HEAT TRACE OF NON-LOCAL OPERATORS

HEAT TRACE OF NON-LOCAL OPERATORS

arXiv:1201.2171v1 [math.SP] 10 Jan 2012

˜ RODRIGO BANUELOS AND SELMA YILDIRIM YOLCU A BSTRACT. This paper extends results of M. van den Berg on two-term asymptotics for the trace of Sch¨odinger operators when the Laplacian is replaced by non-local (integral) operators corresponding to rotationally symmetric stable processes and other closely related L´evy processes.

C ONTENTS 1. Introduction 2. Stable processes, statement of results 3. Proof of Theorems 2.1 and 2.2 4. Alternative proofs of Theorems 2.1 and 2.2 5. Extension to other non-local operators References

1 2 5 9 11 15

1. I NTRODUCTION There is an extensive literature of trace formulæ for heat kernels of the Laplacian and its Schr¨odinger perturbations in spectral and scattering theory. For instance, van den Berg [6] and later Ba˜nuelos and S´a Barreto [5], explicitly compute several coefficients in the asymptotic expansion of the trace of the heat kernel of the Schr¨odinger operator −∆ + V as t ↓ 0 for potentials V ∈ S(Rd ), the class of rapidly decaying functions at infinity. In particular, in [5] a general formula is obtained for these coefficients by using elementary Fourier transform methods. For applications of these results to problems in scattering theory, see [5] and references therein. The results in [5] were extended by Donnelly [20] to certain compact Riemannian manifold. Heat asymptotic results have also been widely used in statistical mechanics, for details we refer the reader to the article [26] of E. H. Lieb on the second virial coefficient of a hard-sphere gas at low temperatures, and to the article [28] of M. D. Penrose, O. Penrose and G. Stell on the sticky spheres in quantum mechanics. We also refer the reader to Datchev and Hezari’s overview article [24] for various other related spectral asymptotic results and applications. For non-local (integral) operators which arise by replacing the Brownian motion with other L´evy processes, many questions concerning spectral asymptotics are wide open at this point. These include, for example, the Weyl two-term asymptotics for the spectral counting function of the fractional Laplacian with Dirichlet boundary conditions and the McKean-Singer [27] result involving the Euler characteristic of the domain in the third-term asymptotics of the trace of the heat semigroup. On the other hand, a two-term trace asymptotics involving the volume of the domain and the surface area of the boundary is proved in Ba˜nuelos and Kulczycki [3] (for C 1,1 domains) and in Ba˜nuelos, Kulczycki and Siudeja [4] (for Lipschitz domains) for the fractional Laplacian associated with symmetric stable processes. These results are in parallel to the results of van den Berg [7] and Brown [10] for the R. Ba˜nuelos is supported in part by NSF Grant # 0603701-DMS. 1

˜ RODRIGO BANUELOS AND SELMA YILDIRIM YOLCU

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Laplacian. For further recent work on the Dirichlet case in domains of Euclidean space, see Frank and Geisinger, [22] and [23]. The purpose of this paper is to obtain analogues of the van den Berg result in [6] for the fractional Laplacian ∆α/2 associated with symmetric α-stable processes, 0 < α < 2, in Rd and for other closely related non-local operators corresponding to sums of stable processes and the relativistic Brownian motion. These are all L´evy processes which are obtained by subordination of Brownian motion. 2. S TABLE

PROCESSES , STATEMENT OF RESULTS

Before we state our results precisely, we recall the basic definitions and elementary notions related to stable processes. Let Xt be the d-dimensional symmetric α–stable process of order α ∈ (0, 2] (α) in Rd . The process Xt has stationary independent increments and its transition density pt (x, y) = (α) pt (x − y), t > 0, x, y ∈ Rd , is determined by its Fourier transform (characteristic function) Z (α) −t|ξ|α iξ·Xt t > 0, ξ ∈ Rd . eiξ·y pt (y)dy, (2.1) e = E(e )= Rd

Px

Ex

Denoting by and the probability and expectation, respectively, of this process starting at x we have that any Borel subset B ⊂ Rd , x ∈ Rd , t > 0, Z (α) x pt (x − y)dy, P (Xt ∈ B) = B

where (α) pt (x)

(2.2) Here we can also write

(α)

pt (x) =

(2.3)

1 = (2π)d Z

∞ 0

Z

α

e−ix·ξ e−t|ξ| dξ. Rd

−|x|2 α/2 1 4s η e t (s) ds, (4πs)d/2

α/2

where ηt (s) is the density for the α/2-stable subordinator. While explicit formula for the transition density of symmetric α-stable processes are only available for α = 1 (the Cauchy process) and α = 2 (the Brownian motion), these processes share many of the basic properties of Brownian motion. It also (α) follows trivially from (2.3) that pt (x) is radial, symmetric and deceasing in x. Thus exactly as in the Brownian motion case we have (2.4)

(α)

(α)

pt (x) ≤ t−d/α p1 (0). (α)

In fact, if we denote by ωd the surface area of the unit sphere in Rd we can compute p1 (0) more explicitly. Z Z ∞ d ωd 1 (α) −|ξ|α e dξ = e−s s( α −1) ds p1 (0) = d d (2π) Rd (2π) α 0 ωd Γ(d/α) (2.5) . = (2π)d α Throughout this paper we will deal with radial transition functions and use the notation (α)

(α)

pt (x, y) = pt (x − y).

HEAT TRACE OF NON-LOCAL OPERATORS

3

(α)

(α)

In particular, pt (x, x) = pt (0). Of importance for us in this paper is the scaling property of these processes. More precisely, by a simple change of variables in (2.2), we see that these processes are self-similar with scaling (α)

(α)

pt (x, y) = t−d/α p1 (t−1/α x, t−1/α y).

(2.6)

(α)

The transition densities pt (x) satisfy the following well-known two sided inequality valid for all x ∈ Rd and t > 0, (2.7)

−1 Cα,d

 t−d/α ∧

t |x|d+α



≤

(α) pt (x)



≤ Cα,d t

−d/α

∧

t |x|d+α



,

where the constant Cα,d only depends on α and d. Here and throughout the paper we use the notation a ∧ b = min{a, b} and a ∨ b = max{a, b} for any a, b ∈ R. For rapidly decaying functions f ∈ S(Rd ), we have the semigroup of the stable processes defined via the the Fourier inversion formula by

Tt f (x) = E x [f (Xt )] = E 0 [f (Xt + x)] Z f (x + y)pt (dy) = pt ∗ f (x) = Rd Z 1 α = eix·ξ e−t|ξ| fb(ξ)dξ. d (2π) Rd

By differentiating this at t = 0 we see that its infinitesimal generator is ∆α/2 in the sense that \ α/2 f (ξ) = −|ξ|α fb(ξ). This is a non-local operator such that for suitable test functions, including all ∆ functions in f ∈ C0∞ (Rd ), we can define it as the principle value integral (2.8)

∆

α/2

f (x) = Ad,α lim

ǫ→0+

Z

{|y|>ǫ}

f (x + y) − f (x) dy, |y|d+α

where Ad,α

Γ d−α = α d/2 2 2 π Γ




α 2


.

We will denote by H0 the fractional Laplacian operator ∆α/2 , α ∈ (0, 2] and by H its Schr¨odinger perturbation H = ∆α/2 + V , where V ∈ L∞ (Rd ). We let e−tH and e−tH0 be the associated heat (α) denote their corresponding transition densities (heat kernels). In fact, semigroups and let pH t and pt (α) pt is just as in (2.1) and the Feynman-Kac formula gives

(2.9)

  Rt (α) − 0 V (Xs )ds t , pH t (x, y) = pt (x, y)Ex,y e

˜ RODRIGO BANUELOS AND SELMA YILDIRIM YOLCU

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t is the expectation with respect to the stable process (bridge) starting at x conditioned to be where Ex,y at y at time t. The main object of study in this paper is the trace difference Z (α) (pH T r(e−tH − e−tH0 ) = t (x, x) − pt (x, x))dx d R Z   Rt (α) t (2.10) e− 0 V (Xs )ds − 1 dx Ex,x = pt (0) Rd Z   Rt (α) t −d/α e− 0 V (Xs )ds − 1 dx, Ex,x = t p1 (0) Rd

(α)

where p1 (0) is the dimensional constant given by the right hand side of (2.5). Before we go further let us observe that this quantity is well defined for all t > 0, provided V ∈ ∞ L (Rd ) ∩ L1 (Rd ). Indeed, the elementary inequality |ez − 1| ≤ |z|e|z| immediately gives that Z  Z t Z  Rt  t − V (X )ds tkV k t s ∞ |V (Xs )|ds dx. Ex,x − 1 dx ≤ e Ex,x e 0 Rd

Rd

However, (2.11)

t Ex,x

Z

t

0

|V (Xs )|ds



Z

=

t

0

t Ex,x |V (Xs )|ds

Z tZ

=

0

0

(α)

(α)

ps (x, y)pt−s (y, x) (α)

pt (x, x)

Rd

(α)

|V (y)|dyds.

(α)

The Chapman–Kolmogorov equations and the fact that pt (x, x) = pt (0, 0) give that Z (α) (α) ps (x, y)pt−s (y, x) dx = 1 (α) Rd pt (x, x) and hence (2.12) It follows then that (2.13)

Z

Rd

t Ex,x

Z

0



t

|V (Xs )|ds dx = tkV k1 .

T r(e−tH − e−tH0 ) ≤ t−d/α+1 p(α) (0)kV k1 etkV k∞ , 1

valid for all t > 0 and all potentials V ∈ L∞ (Rd ) ∩ L1 (Rd ). The previous argument also shows that for all potentials V ∈ L∞ (Rd ) ∩ L1 (Rd ), k Z t Z ∞ X
(−1)k (α) −tH −tH0 t (2.14) Tr e −e = pt (0) V (Xs )ds dx, Ex,x k! 0 Rd k=1

where the sum is absolutely convergent for all t > 0. We have the following two Theorems which parallel the results in van den Berg [6] for the Laplacian. Note however, that here (ii) in Theorem 2.1 (as well as Theorem 2.2) is more general as we do not assume that the potential is nonnegative. Theorem 2.1. (i) Let V : Rd → (−∞, 0], V ∈ L∞ (Rd ) ∩ L1 (Rd ). Then for all t > 0   1 2 (α) tkV k∞ α −tH −tH0 . (2.15) pt (0)tkV k1 ≤ T r(e −e ) ≤ pt (0) tkV k1 + t kV k1 kV k∞ e 2

HEAT TRACE OF NON-LOCAL OPERATORS

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In particular

(2.16)


(α) T r(e−tH − e−tH0 ) = pt (0) tkV k1 + O(t2 )


(α) = t−d/α p1 (0) tkV k1 + O(t2 ) ,

as t ↓ 0. (ii) If we only assume that V ∈ L∞ (Rd ) ∩ L1 (Rd ), then for all t > 0, Z (α) (α) −tH −tH0 V (x)dx ≤ pt (0)Ct2 kV k1 kV k∞ etkV k∞ , −e ) + pt (0)t (2.17) T r(e Rd

for some universal constant C. From this we conclude that   Z (α) 2 −tH −tH0 V (x)dx + O(t ) , ) = pt (0) −t (2.18) T r(e −e Rd

as t ↓ 0.

Theorem 2.2. Suppose V ∈ L∞ (Rd ) ∩ L1 (Rd ) and that it is also uniformly H¨older continuous of order γ (there exists a constant M ∈ (0, ∞) such that |V (x) − V (y)| ≤ M |x − y|γ , for all x, y ∈ Rd ) with 0 < γ < α ∧ 1, whenever 0 < α ≤ 1, and with 0 < γ ≤ 1, whenever 1 < α < 2. Then for all t > 0, Z Z
1 (α) (α) −tH −tH 2 2 0 T r(e V (x)dx − pt (0) t −e ) + pt (0)t |V (x)| dx 2 Rd Rd   (α) (2.19) ≤ Cα,γ,d kV k1 pt (0) |V k2∞ etkV k∞ t3 + tγ/α+2 ,

where the constant Cα,γ,d depends only on α, γ and d. In particular,   Z Z 1 2 (α) 2 γ/α+2 −tH −tH0 |V (x)| dx + O(t ) , V (x)dx + t (2.20) T r(e −e ) = pt (0) −t 2 Rd Rd as t ↓ 0.

Theorems 2.1 and 2.2 are proved in §2. In §3, we provide extensions to other non-local operators. 3. P ROOF

OF

T HEOREMS 2.1

AND

2.2

Proof of Theorem 2.1. Setting a=

Z

t

V (Xs )ds, 0

and

b = tkV k∞ ,

we observe that −b ≤ a ≤ 0. We then use the following elementary inequality from [6]   1 b −a (3.1) − a ≤ e − 1 ≤ −a 1 + be 2 to conclude that   Z t   Z t  Rt 1 tkV k∞ − 0 V (Xs )ds . V (Xs )ds 1 + tkV k∞ e −1 ≤ − V (Xs )ds ≤ e − 2 0 0

t Taking expectations of both sides of this inequality with respect to Ex,x and then integrating on Rd with respect to x, it follows exactly as in the derivation of the general bound (2.13), that (2.15) holds. This concludes the proof of (i) in Theorem 2.1.

˜ RODRIGO BANUELOS AND SELMA YILDIRIM YOLCU

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For (ii), we observe that by (2.14) we have (again using (2.12)) Z (α) −tH −tH0 (3.2) V (x)dx −e ) + pt (0)t T r(e Rd

Z ∞ Z t k X 1 (α) t Ex,x V (Xs )ds dx ≤ pt (0) k! Rd 0 k=2  Z t Z ∞ k−1 k−1 X t kV k∞ (α) ≤ pt (0) Ex,x |V (Xs )|ds dx k! Rd 0 k=2

(α)

= pt (0)tkV k1

∞ k−1 X t kV kk−1 ∞

k=2

k!

(α)

≤ Cpt (0)t2 kV k1 kV k∞ etkV k∞ ,

for some absolute constant C. This proves (ii) and concludes the proof of the theorem.



Remark 3.1. We observe that the above holds for any operator which arises from subordination of a Brownian motion. For such operators the transition probabilities pt are symmetric radial and decreasing, hence pt (x, x) = pt (0). In section 5 below we give examples of other nonlocal operators for which Theorem 2.2 also holds. Proof of Theorem 2.2. We begin by observing that (exactly as in the proof of inequality (3.2)) we have 2 Z t Z t Rt 1 − 0 V (Xs )ds V (Xs )ds − −1+ V (Xs )ds e 2 0 0 Z t (3.3) |V (Xs )|ds, ≤ C(tkV k∞ )2 etkV k∞ 0

t and then intefor some constant C. By taking expectation of both sides of (3.3) with respect to Ex,x grating with respect to x, we obtain

2 ! Z t Z t Rt 1 − 0 V (Xs )ds V (Xs )ds − −1+ V (Xs )ds dx e 2 d 0 R 0  Z t Z t Ex,x |V (Xs )|ds dx ≤ C(tkV k∞ )2 etkV k∞ Z

t Ex,x

Rd

2 tkV k∞

= C(tkV k∞ ) e

0

tkV k1 ,

where we use again (2.12). Returning to the definition of the trace differences in (2.10) we see that this leads to 2 ! Z t Z Z 1 1 −tH −tH0 t V (x)dx − dx −e V (Xs )ds )) + t Ex,x (α) (T r(e p (0) 2 Rd 0 Rd t (3.4)

≤ C(tkV k∞ )2 etkV k∞ tkV k1 .

t ([·]2 ). Since V is uniformly H¨ older with exponent γ and constant It remains to estimate the term Ex,x M , we have

(3.5)

|V (Xs + x) − V (x)| ≤ M |Xs |γ .

HEAT TRACE OF NON-LOCAL OPERATORS

7

Hence, 2 2 Z t Z t 2 Z t t t V (x)ds V (Xs )ds − V (Xs )ds − t2 V 2 (x) = Ex,x Ex,x 0 0 0 !     Z Z 2 2 t t t V (x)ds V (Xs )ds − = Ex,x 0 0  Z t t (3.6) (V (Xs + x) − V (x))ds · = E0,0 ( 0  Z t V (Xs + x) + V (x))ds ). 0

By employing (3.5), we obtain   Z t 2 ! Z t Z t t t 2 2 γ (|V (Xs + x)| + |V (x)|) ds . − t V (x) ≤ M E0,0 |Xs | ds V (Xs )ds Ex,x 0 0 0

Integrating both sides of this inequality with respect to x and using Fubini’s theorem, we have that the value of the second integral becomes 2tkV k1 . Thus we arrive at Z 2 ! Z t Z 2 2 t (3.7) dx − t |V (x)| dx V (Xs )ds Ex,x Rd d R 0  Z t t ≤ 2tM kV k1 E0,0 |Xs |γ ds . 0

Now, it remains to estimate the expectation on the right side of (3.7). As in (2.11) we have  Z t Z t t t γ (3.8) E0,0 (|Xs |γ )ds E0,0 |Xs | ds = 0

0

=

Z tZ 0

(α)

(α)

ps (0, y)pt−s (y, 0) Rd

(α) pt (0, 0)

|y|γ dyds.

To estimate the right hand side we recall the following “5P-inequality” (α)

(3.9)

(α)

ps (x, z)pt (z, y) (α)

ps+t (x, y)

  (α) (z, y) , ≤ Cα,d p(α) (x, z) + p s t

valid for all x, y, z ∈ Rd , s, t > 0 and 0 < α < 2, where Cα,d is a constant depending only on α and d. This inequality is proved by Bogdan and Jakubowski in [9, p.182]. It is derived form the the “3P-inequality” (3.10)

(α)

(α)

p(α) s (x, z) ∧ pt (z, y) ≤ Cα,d ps+t (x, y). (α)

Inequality (3.10) itself is proved by a simple computation replacing ps with the quantity from (2.7). From (3.10) and the fact that for a, b ≥ 0 we have ab = (a ∧ b)(a ∨ b) and (a ∨ b) ≤ (a + b)), (3.9) follows immediately.

˜ RODRIGO BANUELOS AND SELMA YILDIRIM YOLCU

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Returning to (3.8), we have,  Z t Z tZ (α) γ t γ (p(α) E0,0 |Xs | ds ≤ Cα,d s (0, y) + pt−s (y, 0))|y| dyds Rd

0

0

= 2Cα,d = 2Cα,d = 2Cα,d

Z tZ Z

Z

Rd

0

t

0 t 0

γ p(α) s (0, y)|y| dyds

E 0 (|Xs |γ )ds sγ/α E 0 (|X1 |γ )ds

tγ/α+1 , γ/α + 1 where we used the scaling property of the stable process coming from (2.6). We now recall that E 0 (|X1 |γ ) is finite under our assumption that γ < α. (This fact follows trivially from (2.7).) Thus we see that  Z t t γ (3.11) E0,0 |Xs | ds ≤ Cα,γ,d tγ/α+1 , = 2Cα,d E 0 (|X1 |γ )

0

where the constant Cα,γ,d depends only on α, γ and d. We conclude that Z  2 ! Z t Z t Z t γ 2 2 t |Xs | ds dx − t |V (x)| dx ≤ 2tM kV k1 E0,0 V (Xs )ds E Rd x,x 0 Rd 0 ≤ M kV k1 Cα,γ,d tγ/α+2 .

(3.12)

Returning to (3.4) we find that Z 1
−tH −tH0 −e ) +t pα (0) T r(e t

3

≤ Ct kV

(3.13)

1 V (x)dx − t2 2 Rd

k2∞ etkV k∞ kV k1

≤ Cα,γ,d kV k1



+ M kV k1 Cα,γ,d t  |V k2∞ etkV k∞ t3 + tγ/α+2 .

γ/α+2

Z

|V (x)| dx d 2

R

Rewriting this in the form stated in Theorem 2.2 we arrive at the announced bound

(3.14)

Z Z
(α) 21 2 T r(e−tH − e−tH0 ) + pαt (0)t (0)t V (x)dx − p |V (x) dx t 2 Rd Rd   (α) ≤ Cα,γ,d kV k1 pt (0) |V k2∞ etkV k∞ t3 + tγ/α+2 ,

valid for all t > 0.



Remark 3.2. We remark that in the case of Brownian motion, α = 2, one may use the fact (as done in [6]) that (2)

(3.15)

(2)

ps (0, y)pt−s (y, 0) (2) pt (0, 0)

(2)

= p s (t−s) (0, y) t

to explicitly compute the right hand side of (3.8). While the inequality (3.9) does not hold for α = 2, we may estimate the right hand side of (3.8) in the case of Brownian motion without the explicit estimate that comes from (3.15). Indeed, by observing that in law the Brownian bridge started at 0 and

HEAT TRACE OF NON-LOCAL OPERATORS

9

conditioned to return to 0 in time t is the same as Bs − st Bt , where Bs is standard Brownian motion, a simple computation leads to the same conclusion as (3.11) for α = 2. Indeed,  Z t Z t s t γ E0,0 E 0 |Bs − Bt |γ ds |Bs | ds = t 0 0 Z t    s γ (3.16) |Bt |γ ds E 0 |Bs |γ + ≤ Cγ t 0 Z t  s γ  sγ/2 + = Cγ E 0 |B1 |γ tγ/2 ds t 0 = Cγ,d tγ/2+1 ,

where Cγ,d depends only on γ and d. Unfortunately, the simple path construction Xs − st Xt only leads to the stable bridge when α = 2 and we are not able to repeat (3.16) in the case 0 < α < 2, hence our use of (3.9). However, it may be that the construction by Chaumont in [12] (or perhaps some of the estimates of Fitzsimmons and Getoor in [21]) can be used to bypass the estimate (3.9). 4. A LTERNATIVE

PROOFS OF

T HEOREMS 2.1

AND

2.2

Recalling formula (2.14) we have (4.1)

−tH

Tr e

−tH0

−e




=

(α) pt (0)

Z ∞ X (−1)k k=1

k!

Rd

t Ex,x

Z

t

V (Xs )ds

0

k

dx,

where as mentioned there the sum is absolutely convergent for all potentials V ∈ L∞ (Rd ) ∩ L1 (Rd ) and all t > 0. We note also that in fact, for every N ≥ 1, (4.2)

T r e−tH

! k Z t N k Z X
(−1) (α) t − e−tH0 = pt (0) Ex,x V (Xs )ds dx + 0(tN +1 ) , k! Rd 0 k=1

as t ↓ 0. This follows from the fact that k Z t Z ∞ X (−1)k t (4.3) Ex,x V (Xs )ds dx ≤ k! Rd 0 k=N +1

≤

Next observe that

Z

t

0

where Jk =

Z

0≤s1 ≤s2 ≤···≤sk ≤t

V (Xs )ds

k

∞ X

1 k k−1 t kV k∞ kV k1 k!

k=N +1 tN +1 kV kN ∞ kV k1 tkV k∞

(N + 1)!

e

= k!Jk

V (Xs1 )ds1 V (Xs2 )ds2 · · · V (Xsk )dsk .

Recalling that the finite dimensional distributions of the stable bridge are given by t Px,x {Xs1 ∈ dy1 , Xs2 ∈ dy2 , . . . , Xsk ∈ dyk } =

where y0 = yk+1 = x, s0 = 0, sk+1 = t, we can write

1

k Y

(α) psj+1 −sj (yj+1 (α) pt (x, x) j=0

− yj ),

˜ RODRIGO BANUELOS AND SELMA YILDIRIM YOLCU

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∞
X T r e−tH − e−tH0 = (−1)k

(4.4)

k=1

where

I˜k (s1 , s2 , . . . , sk ) =

Z

k Y

(Rd )k+1 j=0

Z

0≤s1 ≤s2 ≤···≤sk ≤t

(α)

psj+1 −sj (yj+1 − yj )

I˜k (s1 , s2 , . . . , sk )ds1 ds2 · · · dsk ,

k Y

j=1

V (yj )dy1 dy2 . · · · dyk dx. (α)

(α)

Here (as several times before) we used the fact that pt (x, x) = pt (0). By Fubini’s theorem we may integrate with respect to x first and use the fact that Z (α) (α) p(α) s1 (y1 − x)pt−sk (yk − x)dx = pt−(sk −s1 ) (yk − y1 ) Rd

to arrive at (4.5)

−tH

Tr e

where

−tH0

−e




=

Ik (s1 , s2 , . . . , sk ) =

∞ X

k

(−1)

k=1

Z

(Rd )k



k−1 Y



j=1

Z



0≤s1 ≤s2 ≤···≤sk ≤t

Ik (s1 , s2 , . . . , sk )ds1 ds2 · · · dsk ,

 (α) pt−(sk −s1 ) (yk − y1 )V (y1 ) · 

(α) psj+1 −sj (yj+1 − yj )V (yj+1 ) dy1 dy2 · · · dyk .

We can then also re-write the right hand side of (4.2) as ! N k Z X (−1) (α) Ik (s1 , s2 , . . . , sk )ds1 ds2 · · · dsk + 0(tN +1 ) . (4.6) pt (0) (α) 0≤s ≤s ≤···≤s ≤t 1 2 k k=1 pt (0)

From these formulas we can also compute coefficients in the asymptotic expansion of the trace as t ↓ 0 fairly directly and in particular give alternative proofs of Theorems 2.1 and 2.2. If k = 1, we see that (α) (α) pt−(sk −s1 ) (yk − y1 ) = pt (0) and

I1 (s1 ) =

(α) pt (0)

Z

V (y)dy, Rd

which immediately leads to (2.18) in Theorem 2.1. If k = 2, we write Z Z (α) (α) (4.7) pt−(s2 −s1 ) (y2 − y1 )p(s2 −s1 ) (y2 − y1 )V (y1 )V (y2 )dy1 dy2 I2 (s1 , s2 ) = d d R R Z Z (α) (α) pt−(s2 −s1 ) (y2 − y1 )p(s2 −s1 ) (y2 − y1 ) (V (y1 ) − V (y2 ) + V (y2 )) V (y2 )dy1 dy2 = d d ZR ZR (α) (α) pt−(s2 −s1 ) (y2 − y1 )p(s2 −s1 ) (y2 − y1 )V 2 (y2 )dy1 dy2 = d d ZR ZR (α) (α) pt−(s2 −s1 ) (y2 − y1 )p(s2 −s1 ) (y2 − y1 ) (V (y1 ) − V (y2 )) V (y2 )dy1 dy2 . + Rd

Rd

HEAT TRACE OF NON-LOCAL OPERATORS

11

Integrating first with respect to y1 we conclude that Z Z Z (α) (α) (α) 2 pt−(s2 −s1 ) (y2 − y1 )p(s2 −s1 ) (y2 − y1 )V (y2 )dy1 dy2 = pt (0) Rd

Rd

Rd

|V (y)|2 dy.

On the other hand, |V (y1 ) − V (y2 )| ≤ M |y1 − y2 |γ an hence Z Z (α) (α) (4.8) d d pt−(s2 −s1 ) (y2 − y1 )p(s2 −s1 ) (y2 − y1 ) (V (y1 ) − V (y2 )) V (y2 )dy1 dy2 R R Z Z (α) (α) pt−(s2 −s1 ) (y2 − y1 )p(s2 −s1 ) (y2 − y1 )|y1 − y2 |γ |V (y2 )|dy1 dy2 ≤ M d d R  ZR Z (α) (α) γ pt−(s2 −s1 ) (y2 − y1 )p(s2 −s1 ) (y2 − y1 )|y2 − y1 | dy1 |V (y2 )|dy2 = M d Rd   Z RZ (α) (α) γ |V (y)|dy . pt−(s2 −s1 ) (y)p(s2 −s1 ) (y)|y| dy = M Rd

Rd

From inequality (3.9) we again obtain Z Z (α) (α) (α) γ pt−(s2 −s1 ) (y)p(s2 −s1 ) (y)|y| dy ≤ Cα,d pt (0) Rd

Thus

Z

I2 (s1 , s2 )ds1 ds2 =

0≤s1 ≤s2 ≤t

where

|Rt | ≤ Cα,d M kV (4.9)

(α) k1 pt (0)

≤ Cα,γ,d M kV

Z

0≤s1 ≤s2 ≤t (α) k1 pt (0)tγ/α+2 .

Rd

  (α) (α) pt−(s2 −s1 ) (y) + p(s2 −s1 ) (y) |y|γ dy.

(α) pt (0)t2

Z

Rd

|V (y)|2 dy + Rt


E 0 |Xt−(s2 −s1 ) |γ + E 0 |X(s2 −s1 ) |γ ds1 ds2

As in (3.13), this gives another proof of (2.20) in Theorem 2.2. 5. E XTENSION

TO OTHER NON - LOCAL OPERATORS

Taking 0 < β < α < 2 and a ≥ 0, we consider the process Zta = Xt + aYt , where Xt and Yt are independent α-stable and β-stable processes, respectively. This process is called the independent sum of the symmetric α-stable process X and the symmetric β-stable process Y with weight a. The infinitesimal generator of Z a is ∆α/2 +aβ ∆β/2 . This again is a non-local operator. Acting on functions f ∈ C0∞ (Rd ) we have Z   ∆α/2 + aβ ∆β/2 f (x) = Ad,α lim ǫ→0+

{|y|>ǫ}



1 aβ + |y|d+α |y|d+β



[f (x + y) − f (x)]dy,

where Ad,α is as in (2.8). Properties of the heat kernel (transition probabilities) for this operator have been studied by several people in recent years. See for example Chen and Kumagai [19], Chen, Kim and Song [17] or Jakubowski and Szczypkowski [25], and references given there. If we denote the heat kernel of this operator by pat (x), we have that Z 1 α β β a pt (x) = (5.1) e−ix·ξ e−t(|ξ| +a |ξ| ) dξ (2π)d Rd Z ∞ −|x|2 1 a 4s η (s) ds, (5.2) = e t (4πs)d/2 0

12

˜ RODRIGO BANUELOS AND SELMA YILDIRIM YOLCU

where ηta (s) is be the density function of the sum of the α/2-stable subordinator and a2 -times the β/2stable subordinator. Again, this density is radial, symmetric, and decreasing in |x|. It is proved in [19] and [17] that there is a constant Cα,β,d depending only on the parameters indicated such that for all x ∈ Rd and t > 0, (5.3) where

−1 fta (x) ≤ pat (x) ≤ Cα,β,d fta (x) Cα,β,d

 aβ t . |x|d+α |x|d+β We note that for a = 0 this is just the estimate in (2.7). For any γ > 0 with 0 < γ < β < α we have for any t > 0,  Z t Z t Z t 0 a γ 0 γ γ 0 γ E (|Zs | )ds ≤ Cγ E (|Xs | )ds + a E (|Ys | )ds 0 0 0 ! γ/α+1 γ/β+1 t t = Cγ E 0 (|X1 |γ ) + aγ E 0 (|Y1 |γ ) (5.4) γ/α + 1 γ/β + 1 ! tγ/β+1 tγ/α+1 + . = Ca,α,β,d γ/α + 1 γ/β + 1    fta (x) = (aβ t)−d/β ∧ t−d/α ∧

t

+

It also follows from the “3P-inequality” (24) in [25] (using again the fact that for any a, b ≥ 0 we have ab = (a ∧ b)(a ∨ b) and (a ∨ b) ≤ (a + b)) that we have the following “5P-inequality” for this process: (5.5)

pas (x, z)pat (z, y) ≤ Ca,α,β,d (pas (x, z) + pat (z, y)) , pas+t (x, y)

valid for all x, y, z ∈ Rd and all s, t > 0. With (5.4) and (5.5), both proofs above for Theorem 2.2 can be repeated to obtain the same result for the operator ∆α/2 + aβ ∆β/2 , replacing ∆α/2 . More precisely we have Theorem 5.1. Let a ≥ 0, 0 < β < α < 2 and let H0a = ∆α/2 + aβ ∆β/2 . Suppose V ∈ L∞ (Rd ) ∩ L1 (Rd ) and that it is also uniformly H¨older continuous of order γ, with 0 < γ < β ∧ 1, whenever 0 < β ≤ 1, and with 0 < γ ≤ 1, whenever 1 < β < 2. Let H a = ∆α/2 + aβ ∆β/2 + V . Then for all t > 0, Z Z
1 2 2 a T r(e−tH a − e−tH0a ) + pat (0)t |V (x)| dx V (x)dx − pt (0) t 2 Rd Rd   (5.6) ≤ Ca,α,β,γ,d kV k1 pat (0) |V k2∞ etkV k∞ t3 + tγ/α+2 + tγ/β+2 ,

where the constant Ca,α,β,γ,d depends only on a, α, β, γ and d. In particular,  Z  Z 1 2 −tH a −tH0a a 2 γ/α+2 V (x)dx + t (5.7) T r(e −e ) = pt (0) −t |V (x)| dx + O(t ) , 2 Rd Rd as t ↓ 0.

We next consider another stochastic process associated with a non-local operator that has been of considerable interest to many people in recent years, the so called relativistic Brownian motion and its more general versions, the α-stable relativistic process. This is again a L´evy process denote by Xtm with characteristic function

HEAT TRACE OF NON-LOCAL OPERATORS

  α/2 −m −t (|ξ|2 +m2/α )

e

(5.8)

iξ·Xtm

= E(e

)=

Z

Rd

13

(m,α)

eiξ·y pt

(y)dy,

for any m ≥ 0 and 0 < α < 2. The case α = 1 is the relativistic Brownian motion and m = 0 yields back the stable processes of order α. Henceforth we assume that m > 0. As in the case of stable processes, Xtm is a subordination of Brownian motion and in fact (m,α) (x) pt

(5.9)

  Z 2 2/α α/2 −m 1 ) −ix·ξ −t (|ξ| +m = = dξ e e (2π)d Rd Z ∞ −|x|2 1 m,α 4s η (s) ds, = e t d/2 (4πs) 0

where ηtm,α (s) is the density of the subordinator with Bernstein function Φ(λ) = (λ + m2/α )α/2 − m. (m,α) (x) is a radial, symmetric, and decreasing in |x|. By a simple change of As before we see that pt variables it is easy to see that the transition densities have the scaling property (m,α)

pt

(5.10)

(1,α)

(x) = md/α pmt (m1/α x).

These properties can all be found in Ryznar [30] where it is also proved that Z ∞ −|x|2 1 (m,α) −m2/α s α/2 mt 4s e ηt (s) ds, e (x) = e (5.11) pt d/2 (4πs) 0 α/2

where ηt

(s) is the density for the α/2-stable subordinator. By scaling α/2

ηt

α/2

(s) = t−2/α η1 (st−2/α ).

Hence changing variables in (5.11) leads to Z ∞ −mt d/α (m,α) (0) = (5.12) lim e t pt t↓0

0

ωd Γ(d/α) 1 α/2 η1 (s) ds = pα1 (0) = , d/2 (2π)d α (4πs)

where the last equality follows from (2.5).
α/2 . The case α = 1 gives the The infinitesimal generator of Xtm is given by m − m2/α − ∆ √ 2 generator m − −∆ + m which is the free relativistic Hamiltonian. For more on these notions we refer the reader to Carmona, Masters and Simon [11]. (m,α) (x) and their Dirichlet counterparts for various Estimates for the global transition probabilities pt domains have been studied by many authors. We refer the reader to [13], [14], [15], [16], [18], [19], [29], [30]. Here we recall the following small time estimate from Chen, Kim and Song [15, Theorem 2.1]. There exists a constant Cα,m,d depending on α, m and d such that for all x ∈ Rd and all t ∈ (0, 1], 1

1

(5.13) where

−1 t−d/α ∧ Cα,m,d

t Ψ(m α |x|) t Ψ(m α |x|) (m,α) (x) ≤ Cα,m,d t−d/α ∧ ≤ pt , d+α |x| |x|d+α −(d+α)

Ψ(r) = 2

Γ



d+α 2

−1 Z

∞

s

d+α −1 2

e−s/4 e−r

0

which is a decreasing function of r 2 with Ψ(0) = 1, Ψ(r) ≤ 1 and with

−r (d+α−1)/2 c−1 ≤ Ψ(r) ≤ c1 e−r r (d+α−1)/2 , 1 e r

2 /s

ds

˜ RODRIGO BANUELOS AND SELMA YILDIRIM YOLCU

14

for all r ≥ 1. (See also Chen and Song,[18, p. 277].) Setting p˜t (x) = t

−d/α

1

t Ψ(m α |x|) ∧ |x|d+α

it follows trivially (exactly as in Bogdan and Jakubowski [9, p.182]) that p˜s (x) ∧ p˜t−s (x) ≤ C p˜t (0),

(5.14)

for all x ∈ Rd and 0 < s < t ≤ 1. Therefore,

(m,α)

(m,α)

ps(m,α) (0, x) ∧ pt−s (x, 0) ≤ Cα,m,d pt

(5.15)

(0, 0)

for all x ∈ Rd and 0 < s < t ≤ 1. This as before leads to the “5P-inequality” (m,α)

ps

(5.16)

(m,α)

(x, 0)pt−s (0, x) (m,α)

pt

(0, 0)

  (m,α) ≤ Cα,m,d ps(m,α) (x, 0) + pt−s (0, x) , (m,α)

(m,α)

valid for all x ∈ Rd , 0 < s < t ≤ 1. (Recall that ps (x, y) = ps (x − y).) Even though this estimate is not as general as the previous “5P-inequalities” above, it suffices for our needs. Next, we need bounds for the γ moments of Xsm . More precisely we want to understand R tto compute γ 0 m the behavior of 0 E |Xr | dr, as t ↓ 0. Let us denote the standard Brownian motion in Rd by Bs and recall that E 0 |B2s |γ = 2γ/2 sγ/2 E 0 |B1 |γ and 2γ/2 E 0 |B1 |γ = Cγ,d , where the latter is an explicit and computable constant whose exact value is not relevant for our calculations here. Using this, (5.10) and (5.11) we have Z t Z t Z mt dγ dγ 0 m γ 0 1 γ ( −1) E |Xr | ds = m α E |Xmr | dr = m α E 0 |Xr1 |γ dr 0 0 0 Z mt Z −1) ( dγ α (5.17) |x|γ p(1,α) (x)dxdr = m r 0 Rd  Z Z mt ∞ ( dγ −1) r 0 γ −s α/2 = m α e E |B2s | e ηr (s)ds dr 0 0  Z mt Z ∞ −1) γ/2 −s α/2 r ( dγ s e ηr (s)ds dr, e = Cγ,d m α 0 0  Z mt Z ∞ α/2 −2/α −1) ( dγ r γ/2 −s −2/α = Cγ,d m α e s e r η1 (r s)ds dr 0 0  Z ∞ Z mt −1) γ/α r γ/2 −sr 2/α α/2 ( dγ α r e = Cγ,d m s e η1 (s)ds dr, 0

0

where we used the scaling property of the stable subordinator for the second to the last line and a change variables for the last line. But Z ∞ Z ∞ α/2 γ/2 −sr 2/α α/2 sγ/2 η1 (s)ds = Cγ E 0 |X1 |γ < ∞, s e η1 (s)ds ≤ 0

0

whenever γ < α. (Here, X1 is the stable process of order α at time 1.) Thus, Z t E 0 |Xrm |γ dr ≤ Cα,γ,m,d tγ/α+1 emt 0

and this is valid for all t > 0. From this and the inequality (5.16), we obtain

HEAT TRACE OF NON-LOCAL OPERATORS

t E0,0

(5.18)

Z

0

t

|Xsm |γ ds



≤ Cα,m,d

Z tZ

= 2Cα,m,d = 2Cα,m,d

0

(m,α)

Rd

(ps(m,α) (y, 0) + pt−s (0, y))|y|γ dyds

Z tZ Z

Rd

0

t

0

15

ps(m,α) (0, y)|y|γ dyds

E 0 (|Xsm |γ )ds

≤ Cα,γ,m,d tγ/α+1 ,

for all t ∈ (0, 1]. With (5.18), the proof of Theorem 2.2 can be repeated to obtain a similar result for the operator associated with the relativistic stable processes. More precisely we have
α/2 . Suppose V ∈ L∞ (Rd ) ∩ L1 (Rd ) and that it is Theorem 5.2. Let H0m = m − m2/α − ∆ also uniformly H¨older continuous of order γ, with 0 < γ < α ∧ 1, whenever 0 < α ≤ 1, and with
α/2 + V . Then, for all t ∈ (0, 1], 0 < γ ≤ 1, whenever 1 < α < 2. Let H m = m − m2/α − ∆ Z Z
1 2 2 α T r(e−tH m − e−tH0m ) + pαt (0)t |V (x)| dx V (x)dx − pt (0) t 2 Rd Rd   (m,α) (5.19) (0) |V k2∞ etkV k∞ t3 + tγ/α+2 , ≤ Cα,γ,m,d kV k1 pt where the constant Cα,γ,m,d depends only on α, γ, m and d. In particular,   Z Z 1 2 (m,α) 2 γ/α+2 −tH0m −tH m |V (x)| dx + O(t ) , V (x)dx + t (0) −t ) = pt −e (5.20) T r(e 2 Rd Rd as t ↓ 0.

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