We study the asymptotic behavior of the free partition function in the t→0+ limit for a diffusion process which consists of d-independent, one-dimensional, symmetric, 2s-stable processes in a hyperrectangular cavity K⊂Rd with an absorbing boundary. Each term of the partition function for this polyhedron in d-dimensions can be represented by a quermassintegral and the geometrical information conveyed by the eigenvalues of the fractional Dirichlet Laplacian for this solvable model is now transparent. We also utilize the intriguing method of images to solve the same problem, in one and two dimensions, and recover identical results to those derived in the previous analysis.

1. Introduction

Trace formulas for heat kernels of the fractional Laplacian (-Δ)s, s∈(0,1) [1], and its Schrödinger perturbations in spectral theory [2] have attracted a lot of attention recently due to the numerous applications to the mathematical physics, mathematical biology, and finance. From a probabilistic point of view the fractional Laplacian on a domain K⊂Rd is a nonlocal operator which arises as the generator of a pure jump Lévy process killed upon exiting K.

In a recent paper [3] we investigated the distribution of eigenvalues of the Dirichlet pseudodifferential operator ∑i=1d(-∂i2)s, s∈(0,1), on an open and bounded subdomain K⊂Rd, which is defined by the principal value integral: (1)-∂i2sψixi=C1,sP.V.∫Rψixi-ψiyixi-yi1+2sdxi=C1,slimϵ→0+∫xi-yi>ϵψixi-ψiyixi-yi1+2sdxi,C1,s=-2cosπsΓ-2s.The ψi(xi) are restrictions of functions that belong to the fractional Sobolev space [4], Hs(R)=Ws,2(R), s∈(0,1), given by (2)HsR=ψi∈L2R:ψixi-ψiyixi-yi1/2+s∈L2R×Rand satisfy ψ(x)=∏i=1dψi(xi)≡0 in Rd∖K-. We have also predicted bounds on the sum of the first N eigenvalues, the counting function, the Riesz means, and the first-term asymptotic expansion of the partition function which was found to be(3)Zt=∫0∞e-EtdNE=12πdΩ2Γ1+1/2sdtd/2s+ot-d/2s,where the counting function N(·) is^{1}(4)NE=♯n∈Z+d:n2s≤Lπ2sE1+o1.

The present paper extends result (3) to all orders in t-β, β>0, in the short-time limit, and provides a novel result for a polyhedron’s partition function which was lacking from the literature even for the ordinary Laplacian. This objective is achieved by using two different methods. The first method relies on the explicit knowledge of the Dirichlet spectrum (20) of the fractional operator and the Euler-Maclaurin summation formula (25). These two ingredients enable us to express the asymptotic behavior of the partition function (29) in terms of the volume of K and the rth quermassintegral. The second method, commonly known as the method of images, suggests that the heat kernel of the cavity, in the short time limit, receives contributions only from the heat kernels of the unbounded space with virtual source points generated by reflecting anticlockwise (or clockwise) the initial point source through the hyperplanes bounding the cavity. Note that only virtual source points clustering around each vertex of the cavity contribute; see (41). Tracing and integrating the heat kernel over the cavity, we recover results identical to those derived by the first method. It is worth noting that this method, though widely known and well established in classical electromagnetism, has not been used before to solve problems falling into this category.

The paper is organized into five sections. Section 2 reviews fundamental notions of convex geometry and applies them to a d-dimensional hyperrectangular parallelepiped in Euclidean space. Section 3 computes the asymptotic behavior of the partition function for the fractional Dirichlet operator by applying the first method. Section 4 uses the alternative approach of images for solving the same problem. Section 5 concludes the work by posing some open problems.

2. Geometric Preliminaries of Convex Bodies in <inline-formula>
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Let C(K) be the family of all convex bodies (nonempty, compact, and convex sets) in the d-dimensional Euclidean vector space. We denote by Bd={x∈Rd:x2≤1} the unit ball and Sd-1={x∈Rd:x2=1} the unit sphere. The Lebesgue measure on Rd is denoted by λd and the spherical Lebesgue measure on Sd-1 is denoted by σd-1. In particular we have ωd:=λd(Bd)=πd/2/Γ(1+d/2) and Od-1:=σd-1(Sd-1)=dωd=2πd/2/Γ(d/2). If K∈C(K), then the parallel body Kρ at distance ρ>0 is given by (5)Kρ≔x∈Rd:distx,K≤ρ,where dist(·,K) denotes the Euclidean distance from K. According to Steiner’s formula the volume of Kρ is given by(6)λρKρ=∑m=0dCdmWmKρm=∑j=0dωd-jVjKρd-j,ρ≥0,Cdm=dm,where Wm:C(K)→R is the mth quermassintegral, or mean cross-sectional measure, introduced by Minkowski, and Vj:C(K)→R is the jth intrinsic volume^{2}. The Wm in (6) is defined by [5] (7)WmK=d-mOm-1⋯O0dOd-2⋯Od-m-1∫Gm,d-mVKd-m′dLmO,where V(Kd-m′) denotes the volume of the convex set of all intersection points of the (d-m)-plane, passing through the fixed point O, with the m-planes orthogonal to it and dLm[O] is the invariant volume element of the Grassmannian manifold Gm,d-m which is the set of m-dimensional planes in Rd-m. Setting m=d-1 into (7) one can define another functional, the so-called mean breadth of K, as follows: (8)b-=2Od-1∫Gd-1,1VK1′dLd-1O=2ωdWd-1K.

If the boundary ∂K of a convex set is a hypersurface Σ of class C2, the quermassintegrals Wm can be expressed by means of the integrals of mean curvature of ∂K. The mth integral of mean curvature Mm(Σ) is defined by (9)MmΣ=1Cd-1m∫ΣSmd-1κdA,where Sm(d-1)(κ)=∑1≤i1<⋯<im≤d-1κi1⋯κim is the mth-elementary symmetric function of the (d-1) principal curvatures and dA is the area element of Σ. The volume of the parallel body Kρ can be then written as (10)λρKρ=VK+∑m=0d-1Cd-1mm+1Mm∂Kρm+1and by comparison with (6) we end up with (11)Mm∂K=dWm+1K.This relation is well defined as long as ∂K is C2. If K does not have a smooth boundary, then we compute (12)limρ→0+Mm∂Kρ=dWm+1K.

In the present paper we focused on the hyperrectangular parallelepiped with edges aj, j=1,…,d. The mth intrinsic volume and quermassintegral are given by (13)VmK=Smda,V0a=1,WmK=ωmCd-1mVd-mK.The mean breadth of K can be computed by combining (9) with (8) for the parallel body Kρ and taking the ρ→0+ limit in the end. A hyperrectangular parallelepiped domain has a total of 2d faces, 2d vertices, and 2d-1d edges. The mean curvature is defined by (14)Hd-2R=1d-1Sd-2d-1R=1d-1∑1≤i1<⋯<id-2≤d-11Ri1⋯Rid-2,where Ri are the principal radii of curvature of ∂Kρ. The boundary of ∂Kρ consists of hyperplanes at the faces, hyperspheres at the vertices, and hypercylinders at the edges ofK. The values ofH for ∂Kρ are (15)Hd-2R=0,hyperplanes,1ρd-2,hyperspheres,1d-1ρd-2,lateral face of hypercylinders.A careful calculation gives (16)Wd-1K=limρ→0+Wd-1∂Kρ=1dωd-1∑i=1daiand therefore (17)b-=2dωd-1ωdV1K.

3. The Partition Function for the Operator <inline-formula>
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Let X(t)={Xi(t)}i=1d, t>0, be a collection of independent, one-dimensional, symmetric 2s-stable processes in R and denote by {PK(t)}t≥0 the semigroup on L2(K) of X(t) killed upon exiting K. Its transition density pK(x,t;y) satisfies (18)PKtfx=∫KpKx,t;yfydy.In [3, 6], performing a slight modification, the following proposition has been proved.

Proposition 1.

On the open and bounded hyperrectangular parallelepiped K⊂Rd of side lengths ai, i=1,…,d, the eigenvalues for the homogeneous Dirichlet problem (19)∑i=1d-∂i2sψnx=Ensψnx,inK;En=EnD2s,ψnx=0onRd∖K-are given by (20)Ens=∑i=1dniπai-1-sπ2ai2s+∑i=1dO1ni,n∈Z+d,where {ψn}n=1∞ forms an orthonormal basis in L2(K) with ψn(x)=cn∏j=1dψnj(xj), none of the indices nj vanishes, and D2s=(ħ2/[M])s is a constant with dimensions [D2s]=[M]s[L]4s/[T]2s.

Arranging the positive, real, and discrete spectrum of D2s in increasing order (including multiplicities), we have (21)0<E1K<E2K≤E3K≤⋯,limn→∞EnK=∞.The simplicity of Enj (the eigenvalue in the j-dimension) is conjectured to hold for s∈(0,1), proved for s=1/2 and s∈[1/2,1) in [7], [6], respectively.

Taking into account the previous proposition, the generator of the semigroup acts on the orthonormal set ψn as (22)e-tD2s∣Kψnx=e-tEnψnx,x∈K,t>0.The transition density is then given by (23)pKx,t;y=∑n∈Z+de-tEnψ-nxψnyand the partition function (or trace of the heat kernel), using (23), satisfies (24)ZKt,s≡Tre-tD2s=∫KpKx,t;xdx=∑n∈Z+de-tEn∫Kψnx2dx=∏j=1d∑nj≥1e-tEnj=∏j=1dZjt,s,t>0.Thus from (24) we observe that the total partition function is written as the disjoint product of partition functions for each spatial dimension. The sum ∑nexp(-tEn) will be calculated utilizing the Euler-Maclaurin summation formula [8] which states the following.

Theorem 2.

Suppose f is a decreasing function with continuous derivatives up to order p. Then(25)∑k=n+1mfk=∫nmfudu+∑l=1p-1lBl0l!fl-1m-fl-1n+-1p-1p!∫nmBpu-ufpudu,where Bp(u) are the Bernoulli polynomials and Bl(0) the Bernoulli numbers.

Applying (25) in one dimension, in the zero time limit, we find for h(s)=1-s/2(26)∑nj=1∞e-π/ajnj-hs2st~ast→0+ajπt1/2sΓ1+12s-s2,s∈0,1.The functions f(t,u,s)=e-π/a(u-h(s))2st belong to the following spaces: (27)f∈Cu10,∞2×12,1Cu∞0,∞2×12C00,∞2×0,12,where the subscript u indicates that the functions are differentiable with respect to u. The case s=1/2 can be summed exactly since it turns out to be a progression. The result is (28)Zjt,12=coshπ/4ajt-sinhπ/4ajtsinhπ/2ajt~t→0+ajπt-14.Substituting (26) into (24), for s∈[1/2,1) we obtain (29)ZKt,s~t→0+∏j=1dajπt1/2sΓ1+12s-s2=∑m=0d1πt1/2sΓ1+12sm-s2d-mVmK=1πt1/2sΓ1+12sdVdK+∑r=0d-11πt1/2sΓ1+12sd-r-s2rd-rOd-r-1dd-rWd-rK,where Vm(K), Wd-r(K) are given by (13).

Remark 3.

(1) The termVd(K)=∏j=1daj is the d-dimensional volume of the hyperrectangle parallelepiped and V1(K) can be written in terms of the mean breadth of K as (30)V1K=dωd2ωd-1b-using (17).

(2) In two dimensions, the trace of the heat kernel for the ordinary Laplacian (s=1), provided that the domain is a rectangle of sides a1,a2, is given by (31)ZK1,t=14Θπta12-1Θπta22-1,where Θ(x) is the familiar Riemann theta function^{3}(32)Θx=ϑ000,-ix=∑n∈Ze-πn2x.The Poisson summation formula^{4} can be casted into the form [9](33)Θx=1xΘ1xand utilized to determine the asymptotic behavior of the partition function in the t→0+ limit. Therefore, we reproduce the well-known result (34)ZKt,s=1~14πtV2K-18πtV1K+14,where V2(K) is the surface area of K and V1(K) is the length of the boundary ∂K. An independent calculation using (29) gives identical result. Kac, in [10], extended the dimensionless corner correction for a closed polygon with obtuse angles through a complicated integral. Later it was reported in [11] that D. B. Ray obtained the correction (35)∑i=1nπ2-ϕi224πϕi,0<ϕi<2πfor arbitrary angles by expressing Green’s function as a Kontorovich-Lebedev transform. It is noteworthy that if we approximate a circle by an inscribed regular polygon then (35) becomes (36)∑i=1nπ2-ϕi224πϕi=16n-1n-2.In (36) as the number of edges of the polygon tends to infinity the sum converges to the topological invariant constant 1/6. For simply connected, open with compact boundary two-dimensional Riemannian manifolds (M,g), where g is the metric tensor, the partition function contains the t-independent term given by [11] (37)E6=112π∫MRicdetgdx,where E is the Euler characteristic and Ric is the Ricci scalar of the manifold M.

4. A New Alternative Approach Based on the Image Method

The transition function pK (or elementary solution) is the solution of the following diffusion problem in K=(0,L)⊂R described by (38)∂pKx,t;y∂t=-d2dx2spKx,t;y,inK×R+with initial and Dirichlet boundary conditions(39a)limt→0+∫KpKx,t;ydx=∫Kδx-ydx=1,(39b)limx→q∈∂KpKx,t;y=0,∀y∈K.The physical context of (39a) is the existence of a point source initially described by a generalized Dirac-δ function while (39b) dictates that the process is killed upon reaching the boundary ∂K. The solution of (38) with the initial condition (39a) on R is given by [12](40)pRx,t;y=12π∫Re-tk-signkπhs/L2s+ik-signkπhs/Lx-ydk,hs=1-s2,where the appearance of the function sign(k)πh(s)/L will be apparent shortly. Note that due to the translational invariance of the integral it can be absorbed in k and written in the usual way.

In general the asymptotic behavior of the Dirichlet partition function in the t→0+ limit is dominated by the diagonal elements of the heat kernel which receive contributions from the heat kernels of the unbounded space for the virtual source points clustering around each vertex of the polytope K. The corresponding expression is(41)limx→ypKvx,t;y~t→0+∑ρ=0Gv-1ρpRI-Raρvy,t,where Gv is the order of the reflection group at vertex v. Thus integrating (41) over a suitable subdomain of K and then taking the t→0+ limit we recover the desired result.

In d=1, G coincides with the infinite dihedral group Dih∞ with defining relations(42)Ra12=Ra22=I,where Rai’s are involuntary transformations and represent reflections with respect to the boundary points of K. For every y∈K we generate the following two infinite sequences of virtual source points depending on whether we start reflection from the left or the right fixed point of the isometry Ri (as shown in Table 1). In Figure 1, we graphically depict the virtual domains in which the corresponding image source points belong to.

The virtual image points of y

Left fixed point

Right fixed point

Group element

Location of virtual image point

Group element

Location of virtual image point

Ra2

-y

Ra1

-y+2L

Ra1⋅Ra2

y+2L

Ra2⋅Ra1

y-2L

Ra2⋅Ra1⋅Ra2

-y-2L

Ra1⋅Ra2⋅Ra1

-y+4L

(Ra1⋅Ra2)2

y+4L

(Ra2⋅Ra1)2

y-4L

⋮

⋮

⋮

⋮

The fundamental region K=(0,L) and the virtual domains generated by the elements Ri≡Rai of the infinite dihedral group.

The solution, using (41), is given by(43)pKx-y,t=∑n∈ZpRx-y-2nLj,t-∑n∈ZpRx+y-2nLj,t.One can check that both the initial and boundary conditions are fulfilled and moreover the x→y limit of (43) gives(44)limx→ypKx-y,t=pR0,t+2∑n∈N∖0pR2nL,t-∑n∈ZpR2y-nL,tand the contribution of each term to the partition function ZK(t)=∫KpK(y,t;y)dy is (45)∫0LpR0,tdy=Lπt1/2sΓ1+12s+hs2limt→0+∑n∈N∖0∫0LpR2nL,tdy=∑n∈N∖0δn=0limt→0+∑n∈Z∫0LpR2y-nL,tdy=12.Therefore, (46)ZKt~t→0+Lπt1/2sΓ1+12s+-s2as promised.

The diffusion initial-boundary value problem for K=(0,L1)×(0,L2) with s=1 can also be solved applying the method of images in a more general setting [13]. Let θ be the angle between two intersecting mirrors in R2. If we require the absence of a virtual mirror between the two given ones, after successive reflections of the fundamental domain, then θ=π/q where q∈N∖{1}. This remark facilitates the enumeration of bounded tessellations of the plane through reflections [14]. One of the admissible polygons apart from the congruent equilateral triangles, the isosceles right triangles and the bisected equilateral triangles, is the rectangle. Generally speaking, if the angle θ=π/m, m∈N∖{1}, then there is a unique, up to isomorphism, group generated by two involutions Rα1,Rα2 such that their product Rα1·Rα2 has order m. The group is denoted by Dih2m, called the dihedral group of order 2m and has the presentation(47)Dih2m=Rα1,Rα2∣Rα12=Rα22=Rα1·Rα2m=I.In our case m=2 and each wedge produces the transition function(48)limx→ypKx-y,t=pR0,t-pRI-Rα1y,t+pRI-Rα2·Rα1y,t-pRI-Rα2y,t.Integrating (48) over a suitable subdomain Di⊂K once, multiplying the result by a factor of four in order to take into account the contribution from all four wedges of the rectangle, and taking the t→0+ limit we recover (34).

5. Discussion

The nonlocal fractional operator D2s, considered in this paper, is the infinitesimal generator of time translations for a symmetric Lévy process with index (or characteristic exponent) s, killed upon exiting the cavity. The resulting partition function turns out to depend not only on time t and the dimensionality d of the space but also on s. If one investigates the same problem considering processes with stable law parameters, the skewness β, the scale c, and the location τ, then extra contributions should be expected.

In two dimensions, the topological term given by (35), for the ordinary diffusion (s=1), predicts the value 1/4. Nevertheless, the calculation of the partition function for the fractional operator suggests the s-dependent value (-s/2)2. This is an interesting result that signals the connection of the corner corrections of a bounded domain with the s-stable Lévy process. In the same spirit, relation (37) requires further investigation in order to reveal a possible similar dependence. Unfortunately, the absence of a relation between (37) and the corresponding term of the partition function, in the case of a symmetric Lévy flight, puts severe constraints in this direction.

Another challenging open problem would be to predict the partition function for an open manifold with convex and compact boundary knowing the sequence of polyhedral partition functions which approximates it. This issue is related to the geometric measure theory and fully resolving it will lead to the discovery of new results in higher dimensions than those presented in [11].

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to thank the anonymous reviewer for constructive critisism that improved the presentation of the paper.

Endnotes

We denote by x2s=∑i=1d|xi|2s the 2s-norm and by ·2 the Euclidean norm.

The word intrinsic here means that the quantity under study does not depend on the dimension of the ambient space.

The Jacobi theta function is defined by ∗ϑabz,τ=∑n∈Zeiπn+a2τ+2iπn+az+b.

The Poisson formula states ∗∗∑n∈Ze-πn2A+2πnsA=1Aeπs2A∑m∈Ze-πm2A-1-2iπmsand can be proved using ∑me2iπrm=∑nδr-n.

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