( t ) e mt t d/α , we have that Z D ( t ) − C ( t ) e mt | D | t d/α = Z D p D ( t, x, x ) dx − Z D p ( t, x, x ) dx = − Z D r D ( t, x, x ) dx, (3.1)where C ( t ) is as stated in the theorem. Therefore we must estimate (3.1). We breakour domain into two pieces, D R/ and its complement D CR/ . We will first considerthe contribution of D R/ . Claim 1:
For t /α ≤ R/ Z D R/ r D ( t, x, x ) dx ≤ ce mt | D | t /α R t d/α . Proof of Claim 1:
By Lemma 2.6 we have(3.3) Z D R/ r D ( t, x, x ) dx ≤ e mt Z D R/ ˜ r D ( t, x, x ) dx, and by scaling of the stable density the right hand side of (3.3) equals(3.4) e mt t d/α Z D R/ ˜ r D/t /α (1 , xt /α , xt /α ) dx. For x ∈ D R/ we have δ D/t /α ( x/t /α ) ≥ R/ (2 t /α ) ≥ . By [3, Lemma 2.1], we get˜ r D/t /α (cid:18) , xt /α , xt /α (cid:19) ≤ cδ d + αD/t /α ( x/t /α ) ≤ cδ D/t /α ( x/t /α ) ≤ ct /α R . Using the above inequality, we get Z D R/ r D ( t, x, x ) dx ≤ e mt t d/α Z D R/ ct /α R dx ≤ ce mt | D | t /α R t d/α , which proves (3.2).Now we will introduce the following notation. Since D has R -smooth boundary,for any point y ∈ ∂D there are two open balls B and B both of radius R suchthat B ⊂ D, B ⊂ R d \ ( D ∪ ∂D ) , ∂B ∩ ∂B = y. For any x ∈ D R/ there exist a nique point x ∗ ∈ ∂D such that δ D ( x ) = | x − x ∗ | . Let B = B ( z , R ) , B = B ( z , R )be inner/outer balls for the point x ∗ . Let H ( x ) be the half-space containing B suchthat ∂H ( x ) contains x ∗ and is perpendicular to the segment z z . We will need the following very important proposition in the proof of Theorem 1.2.Such a proposition has been proved for the stable process in [3, Proposition 3.1].
Proposition 3.1.
Let D ⊂ R d , d ≥ , be an open bounded set with R-smooth boundary ∂D . Then for any x ∈ D CR/ and t > such that t /α ≤ R/ we have (3.5) | r D ( t, x, x ) − r H ( x ) ( t, x, x ) | ≤ ce mt t /α Rt d/α (cid:18)(cid:18) t /α δ D ( x ) (cid:19) d + α/ − ∧ (cid:19) . Proof.
Exactly as in [3], let x ∗ ∈ ∂D be a unique point such that | x − x ∗ | = dist( x, ∂D )and B and B be balls with radius R such that B ⊂ D, B ⊂ R d \ ( D ∪ ∂D ) , ∂B ∩ ∂B = x ∗ . Let us also assume that x ∗ = 0 and choose an orthonormal coordinatesystem ( x , x , ..., x d ) so that the positive axis 0 x is in the direction of −→ p where p isthe center of the ball B . Note that x lies on the interval 0 p so x = ( | x | , , , ..., . Note also that B ⊂ D ⊂ ( B ) c and B ⊂ H ( x ) ⊂ ( B ) c . For any open sets A , A such that A ⊂ A we have r A ( t, x, y ) ≥ r A ( t, x, y ) so | r D ( t, x, x ) − r H ( x ) ( t, x, x ) | ≤ r B ( t, x, x ) − r ( B ) c ( t, x, x ) . So in order to prove the proposition it suffices to show that r B ( t, x, x ) − r ( B ) c ( t, x, x ) ≤ ce mt t /α Rt d/α (cid:18)(cid:18) t /α δ D ( x ) (cid:19) d + α/ − ∧ (cid:19) , for any x = ( | x | , , ..., , | x | ∈ (0 , R/ . Such an estimate was proved for the case m = 0 in [3]. In order to complete the proof it is enough to prove that r B ( t, x, x ) − r ( B ) c ( t, x, x ) ≤ ce mt n ˜ r B ( t, x, x ) − ˜ r ( B ) c ( t, x, x ) o . o show this given the ball B , we set U = ( B ) c . Now using the generalizedIkeda-Watanabe formula, Proposition (2.5) and Lemma (2.6) we have r B ( t, x, x ) − r U ( t, x, x )= E x [ t > τ B , X ( τ B ) ∈ U \ B ; p U ( t − τ B , X ( τ B ) , x )]= Z B Z t p B ( s, x, y ) ds Z U \ B v ( y − z ) p U ( t − s, z, x ) dzdy ≤ e mt Z B Z t ˜ p B ( s, x, y ) ds Z U \ B ˜ v ( y − z )˜ p U ( t − s, z, x ) dzdy ≤ ce mt E x h t > ˜ τ B , ˜ X (˜ τ B ) ∈ U \ B ; ˜ p U ( t − ˜ τ B , ˜ X (˜ τ B ) , x ) i = ce mt (˜ r B ( t, x, x ) − ˜ r U ( t, x, x )) ≤ ce mt t /α Rt d/α (cid:18)(cid:18) t /α δ D ( x ) (cid:19) d + α/ − ∧ (cid:19) . The last inequality follows by Proposition 3.1 in [3]. (cid:3)
Now using this proposition we estimate the contribution from D \ D R/ to the inte-gral of r D ( t, x, x ) in (3.1). Claim 2:
For t /α ≤ R/ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D \ D R/ r D ( t, x, x ) dx − Z D \ D R/ r H ( x ) ( t, x, x ) dx (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ce mt | D | t /α R t d/α . Proof of Claim 2:
By Proposition 3.1 the left hand side of (3.6) is bounded aboveby ce mt t /α Rt d/α Z R/ | ∂D q | (cid:18)(cid:18) t /α q (cid:19) d + α/ − ∧ (cid:19) dq. By Corollary 2.8, (i), the last quantity is smaller than or equal to ce mt t /α | ∂D | Rt d/α Z R/ (cid:18)(cid:18) t /α q (cid:19) d + α/ − ∧ (cid:19) dq. he integral in the last quantity is bounded by ct /α . To see this observe that since t /α ≤ R/ Z t /α (cid:18)(cid:18) t /α q (cid:19) d + α/ − ∧ (cid:19) dq + Z R/ t /α (cid:18)(cid:18) t /α q (cid:19) d + α/ − ∧ (cid:19) dq = Z t /α dq + Z R/ t /α (cid:18) t /α q (cid:19) d + α/ − dq ≤ ct /α . Using this and Corollary (2.8), (ii), we get (3.6).Recall that H = { ( x , · · · , x d ) ∈ R d : x > } . For abbreviation let us denote f H ( t, q ) = r H ( t, ( q, , · · · , , ( q, , · · · , , t, q > . Of course we have r H ( x ) ( t, x, x ) = f H ( t, δ H ( x )) . In the next step we will show that(3.7) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z D \ D R/ r H ( x ) ( t, x, x ) dx − | ∂D | Z R/ f H ( t, q ) dq (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ce mt | D | t /α R t d/α . We have Z D \ D R/ r H ( x ) ( t, x, x ) dx = Z R/ | ∂D q | f H ( t, q ) dq. Hence the left hand side of (3.7) is bounded above by Z R/ || ∂D q | − | ∂D || f H ( t, q ) dq. By Corollary 2.8, (iii), this is smaller than c | D | R Z R/ qf H ( t, q ) dq ≤ c | D | e mt R Z R/ q ˜ f H ( t, q ) dq = c | D | e mt R Z R/ qt − d/α ˜ f H (1 , qt − /α ) dq = c | D | e mt R t d/α Z R/ t /α qt /α ˜ f H (1 , q ) dq ≤ c | D | e mt t /α R t d/α Z ∞ q (cid:0) q − d − α ∧ (cid:1) dq ≤ c | D | e mt t /α R t d/α . his shows (3.7). Finally, we have (cid:12)(cid:12)(cid:12)(cid:12) | ∂D | Z R/ f H ( t, q ) dq − | ∂D | Z ∞ f H ( t, q ) dq (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ∂D | Z ∞ R/ f H ( t, q ) dq ≤ c | D | R Z ∞ R/ f H ( t, q ) dq by Corollary 2.8, (ii) ≤ c | D | e mt Rt d/α Z ∞ R/ ˜ f H (1 , qt − /α ) dq = c | D | e mt t /α Rt d/α Z ∞ R/ t /α ˜ f H (1 , q ) dq. Since R/ t /α ≥ , so for q ≥ R/ t /α ≥ f H (1 , q ) ≤ cq − d − α ≤ cq − . Therefore, Z ∞ R/ t /α ˜ f H (1 , q ) dq ≤ c Z ∞ R/ t /α dqq ≤ ct /α R .
Hence,(3.8) (cid:12)(cid:12)(cid:12)(cid:12) | ∂D | Z R/ f H ( t, q ) dq − | ∂D | Z ∞ f H ( t, q ) dq (cid:12)(cid:12)(cid:12)(cid:12) ≤ c | D | e mt t /α R t d/α . Note that the constant C ( t ) which appears in the formulation of Theorem 1.4satisfies C ( t ) = R ∞ f H ( t, q ) dq. Now equations (3.1), (3.2), (3.6), (3.7), (3.8) give(1.11). (cid:3)
References ∼ ∼ banuelos/Lectures/SPA2010.pdf[3] R. Ba˜nuelos, T. Kulczycki. Trace estimates for stable processes. Probab. Theory and Rel. Fields. (2008), no. 3-4, 313-338.[4] R. Ba˜nuelos, T. Kulczycki and B. Siudeja. On the Trace of symmetric stable processes onLipschitz domains, J. Funct. Anal. (2009), no. 10, 33293352.[5] M. van den Berg. On the asymptotics of the heat equation and bounds on traces associatedwith Dirichlet Laplacian. J. Funct. Anal. (1987) 279-293[6] R.M. Blumenthal, R.K. Getoor, and D.B. Ray. On the distribution of first hits for the symmetricstable processes. Trans. Amer. Math. Soc. (1961), 540554.[7] K. Bogdan. The boundary Harnack principle for the fractional Laplacian. Stud. Math. (1986) 103-122.
8] J. Brossard and R. Carmona. Can one hear the dimension of a fractal? Comm. Math. Phys. (1986) 103-122.[9] Z-Q. Chen, P. Kim, R. Song. Green function estimates for relativistic stable processes in half-space-like open sets.Stochastic Process. Appl. (2011), no. 5, 11481172.[10] E.B. Davies. Heat Kernels and Spectral Theory. Cambridge Univ. Press, Cambridge, (1989).[11] W. Feller. An Introduction to Probability Theory and its Applications. Vol. II. Wiley, New York(1971).[12] N. Ikeda and S. Watanabe. : On some relations between the harmonic measure and the L´evymeasure for a certain class of Markov processes, J. Math. Kyoto Univ. (1962), 79-95.[13] M. Ryznar (2002) Estimates of Green function for relativistic α -stable process, Pot. Anal. (2006), 5025-5057. Rodrigo Ba˜nuelos, Department of Mathematics Purdue University 150 North Uni-versity Street West Lafayete, Indiana 47907-2067
E-mail address : [email protected] URL : Jebessa Mijena, 221 Parker Hall, Department of Mathematics and Statistics,Auburn University, Auburn, Al 36849
E-mail address : [email protected] Erkan Nane, 221 Parker Hall, Department of Mathematics and Statistics, AuburnUniversity, Auburn, Al 36849
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