Left-Inverses of Fractional Laplacian and Sparse Stochastic Processes
aa r X i v : . [ c s . I T ] S e p LEFT-INVERSES OF FRACTIONAL LAPLACIAN AND SPARSESTOCHASTIC PROCESSES
QIYU SUN AND MICHAEL UNSER
Abstract.
The fractional Laplacian ( −△ ) γ/ commutes with the primarycoordination transformations in the Euclidean space R d : dilation, translationand rotation, and has tight link to splines, fractals and stable Levy processes.For 0 < γ < d , its inverse is the classical Riesz potential I γ which is dilation-invariant and translation-invariant. In this work, we investigate the functionalproperties (continuity, decay and invertibility) of an extended class of differen-tial operators that share those invariance properties. In particular, we extendthe definition of the classical Riesz potential I γ to any non-integer number γ larger than d and show that it is the unique left-inverse of the fractional Lapla-cian ( −△ ) γ/ which is dilation-invariant and translation-invariant. We observethat, for any 1 ≤ p ≤ ∞ and γ ≥ d (1 − /p ), there exists a Schwartz function f such that I γ f is not p -integrable. We then introduce the new unique left-inverse I γ,p of the fractional Laplacian ( −△ ) γ/ with the property that I γ,p isdilation-invariant (but not translation-invariant) and that I γ,p f is p -integrablefor any Schwartz function f . We finally apply that linear operator I γ,p with p = 1 to solve the stochastic partial differential equation ( −△ ) γ/ Φ = w withwhite Poisson noise as its driving term w . Introduction
Define the Fourier transform F f (or ˆ f for brevity) of an integrable function f on the d -dimensional Euclidean space R d by(1.1) F f ( ξ ) := Z R d e − i h x ,ξ i f ( x ) d x , and extend the above definition to all tempered distributions as usual. Here wedenote by h· , ·i and k · k the standard inner product and norm on R d respectively.Let S := S ( R d ) be the space of all Schwartz functions on R d and S ′ := S ′ ( R d )the space of all tempered distributions on R d . For γ >
0, define the fractionalLaplacian ( −△ ) γ/ by(1.2) F (( −△ ) γ/ f )( ξ ) := k ξ k γ F f ( ξ ) , f ∈ S . The fractional Laplacian has the remarkable property of being dilation-invariant.It plays a crucial role in the definition of thin plate splines [4], is intimately tied tofractal stochastic processes (e.g., fractional Brownian fields) [8, 12] and stable Levyprocesses [3], and has been used in the study of singular obstacle problems [2, 10].In this paper, we present a detailed mathematical investigation of the functionalproperties of dilation-invariant differential operators together with a characteriza-tion of their inverses. Our primary motivation is to provide a rigorous operatorframework for solving the stochastic partial differential equation(1.3) ( −△ ) γ/ Φ = w with white noise w as its driving term. We will show that this is feasible via thespecification of a novel family of dilation-invariant left-inverses of the fractionalLaplacian ( −△ ) γ/ which have appropriate L p -boundedness properties.We say that a continuous linear operator I from S to S ′ is dilation-invariant ifthere exists a real number γ such that(1.4) I ( δ t f ) = t γ δ t ( If ) for all f ∈ S and t > , and translation-invariant if(1.5) I ( τ x f ) = τ x ( If ) for all f ∈ S and x ∈ R d , where the dilation operator δ t , t > translation operator τ x , x ∈ R d aredefined by ( δ t f )( x ) = f ( t x ) and τ x f ( x ) = f ( x − x ) , f ∈ S , respectively. Onemay verify that the fractional Laplacian ( −△ ) γ/ , γ >
0, is dilation-invariant andtranslation-invariant, a central property used in the definition of thin plate splines[4].Next, we define the
Riesz potential I γ ([9]) by(1.6) I γ f ( x ) = π − d/ − γ Γ(( d − γ ) / γ/ Z R d k x − y k γ − d f ( y ) d y , f ∈ S , where 0 < γ < d . Here the Gamma function Γ is given by Γ( z ) = R ∞ t z − e − t dt when the real part Re z is positive, and is extended analytically to a meromorphicfunction on the complex plane. For any Schwartz function f , I γ f is continuous andsatisfies(1.7) | I γ f ( x ) | ≤ C ǫ (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) d + ǫ (cid:17) (1 + k x k ) γ − d for all x ∈ R d , where ǫ > C ǫ is a positive constant, see also Theorem 2.1. Then the Rieszpotential I γ is a continuous linear operator from S to S ′ . Moreover one may verifythat I γ is dilation-invariant and translation-invariant, and also that I γ , < γ < d ,is the inverse of the fractional Laplacian ( −△ ) γ/ ; i.e.,(1.8) I γ ( −△ ) γ/ f = ( −△ ) γ/ I γ f = f for all f ∈ S because(1.9) F ( I γ f )( ξ ) = k ξ k − γ F f ( ξ ) , f ∈ S . A natural question then is as follows:
Question 1 : For any γ > , is there a continuous linear operator I from S to S ′ that is translation-invariant and dilation-invariant, and that is an inverse of thefractional Laplacian ( −△ ) γ/ ? In the first result of this paper (Theorem 1.1), we give an affirmative answerto the above existence question for all positive non-integer numbers γ with theinvertibility replaced by the left-invertibility, and further prove the uniqueness ofsuch a continuous linear operator.To state that result, we recall some notation and definitions. Denote the dual pairbetween a Schwartz function and a tempered distribution using angle bracket h· , ·i ,which is given by h f, g i = R R d f ( x ) g ( x ) d x when f, g ∈ S (we remark that the dualpair between two complex-valued square-integrable functions is different from theirstandard inner product). A tempered distribution f is said to be homogeneousof degree γ if h f, δ t g i = t − γ − d h f, g i for all Schwartz functions g and all positivenumbers t . We notice that the multiplier k ξ k − γ in the Riesz potential I γ , see (1.9), RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 3 is a homogenous function of degree − γ ∈ ( − d, I γ to any non-integer number γ > d asfollows: I γ f ( x ) := (2 π ) − d Γ( d − γ )Γ( d + k − γ ) Z S d − Z ∞ r k − γ + d − × (cid:16) − ddr (cid:17) k (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) (cid:17) drdσ ( ξ ′ ) , f ∈ S , (1.10)where S n − = { ξ ′ ∈ R d : k ξ ′ k = 1 } is the unit sphere in R d , dσ is the area elementon S n − , and k is a nonnegative integer larger than γ − d . Integration by partsshows that the above definition (1.10) of I γ f is independent on the nonnegativeinteger k as long as it is larger than γ − d , and also that it coincides with theclassical Riesz potential when 0 < γ < d by letting k = 0 and recalling that theinverse Fourier transform F − f of an integrable function f is given by(1.11) F − f ( x ) := (2 π ) − d Z R d e i h x ,ξ i f ( ξ ) dξ. Because of the above consistency of definition, we call the continuous linear operator I γ , γ ∈ (0 , ∞ ) \ ( Z + + d ) in (1.10) the generalized Riesz potential , where Z + is theset of all nonnegative integers. Theorem 1.1.
Let γ be a positive number with γ − d Z + , and let I γ be the linearoperator defined by (1.10) . Then I γ is the unique continuous linear operator from S to S ′ that is dilation-invariant and translation-invariant, and that is a left inverseof the fractional Laplacian ( −△ ) γ/ . Let L p := L p ( R d ) , ≤ p ≤ ∞ , be the space of all p -integrable functions on R d with the standard norm k·k p . The Hardy-Littlewood-Sobolev fractional integrationtheorem ([11]) says that the Riesz potential I γ is a bounded linear operator from L q to L p when 1 < p ≤ ∞ , < γ < d (1 − /p ) and q = pd/ ( d + γp ). Hence I γ f ∈ L p for any Schwartz function f when 0 < γ < d (1 − /p ). We observethat for any non-integer number γ larger than or equal to d (1 − /p ), there existsa Schwartz function f such that I γ f L p , see Corollary 2.16. An implicationof this negative result, which will become clearer in the sequel (cf. Section 4),is that we cannot generally use the translation-invariant inverse I γ to solve thestochastic partial differential equation (1.3). What is required instead is a specialleft-inverse of the fractional Laplacian that is dilation-invariant and p -integrable.Square-integrability in particular ( p = 2) is a strict requirement when the drivingnoise is Gaussian and has been considered in prior work [12]; it leads to a fractionalBrownian field solution, which is the multi-dimensional extension of Mandelbrot’scelebrated fractional Brownian motion [1, 8]. Our desire to extend this method ofsolution for non-Gaussian brands of noise leads to the second question. Question 2 : Let ≤ p ≤ ∞ and γ > . Is there a continuous linear operator I from S to S ′ that is dilation-invariant and a left-inverse of the fractional Laplacian ( −△ ) γ/ such that If ∈ L p for all Schwartz functions f ? In the second result of this paper (Theorem 1.2), we give an affirmative answerto the above question when both γ and γ − d (1 − /p ) are not integers, and showthe uniqueness of such a continuous linear operator.To state that result, we introduce some additional multi-integer notation. For x = ( x , . . . , x d ) ∈ R d and j = ( j , . . . , j d ) ∈ Z d + (the d -copies of the set Z + ), QIYU SUN AND MICHAEL UNSER we set | j | := | j | + · · · + | j d | , j ! := j ! · · · j d ! with 0! := 1, x j := x j · · · x j d d and ∂ j f ( x ) := ∂ j x · · · ∂ j d x d f ( x ). For 1 ≤ p ≤ ∞ and γ >
0, we define the linear operator I γ,p from S to S ′ with the help of the Fourier transform:(1.12) F ( I γ,p f )( ξ ) = (cid:16) F f ( ξ ) − X | j |≤ γ − d (1 − /p ) ∂ j ( F f )( ) j ! ξ j (cid:17) k ξ k − γ , f ∈ S , which is the natural L p extension of the fractional integral operator that was intro-duced in [1, 12, 13] for p = 2 and γ Z / I γ,p the p -integrable Riesz potential of degree γ , or the integrable Rieszpotential for brevity. Indeed, when both γ and γ − d (1 − /p ) are non-integers, thelinear operator I γ,p is the unique left-inverse of the fractional Laplacian ( −△ ) γ/ that enjoys the following dilation-invariance and stability properties. Theorem 1.2.
Let ≤ p ≤ ∞ , and γ is a positive number such that both γ and γ − d + d/p are not nonnegative integers. Then I γ,p in (1.12) is the unique dilation-invariant left-inverse of the fractional Laplacian ( −△ ) γ/ such that its image of theSchwartz space S is contained in L p . One of the primary application of the p -integrable Riesz potentials is the con-struction of generalized random processes by suitable functional integration of whitenoise [12, 13, 14]. These processes are defined by the stochastic partial differentialequation (1.3), the motivation being that the solution should essentially display thesame invariance properties as the defining operator (fractional Laplacian). In par-ticular, these processes will exhibit some level of self-similarity (fractality) because I γ,p is dilation-invariant. However, they will in general not be stationary becausethe requirement for a stable inverse excludes translation invariance. It is this lastaspect that deviates from the classical theory of stochastic processes and requiresthe type of mathematical safeguards that are provided in this paper. While thecase of a white Gaussian noise excitation is fairly well understood [12], it is not yetso when the driving term is impulse Poisson noise which leads to the specificationof sparse stochastic processes with a finite rate of innovation. The current statushas been to use the operator I γ, to specify sparse processes with the restrictionthat the impulse amplitude distribution must be symmetric [14, Theorem 2]. Ourpresent contribution is to show that one can lift this restriction by considering theoperator I γ, , which is the proper inverse to handle general impulsive Poisson noise.To state our third result, we recall some concepts about generalized randomprocesses and Poisson noises. Let D be the space of all compactly supported C ∞ functions with standard topology. A generalized random process is a random func-tional Φ defined on D (i.e., a random variable Φ( f ) associated with every f ∈ D )which is linear, continuous and compatible [6].The white Poisson noise(1.13) w ( x ) := X k ∈ Z a k δ ( x − x k )is a generalized random process such that the random variable associated with afunction f ∈ D is given by(1.14) w ( f ) := X k ∈ Z a k f ( x k ) , RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 5 where the a k ’s are i.i.d. random variables with probability distribution P ( a ), andwhere the x k ’s are random point locations in R n which are mutually independentand follow a spatial Poisson distribution with Poisson parameter λ >
0. Therandom point locations x k in R n follow a spatial Poisson distribution with Poissonparameter λ > E with finite Lebesguemeasure | E | , the probability of observing n events in E (i.e., the cardinality ofthe set { k | x k ∈ E } is equal to n ) is exp( − λ | E | )( λ | E | ) n /n !. Thus, the Poissonparameter λ represents the average number of random impulses per unit.As the white Poisson noise w is a generalized random process, the stochasticpartial differential equation (1.3) can be interpreted as the following:(1.15) h Φ , ( −△ ) γ/ f i = h w, f i for all f ∈ D . So if I is a left-inverse of the fractional Laplacian operator ( −△ ) γ/ , then(1.16) Φ = I ∗ w is literally the solution of the stochastic partial differential equation (1.3) as(1.17) h I ∗ w, ( −△ ) γ/ f i = h w, I ( −△ ) γ/ f i = h w, f i for all f ∈ D , where I ∗ is the conjugate operator of the continuous linear operator I from S to S ′ defined by h I ∗ f, g i := h f, Ig i for all f, g ∈ S . The above observation is usable only if we can specify a left-inverse (or equivalentlywe can impose appropriate boundary condition) so that I ∗ w defines a bona fidegeneralized random process in the sense of Gelfand and Vilenkin; mathematically,the latter is equivalent to providing its characteristic functional by the Minlos-Bochner Theorem (cf. Section 4). The following result establishes that P γ w := I ∗ γ, w is a proper solution of the stochastic partial differential equation (1.3), where w is the Poisson noise defined by (1.13). Theorem 1.3.
Let γ be a positive non-integer number, λ be a positive number, P ( a ) be a probability distribution with R R | a | dP ( a ) < ∞ , and I γ, be defined as in (1.12) . For any f ∈ D , define the random variable P γ w associated with f by (1.18) P γ w ( f ) := X k a k I γ, ( f )( x k ) where the a k ’s are i.i.d. random variables with probability distribution P ( a ) , and the x k ’s are random point locations in R n which are mutually independent and follow aspatial Poisson distribution with Poisson parameter λ . Then P γ w is the generalizedrandom process associated with the characteristic functional (1.19) Z P γ w ( f ) = exp (cid:16) λ Z R d Z R (cid:0) e − ia ( I γ, f )( x ) − (cid:1) dP ( a ) d x (cid:17) , f ∈ D . The organization of the paper is as follows. In Section 2, we first introduce alinear operator J Ω for any homogeneous function Ω ∈ C ∞ ( R d \{ } ) of degree − γ ,where γ − d Z + . The linear operator J Ω becomes the generalized Riesz potential I γ in (1.10) when Ω( ξ ) = k ξ k − γ ; conversely, any derivative of the generalized Rieszpotential I γ is a linear operator J Ω associated with some homogeneous function Ω: ∂ j I γ f = J Ω j f for all f ∈ S and j ∈ Z d + , QIYU SUN AND MICHAEL UNSER where Ω j ( ξ ) = ( iξ ) j k ξ k − γ . We then study various properties of the above linearoperator J Ω , such as polynomial decay property, dilation-invariance, translation-invariance, left-invertibility, and non-integrability in the spatial domain and in theFourier domain. The proof of Theorem 1.1 is given at the end of Section 2.In Section 3, we introduce a linear operator U Ω ,p for any homogeneous functionΩ ∈ C ∞ ( R d \{ } ) of degree − γ , where 1 ≤ p ≤ ∞ . The above linear operator U Ω ,p becomes the operator I γ,p in (1.12) when Ω( ξ ) = k ξ k − γ , and the operator J Ω in (2.1) when 0 < γ < d (1 − /p ). We show that the linear operator U Ω ,p isdilation-invariant, translation-variant and p -integrable, and is a left-inverse of thefractional Laplacian ( −△ ) γ/ when Ω( ξ ) = k ξ k − γ . The proof of Theorem 1.2 isgiven at the end of Section 3.In Section 4, we give the proof of Theorem 1.3 and show that the generalizedrandom process P γ w can be evaluated pointwise in the sense that we can replacethe function f in (1.18) by the delta functional δ .In this paper, the capital letter C denotes an absolute positive constant whichmay vary depending on the occurrence.2. Generalized Riesz Potentials
Let γ be a real number such that γ − d Z + , and let Ω ∈ C ∞ ( R d \{ } ) bea homogeneous function of degree − γ . Following the definition of homogenoustempered distributions in [7], we define the linear operator J Ω from S to S ′ by J Ω f ( x ) := (2 π ) − d Γ( d − γ )Γ( d + k − γ ) Z S d − Z ∞ Ω( ξ ′ ) r k − γ + d − × (cid:16) − ddr (cid:17) k (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) (cid:17) drdσ ( ξ ′ ) , f ∈ S , (2.1)where S n − = { ξ ′ ∈ R d : k ξ ′ k = 1 } is the unit sphere in R d , dσ is the area elementon S n − , and k is a nonnegative integer larger than γ − d .Note that the linear operator J Ω in (2.1) becomes the generalized Riesz potential I γ in (1.10) when Ω( ξ ) = k ξ k − γ and γ >
0. Therefore we call the linear operator J Ω in (2.1) the generalized Riesz potential associated with the homogeneous function Ω of degree − γ , or the generalized Riesz potential for brevity.The above definition of the generalized Riesz potential J Ω is independent on thenonnegative integer k as long as it satisfies k > γ − d , that can be shown byintegration by parts. Then, for γ ∈ ( −∞ , d ), we may take k = 0 and reformulate(2.1) as follows:(2.2) J Ω f ( x ) = (2 π ) − d Z R d e i h x ,ξ i Ω( ξ ) ˆ f ( ξ ) dξ for all f ∈ S , or equivalently(2.3) d J Ω f ( ξ ) = Ω( ξ ) ˆ f ( ξ ) for all f ∈ S , so that the role of the homogeneous function Ω( ξ ) in (2.1) is essentially that of theFourier symbol for a conventional translation-invariant operator.Let S ∞ be the space of all Schwartz functions f such that ∂ i ˆ f ( ) = 0 for all i ∈ Z d + , or equivalently that R R d x j f ( x ) d x = 0 for all j ∈ Z d + . Given a homogenousfunction Ω ∈ C ∞ ( R d \{ } ), define the linear operator i Ω on S ∞ by(2.4) d i Ω f ( ξ ) = Ω( ξ ) ˆ f ( ξ ) , f ∈ S ∞ . RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 7
Clearly i Ω is a continuous linear operator on the closed linear subspace S ∞ of S .For any function f ∈ S ∞ , applying the integration-by-parts technique k times andnoticing that lim ǫ → ǫ − γ | ∂ i ˆ f ( ǫξ ′ ) | = 0 for all ξ ′ ∈ S d − and i ∈ Z d + , we obtain that J Ω f ( x ) = (2 π ) − d Γ( d − γ )Γ( d + k − γ ) lim ǫ → Z S d − Z ∞ ǫ r k + d − γ − Ω( ξ ′ ) × (cid:16) − ddr (cid:17) k (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) (cid:17) drdσ ( ξ ′ )= (2 π ) − d lim ǫ → Z S d − Z ∞ ǫ Ω( ξ ′ ) r d − γ − e ir h x ,ξ ′ i ˆ f ( rξ ′ ) drdσ ( ξ ′ )= (2 π ) − d Z R d e i h x ,ξ i Ω( ξ ) ˆ f ( ξ ) dξ = i Ω f ( x ) . (2.5)Hence the generalized Riesz potential J Ω is the extension of the linear operator i Ω from the closed subspace S ∞ to the whole space S .In the sequel, we will study further properties of the generalized Riesz potential J Ω , such as the polynomial decay property (Theorem 2.1), the continuity as alinear operator from S to S ′ (Corollary 2.3), the translation-invariance and dilation-invariance (Theorem 2.7), the composition and left-inverse property (Theorem 2.8and Corollary 2.9), the uniqueness of various extensions of the linear operator i Ω from the closed subspace S ∞ to the whole space S (Theorems 2.11 and 2.14), thenon-integrability in the spatial domain (Theorem 2.15), and the non-integrabilityin the Fourier domain (Theorem 2.17). Some of those properties will be used toprove Theorem 1.1, which is included at the end of this section.2.1. Polynomial decay property and continuity.Theorem 2.1.
Let γ be a positive number with γ − d Z + , k be the smallestnonnegative integer larger than γ − d , and let Ω ∈ C ∞ ( R d \{ } ) be a homogeneousfunction of degree − γ . If there exist positive constants ǫ and C ǫ such that (2.6) | f ( x ) | ≤ C ǫ (1 + k x k ) − k − d − ǫ for all x ∈ R d , then there exists a positive constant C such that | J Ω f ( x ) | ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d + ǫ (cid:17) (1 + k x k ) γ − d , x ∈ R d . (2.7) Proof.
Noting that (cid:0) ddr (cid:1) s e ir h x ,ξ ′ i = s ! (cid:16) P | i | = s ( i x ) i ξ ′ i i ! (cid:17) e ir h x ,ξ ′ i and (cid:0) ddr (cid:1) k − s ˆ f ( rξ ′ ) =( k − s )! P | j | = k − s ( ξ ′ ) j ∂ j ˆ f ( rξ ′ ) j ! for all 0 ≤ s ≤ k , we obtain from the Leibniz rulethat (cid:16) ddr (cid:17) k (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) (cid:17) = k X s =0 (cid:18) k s (cid:19)n(cid:16) ddr (cid:17) k − s e ir h x ,ξ ′ i o · n(cid:16) ddr (cid:17) k ˆ f ( rξ ′ ) o = (cid:16) X | i | + | j | = k k ! i ! j ! ( i x ) i ( ξ ′ ) i + j ∂ j ˆ f ( rξ ′ ) (cid:17) e ir h x ,ξ ′ i . QIYU SUN AND MICHAEL UNSER
Substituting the above expression into (2.1) we get J Ω f ( x ) = ( − k X | i | + | j | = k k ! i ! j ! ( i x ) i n (2 π ) − d Γ( d − γ )Γ( d + k − γ ) × Z R d e i h x ,ξ i (cid:0) ξ i + j Ω( ξ ) (cid:1) ∂ j ˆ f ( ξ ) dξ o = Γ( d − γ )Γ( d + k − γ ) X | i | + | j | = k k ! i ! j ! ( − x ) i J Ω i + j ( f j )( x ) , (2.8)where Ω i + j ( ξ ) = ( iξ ) i + j Ω( ξ ) and f j ( x ) = x j f ( x ). Denote the inverse Fourier trans-form of Ω k , | k | = k , by K k . Then K k ∈ C ∞ ( R d \{ } ) is a homogeneous functionof degree γ − k − d ([7, Theorems 7.1.16 and 7.1.18]), and hence there exists apositive constant C such that(2.9) | K k ( x ) | ≤ C k x k γ − k − d for all x ∈ R d \{ } . For any ǫ > β ∈ (0 , d ), we have Z R d k x − y k − β (1 + k y k ) − d − ǫ d y ≤ (cid:16) Z k y k≤ ( k x k +1) / + Z ( k x k +1) / ≤k y k≤ k x k +1) + Z k y k≥ k x k +1) (cid:17) k x − y k − β (1 + k y k ) − d − ǫ d y ≤ C (1 + k x k ) − β . (2.10)Combining (2.8), (2.9) and (2.10) yields | J Ω f ( x ) | ≤ C X | i | + | j | = k | x | | i | (cid:12)(cid:12)(cid:12) Z R d K i + j ( x − y ) y j f ( y ) (cid:12)(cid:12)(cid:12) d y ≤ C (1 + k x k ) k Z R d k x − y k γ − k − d (1 + k y k ) k | f ( y ) | d y ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d + ǫ (cid:17) (1 + k x k ) γ − d . This proves the desired polynomial decay estimate (2.7).
RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 9
For any f ∈ S and j ∈ Z d + with | j | = 1, it follows from (2.1) that ∂ j ( J Ω f )( x ) = J Ω ( ∂ j f )( x )= (2 π ) − d Γ( d − γ )Γ( d + k − γ ) Z S d − Z ∞ Ω( ξ ′ )( iξ ′ ) j r k + d − γ − × (cid:16) − ddr (cid:17) k (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) r (cid:17) drdσ ( ξ ′ )= (2 π ) − d Γ( d − γ )Γ( d + k − γ ) Z S d − Z ∞ Ω( ξ ′ )( iξ ′ ) j r k + d − γ − × n r (cid:16) − ddr (cid:17) k (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) (cid:17) − k (cid:16) − ddr (cid:17) k − (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) (cid:17)o drdσ ( ξ ′ )= (cid:16) d + k − γd − γ − k d − γ (cid:17) J Ω j f ( x ) = J Ω j f ( x ) , where Ω j ( ξ ) = ( iξ ) j Ω( ξ ). Applying the argument inductively leads to(2.11) ∂ j ( J Ω f ) = J Ω ( ∂ j f ) = J Ω j f for all f ∈ S and j ∈ Z d + , where Ω j ( ξ ) = ( iξ ) j Ω( ξ ). This together with Theorem 2.1 shows that J Ω f is asmooth function on R d for any Schwartz function f . Corollary 2.2.
Let γ, k and Ω be as in Theorem 2.1. If f satisfies (2.6) for somepositive constants ǫ and C ǫ , then for any j ∈ Z d + with | j | < γ there exists a positiveconstant C j such that (2.12) | ∂ j ( J Ω f )( x ) | ≤ C j (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d + ǫ (cid:17) (1 + k x k ) γ −| j |− d , x ∈ R d . An easy consequence of the above smoothness result about J Ω f is the continuityof the generalized Riesz potential J Ω from S to S ′ . Corollary 2.3.
Let γ be a positive number with γ − d Z + , and let Ω ∈ C ∞ ( R d \{ } ) be a homogeneous function of degree − γ . Then the generalized Riesz potential J Ω associated with the homogeneous function Ω is a continuous linear operator from S to S ′ . Now consider the generalized Riesz potential J Ω when Ω is a homogeneous func-tion of positive degree α . In this case, J Ω f ( x ) = (2 π ) − d Z R d e i h x ,ξ i Ω( ξ ) ˆ f ( ξ ) dξ for all f ∈ S by (2.2). Applying the integration-by-parts technique then gives J Ω f ( x ) = (2 π ) − d ( − i x i ) − X j + k = i i ! j ! k ! Z R d e i h x ,ξ i ∂ j Ω( ξ ) ∂ k ˆ f ( ξ ) dξ for any i ∈ Z d + . This, together with the identity1 = X | l | = ⌈ α ⌉−| j | ( ⌈ α ⌉ − | j | )! l ! (cid:16) iξ k ξ k (cid:17) l ( − iξ ) l , | j | ≤ ⌈ α ⌉ , leads to the following estimate of J Ω f ( x ): | J Ω f ( x ) | ≤ C (1 + k x k ) −⌈ α ⌉ X | j | + | k |≤⌈ α ⌉ , | l | = ⌈ α ⌉−| j | (cid:12)(cid:12)(cid:12) Z R d e i h x ,ξ i Ω j , l ( ξ ) ξ l ∂ k ˆ f ( ξ ) dξ (cid:12)(cid:12)(cid:12) ≤ C (1 + k x k ) −⌈ α ⌉ X | j | + | k |≤⌈ α ⌉ , | l | + | j | = ⌈ α ⌉ | I Ω j , l f l , k ( x ) | , where ⌈ α ⌉ is the smallest integer larger than α , Ω j , l ( ξ ) = ∂ j Ω( ξ )( iξ/ k ξ k ) l , and d f l , k ( ξ ) = ( − iξ ) l ∂ k ˆ f ( ξ ). Note that Ω j , l ∈ C ∞ ( R d \{ } ) is a homogeneous function ofdegree α − ⌈ α ⌉ < | j | + | l | = ⌈ α ⌉ , and also that functions f l , k ( x ) , | k | , | l | ≤ ⌈ α ⌉ are linear combinations of x i ∂ j f ( x ) , | i | , | j | ≤ ⌈ α ⌉ . We then apply Theorem 2.1 toobtain the following polynomial decay estimate of J Ω f when Ω is a homogeneousfunction of positive degree: Proposition 2.4.
Let α be a positive non-integer number, and Ω ∈ C ∞ ( R d \{ } ) be a homogeneous function of degree α . If there exist positive constants ǫ and C ǫ such that X | i |≤⌈ α ⌉ | ∂ i f ( x ) | ≤ C ǫ (1 + k x k ) −⌈ α ⌉− d − ǫ for all x ∈ R d , then there exists a positive constant C such that (2.13) | J Ω f ( x ) | ≤ C (cid:16) X | i |≤⌈ α ⌉ sup z ∈ R d | ∂ i f ( z ) | (1 + k z k ) ⌈ α ⌉ + d + ǫ (cid:17) (1 + k x k ) − α − d for all x ∈ R d . The estimates in (2.7) and (2.13) indicate that the generalized Riesz potential J Ω f has faster polynomial decay at infinity when the degree of the homogeneousfunction Ω becomes larger. Next, we show that the generalized Riesz potential J Ω f has faster polynomial decay at infinity when f has vanishing moments up to someorder; i.e.,(2.14) Z R d x i f ( x ) d x = 0 , | i | ≤ m where m ≥
0. In this case, ∂ i ˆ f ( ) = 0 for all | i | ≤ m , and hence(2.15) ˆ f ( ξ ) = X | k | = m +1 m + 1 k ! Z ξ k ∂ k ˆ f ( tξ )(1 − t ) m dt by the Taylor expansion to ˆ f at the origin. Now we assume that Ω ∈ C ∞ ( R d \{ } )is a homogeneous function of degree α ∈ ( − m − , ∞ ) \ Z . Then | J Ω f ( x ) | ≤ C X | k | = m +1 Z Z k ξ k≤ | ξ | α + m +1 | ∂ k ˆ f ( tξ ) | dξdt + C Z | ξ |≥ | ξ | α | ˆ f ( ξ ) | dξ ≤ C X | i |≤ m +1 sup ξ ∈ R d (cid:0) (1 + k ξ k ) ⌈ α ⌉ + d | ∂ i ˆ f ( ξ ) | (cid:1) (2.16) RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 11 for all x ∈ R d with k x k ≤
1, and | J Ω f ( x ) | ≤ C X | k | = m +1 Z (cid:12)(cid:12)(cid:12) Z R d e − i h x ,ξ i φ ( k x k ξ ) ξ k Ω( ξ ) ∂ k ˆ f ( tξ ) dξ (cid:12)(cid:12)(cid:12) dt + C X | k | = m +1 Z (cid:12)(cid:12)(cid:12) Z R d e − i h x ,ξ i (cid:0) φ ( ξ ) − φ ( k x k ξ ) (cid:1) ξ k Ω( ξ ) ∂ k ˆ f ( tξ ) dξ (cid:12)(cid:12)(cid:12) dt + C (cid:12)(cid:12)(cid:12) Z R d e − i h x ,ξ i (cid:0) − φ ( ξ ) (cid:1) Ω( ξ ) ˆ f ( ξ ) dξ (cid:12)(cid:12)(cid:12) ≤ C (1 + k x k ) −⌈ α ⌉− m − d n X | k | = m +1 , | j |≤⌈ α ⌉ + m + d Z Z R d (cid:12)(cid:12)(cid:12) ∂ j (cid:0) φ ( k x k ξ ) ξ k Ω( ξ ) ∂ k ˆ f ( tξ ) (cid:1)(cid:12)(cid:12)(cid:12) dξdt o + C (1 + k x k ) −⌈ α ⌉− m − d − n X | k | = m +1 , | j |≤⌈ α ⌉ + m + d +1 Z Z R d (cid:12)(cid:12)(cid:12) ∂ j (cid:0) ( φ ( ξ ) − φ ( k x k ξ )) ξ k Ω( ξ ) ∂ k ˆ f ( tξ ) (cid:1)(cid:12)(cid:12)(cid:12) dξdt o + C (1 + k x k ) −⌈ α ⌉− m − d − × n X | j |≤⌈ α ⌉ + m + d +1 Z R d (cid:12)(cid:12)(cid:12) ∂ j (cid:0) (1 − φ ( ξ ))Ω( ξ ) ˆ f ( ξ ) (cid:1)(cid:12)(cid:12)(cid:12) dξ o ≤ C (cid:16) X | i |≤⌈ α ⌉ +2 m + d +2 sup ξ ∈ R d (1 + k ξ k ) ⌈ α ⌉ + d | ∂ i ˆ f ( ξ ) | (cid:17) (1 + k x k ) − α − m − d − (2.17)for all x ∈ R d with k x k ≥
1, where φ is a C ∞ function such that φ ( ξ ) = 1 forall ξ in the unit ball B ( ,
1) centered at the origin, and φ ( ξ ) = 0 for all ξ not inthe ball B ( ,
2) with radius 2 and center at the origin. This proves the followingresult about the generalized Riesz potential J Ω f when f has vanishing momentsupto some order. Proposition 2.5.
Let m ≥ , α ∈ ( − m − , ∞ ) \ Z , and Ω ∈ C ∞ ( R d \{ } ) be ahomogeneous function of degree α . Then the following statements hold. (i) If f satisfies (2.14) and (2.18) X | i |≤⌈ α ⌉ +2 m + d +2 sup ξ ∈ R d (1 + k ξ k ) ⌈ α ⌉ + d | ∂ i ˆ f ( ξ ) | < ∞ , then there exists a positive constant C such that | J Ω f ( x ) | ≤ C (cid:16) X | i |≤⌈ α ⌉ +2 m + d +2 sup ξ ∈ R d (1 + k ξ k ) ⌈ α ⌉ + d | ∂ i ˆ f ( ξ ) | (cid:17) × (1 + k x k ) − α − m − d − for all x ∈ R d . (2.19)(ii) If f satisfies (2.14) and (2.20) X | i |≤ max( ⌈ α ⌉ + d, sup z ∈ R d (cid:0) (1 + k z k ) ⌈ α ⌉ +2 m +2 d +2+ ǫ | ∂ i f ( z ) | (cid:1) < ∞ for some ǫ > , then | J Ω f ( x ) | ≤ C (cid:16) X | i |≤ max( ⌈ α ⌉ + d, sup z ∈ R d (cid:0) (1 + k z k ) ⌈ α ⌉ +2 m +2 d +2+ ǫ | ∂ i f ( z ) | (cid:17) × (1 + k x k ) − α − m − d − for all x ∈ R d . (2.21)The conclusions in Proposition 2.5 do not apply to the generalized Riesz potential J Ω f where Ω ∈ C ∞ ( R d \{ } ) is a homogeneous function of degree zero. In this case,applying the argument used to establish (2.16) and (2.17), we have that | J Ω f ( x ) | ≤ C X | i |≤ m +1 sup ξ ∈ R d (cid:0) (1 + k ξ k ) d + ǫ | ∂ i ˆ f ( ξ ) | (cid:1) (2.22)for all x ∈ R d with k x k ≤
1, and | J Ω f ( x ) | ≤ C (1 + k x k ) − m − d × n X | k | = m +1 , | j |≤ m + d Z Z R d (cid:12)(cid:12) ∂ j (cid:0) φ ( k x k ξ ) ξ k Ω( ξ ) ∂ k ˆ f ( tξ ) (cid:1)(cid:12)(cid:12) dξdt o + C (1 + k x k ) − m − d − X | k | = m +1 , | j | + | l |≤ m + d +1 , | j |≤ m + d Z Z R d (cid:12)(cid:12) ∂ j (cid:0) ( φ ( ξ ) − φ ( k x k ξ )) ξ k Ω( ξ ) (cid:1)(cid:12)(cid:12) × (cid:12)(cid:12) ∂ k + l ˆ f ( tξ ) (cid:1)(cid:12)(cid:12) dξdt o + C (1 + k x k ) − m − d − X | k | = m +1 , | j | + | l |≤ m + d +2 , | l |≤ Z Z R d (cid:12)(cid:12) ∂ j (cid:0) ( φ ( ξ ) − φ ( k x k ξ )) ξ k Ω( ξ ) (cid:1)(cid:12)(cid:12) × (cid:12)(cid:12) ∂ k + l ˆ f ( tξ ) (cid:1)(cid:12)(cid:12) dξdt o + C (1 + k x k ) − m − d − X | j |≤ m + d +1 Z R d (cid:12)(cid:12) ∂ j (cid:0) (1 − φ ( ξ ))Ω( ξ ) ˆ f ( ξ ) (cid:1)(cid:12)(cid:12) dξ ≤ C (cid:16) X | i |≤ m + d +2 sup ξ ∈ R d (1 + k ξ k ) d + ǫ | ∂ i ˆ f ( ξ ) | (cid:17) (1 + k x k ) − m − d − (2.23)for all x ∈ R d with k x k ≥
1, where ǫ ∈ (0 , Proposition 2.6.
Let Ω ∈ C ∞ ( R d \{ } ) be a homogeneous function of degree zero.Then the following statements hold. (i) If f satisfies (2.14) for some m ≥ and X | i |≤ m + d +2 sup ξ ∈ R d (1 + k ξ k ) d + ǫ | ∂ i ˆ f ( ξ ) | < ∞ for some ǫ > , then there exists a positive constant C such that | J Ω f ( x ) | ≤ C (cid:16) X | i |≤ m + d +2 sup ξ ∈ R d (1+ k ξ k ) d + ǫ | ∂ i ˆ f ( ξ ) | (cid:17) (1+ k x k ) − m − d − for all x ∈ R d . (ii) If f satisfies (2.14) for some m ≥ and X | i |≤ d +1 sup z ∈ R d (cid:0) (1 + k z k ) m +2 d +2+ ǫ | ∂ i f ( z ) | (cid:1) < ∞ RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 13 for some ǫ > , then | J Ω f ( x ) | ≤ C (cid:16) X | i |≤ d +1 sup z ∈ R d (cid:0) (1 + k z k ) m +2 d +2+ ǫ | ∂ i f ( z ) | (cid:17) × (1 + k x k ) − m − d − for all x ∈ R d . Translation-invariance and dilation-invariance.
In this subsection, weshow that the generalized Riesz potential J Ω from S to S ′ is dilation-invariant andtranslation-invariant, and that its restriction on the closed subspace S ∞ of S is thesame as the linear operator i Ω on S ∞ . Theorem 2.7.
Let γ ∈ R with γ − d Z + , Ω ∈ C ∞ ( R d \{ } ) be a homogeneousfunction of degree − γ , and let J Ω be defined by (2.1) . Then (i) J Ω is dilation-invariant; (ii) J Ω is translation-invariant; and (iii) d J Ω f ( ξ ) = Ω( ξ ) ˆ f ( ξ ) for any function f ∈ S ∞ .Proof. (i) For any f ∈ S and any t > J Ω ( δ t f )( x ) = (2 πt ) − d Γ( d − γ )Γ( d + k − γ ) Z S d − Z ∞ Ω( ξ ′ ) r k − γ + d − × (cid:16) − ddr (cid:17) k (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ /t ) (cid:17) drdσ ( ξ ′ ) = t − γ δ t ( J Ω f )( x ) , where the first equality follows from c δ t f ( ξ ) = t − d ˆ f ( ξ/t ) and the second equalityis obtained by change of variables. This leads to the dilation-invariance of thegeneralized Riesz potential J Ω . (ii) For any f ∈ S and a vector x ∈ R d , we obtain from (2.1) that J Ω ( τ x f )( x ) = (2 π ) − d Γ( d − γ )Γ( d + k − γ ) Z S d − Z ∞ r k − γ + d − Ω( ξ ′ ) × (cid:0) − ddr (cid:1) k (cid:16) e ir h x − x ,ξ ′ i ˆ f ( rξ ′ ) (cid:17) drdσ ( ξ ′ ) = J Ω f ( x − x ) , where k is a nonnegative integer larger than γ − d . This shows that the generalizedRiesz potential J Ω is translation-invariant. (iii) The third conclusion follows by taking Fourier transform of the equation(2.5) on both sides.2.3.
Composition and left-inverse.
In this subsection, we consider the compo-sition and left-inverse properties of generalized Riesz potentials.
Theorem 2.8.
Let γ and γ ∈ R satisfy γ < d, γ + γ < d and γ − d Z + ,and let Ω , Ω ∈ C ∞ ( R d \{ } ) be homogeneous functions of degree − γ and − γ respectively. Then (2.24) J Ω ( J Ω f ) = J Ω Ω f for all f ∈ S . As a consequence of Theorem 2.8, we have the following result about left-invertibility of the generalized Riesz potential J Ω . Corollary 2.9.
Let γ ∈ ( − d, ∞ ) with γ − d Z + and Ω ∈ C ∞ ( R d \{ } ) behomogeneous of degree − γ with Ω( ξ ) = 0 for all ξ ∈ S d − . Then J Ω J Ω − is anidentity operator on S . If we further assume that γ ∈ ( − d, d ) , then both J Ω − J Ω and J Ω J Ω − are identity operators on S . Taking Ω( ξ ) = k ξ k − γ in the above corollary yields that the linear operator I γ in(1.10) is a left-inverse of the fractional Laplacian ( −△ ) γ/ . Corollary 2.10.
Let γ be a positive number with γ − d Z + . Then I γ is aleft-inverse of the fractional Laplacian ( −△ ) γ/ .Proof of Theorem 2.8. Let k be the smallest nonnegative integer such that k − γ + d >
0, and set Ω( ξ ) = Ω ( ξ )Ω ( ξ ). If k = 0, then the conclusion (2.24) followsfrom (2.2). Now we assume that k ≥
1. Then J Ω ( J Ω f )( x ) = (2 π ) d Γ( d − γ )Γ( d + k − γ ) lim ǫ → Z S d − Z ∞ ǫ Ω( ξ ′ ) r k + d − γ − × n r (cid:16) − ddr (cid:17) k (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) r − γ − (cid:17) − k (cid:16) − ddr (cid:17) k − (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) r − γ − (cid:17)o drdσ ( ξ ′ )= (2 π ) d Γ( d + 1 − γ )Γ( d + k − γ ) lim ǫ → Z S d − Z ∞ ǫ Ω( ξ ′ ) r k + d − γ − × (cid:16) − ddr (cid:17) k − (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) r − γ − (cid:17) drdσ ( ξ ′ )= · · · = (2 π ) − d Γ( d + k − γ )Γ( d + k − γ ) lim ǫ → Z S d − Z ∞ ǫ Ω( ξ ′ ) r k + d − γ − × (cid:16) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) r − γ − k (cid:17) drdσ ( ξ ′ )= J Ω Ω f ( x ) for all x ∈ R d , where the second equality is obtained by applying the integration-by-parts tech-nique and using the fact that ǫ k + d − γ (cid:0) ddr (cid:1) k − (cid:0) e ir h x ,ξ ′ i ˆ f ( rξ ′ ) r − γ − (cid:1)(cid:12)(cid:12) r = ǫ convergesto zero uniformly on ξ ∈ S d − under the assumption that γ + γ < d . The con-clusion (2.24) then follows.2.4. Translation-invariant and dilation-invariant extensions of the linearoperator i Ω . In this subsection, we show that the generalized Riesz potential J Ω in (2.1) is the only continuous linear operator from S to S ′ that is translation-invariant and dilation-invariant, and that is an extension of the linear operator i Ω in (2.4) from the closed subspace S ∞ to the whole space S . Theorem 2.11.
Let γ be a positive number with γ − d Z + , Ω ∈ C ∞ ( R d \{ } ) bea nonzero homogeneous function of degree − γ , and let J Ω be defined by (2.1) . Then I is a continuous linear operator from S to S ′ such that I is dilation-invariant andtranslation-invariant, and that the restriction of I on S ∞ is the same as the linearoperator i Ω in (2.4) if and only if I = J Ω . To prove Theorem 2.11, we need two technical lemmas about extensions of thelinear operator i Ω on S ∞ . Lemma 2.12.
Let γ be a positive number with γ − d Z + , Ω ∈ C ∞ ( R d \{ } ) be a homogeneous function of degree − γ , and let J Ω be defined by (2.1) . Then a RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 15 continuous linear operator I from S to S ′ is an extension of the linear operator i Ω on S ∞ if and only if (2.25) If = J Ω f + X | i |≤ N ∂ i ˆ f ( ) i ! H i for some integer N and tempered distributions H i , i ∈ Z d + with | i | ≤ N .Proof. The sufficiency follows from Theorem 2.7 and the assumption that H i , | i | ≤ N , in (2.25) are tempered distributions. Now the necessity. By Corollary 2.3 andTheorem 2.7, I − J Ω is a continuous linear operator from S to S ′ that satisfies that( I − J Ω ) f = 0 for all f ∈ S ∞ . This implies that the inverse Fourier transform ofthe tempered distribution ( I − J Ω ) ∗ g is supported on the origin for any Schwartzfunction g . Hence there exist an integer N and tempered distribution H i , | i | ≤ N ,such that F − (( I − J Ω ) ∗ g ) = P | i |≤ N h g, H i i δ ( i ) / i !, where the tempered distributions δ ( i ) , i ∈ Z d + , are defined by h δ ( i ) , f i = ∂ i f ( ) [7, Theorem 2.3.4]. Then h ( I − J Ω ) f, g i = h ˆ f , F − ( I − J Ω ) ∗ g i = P | i |≤ N h H i , g i ∂ i ˆ f ( ) / i ! for all Schwartz functions f and g , and hence (2.25) is established. Lemma 2.13.
Let γ be a positive number with γ − d Z + , and consider thecontinuous linear operator K from S to S ′ : (2.26) Kf = X | i |≤ N ∂ i ˆ f ( ) i ! H i , f ∈ S where N ∈ Z + and H i , | i | ≤ N , are tempered distributions, Then the followingstatements hold. (i) The equation (2.27) K ( δ t f ) = t − γ δ t ( Kf ) holds for any f ∈ S and t > if and only if for every i ∈ Z d + with | i | ≤ N , H i is homogeneous of degree γ − d − | i | . (ii) The linear operator K is translation-invariant if and only if there exists apolynomial P of degree at most N such that H i = ( − i∂ ) i P for all i ∈ Z d + with | i | ≤ N . (iii) The linear operator K is translation-invariant and satisfies (2.27) if andonly if H i = 0 for all i ∈ Z d + with | i | ≤ N .Proof. (i) The sufficiency follows from the homogeneous assumption on H i , | i | ≤ N , and the observation that(2.28) ∂ i c δ t f ( ) = t − d −| i | ∂ i ˆ f ( ) for all f ∈ S and i ∈ Z d + . Now the necessity. Let φ be a C ∞ function such that φ ( ξ ) = 1 for all ξ ∈ B ( , φ ( ξ ) = 0 for all ξ B ( , B ( x , r ) is the ball with center x ∈ R d andradius r >
0. Define ψ i ∈ S , i ∈ Z d + , with the help of the Fourier transform by(2.29) b ψ i ( ξ ) = ξ i i ! φ ( ξ ) . One may verify that(2.30) ∂ i ′ b ψ i ( ) = (cid:26) i ′ = i , i ′ = i . For any i ∈ Z d + with | i | ≤ N , the homogeneous property of the tempered distribu-tion H i follows by replacing f in (2.27) by ψ i and using (2.30). (ii) ( ⇐ =) Given f ∈ S and x ∈ R d , K ( τ x f )( x ) = X | i |≤ N X j + k = i ( − i x ) k k ! ∂ j ˆ f ( ) j ! ( − i∂ ) i P ( x )= X | j |≤ N ( − i ) j ∂ j ˆ f ( ) j ! (cid:16) X | k |≤ N −| j | ∂ j + k P ( x ) k ! ( − x ) k (cid:17) = X | j |≤ N ( − i ) j ∂ j ˆ f ( ) j ! ∂ j P ( x − x ) = Kf ( x − x ) , (2.31)where the first equality follows from(2.32) ∂ i d τ x f ( ) = X j ≤ i (cid:18) ij (cid:19) ( − i x ) i − j ∂ j ˆ f ( ) , and the third equality is deducted from the Taylor expression of the polynomial ∂ j P of degree at most N − | j | .(= ⇒ ) By (2.32) and the translation-invariance of the linear operator K ,(2.33) X | i |≤ N X j + k = i ( − i x ) k k ! ∂ j ˆ f ( ) j ! H i = X | i |≤ N ∂ j ˆ f ( ) j ! τ x H j holds for any Schwartz function f and x ∈ R d . Replacing f in the above equationby the function ψ in (2.29) and then using (2.30), we get(2.34) τ x H = X | i |≤ N ( − i x ) i i ! H i . This implies that h H , g ( · + x ) i = P | i |≤ N ( − i x ) i i ! h H i , g i for any Schwartz function g . By taking partial derivatives ∂ k , | k | = N + 1, with respect to x of both sides ofthe above equation, using the fact that ∂ k x i = 0 for all k ∈ Z + with | k | = N + 1,and then letting x = , we obtain that h H , ∂ k g i = 0 holds for any g ∈ S and k ∈ Z + with | k | = N + 1. Hence H = P for some polynomial P of degree atmost N . The desired conclusion about H i , | i | ≤ N , then follows from (2.34) and τ x H ( x ) = P | i |≤ N ( − x ) i i ! ∂ i P ( x ) by the Taylor expansion of the polynomial P . (iii) Clearly if H i = 0 for all | i | ≤ N , then Kf = 0 for all f ∈ S and hence K is translation-invariant and satisfies (2.27). Conversely, if K is translation-invariantand satisfies (2.27), it follow from the conclusions (i) and (ii) that for every i ∈ Z d + with | i | ≤ N , H i is homogeneous of degree γ − d − | i | 6∈ Z and also a polynomialof degree at most N − | i | . Then H i = 0 for all i ∈ Z d + with | i | ≤ N becausethe homogeneous degree of any nonzero polynomial is a nonnegative integer if it ishomogeneous.We now have all of ingredients to prove Theorem 2.11. Proof of Theorem 2.11.
The sufficiency follows from Corollary 2.3 and Theorem2.7. Now the necessity. By Lemma 2.12, there exist an integer N and tempered RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 17 distributions H i , | i | ≤ N , such that (2.25) holds. Define Kf = P | i |≤ N ∂ i ˆ f ( ) i ! H i forany f ∈ S . Then Kf is a continuous linear operator from S to S ′ and(2.35) If = J Ω f + Kf, f ∈ S . Moreover the linear operator K satisfies (2.27) and is translation-invariant by (2.35),Theorem 2.7 and the assumption on I . Then Kf = 0 for all f ∈ S by Lemma 2.13.This together with (2.35) proves the desired conclusion that I = J Ω .2.5. Translation-invariant extensions of the linear operator i Ω with ad-ditional localization in the Fourier domain. Given a nonzero homogeneousfunction Ω ∈ C ∞ ( R d \{ } ) of degree − γ , we recall from (2.2) and Theorem 2.7 that J Ω is translation-invariant and the Fourier transform of J Ω f belongs to K when γ ∈ (0 , d ), where(2.36) K = n h : Z R d | h ( ξ ) | (1 + k ξ k ) − N dξ < ∞ for some N ≥ o . In fact, the generalized Riesz potential J Ω is the only extension of the linear oper-ator i Ω on S ∞ to the whole space S with the above two properties. Theorem 2.14.
Let γ > with γ − d Z + , Ω ∈ C ∞ ( R d \{ } ) be a nonzerohomogeneous function of degree − γ , and the continuous linear operator I from S to S ′ be an extension of the linear operator i Ω on S ∞ such that the Fourier transformof If belongs to K for all f ∈ S . Then I is translation-invariant if and only if I = J Ω and γ ∈ (0 , d ) .Proof. The sufficiency follows from (2.2) and Theorem 2.7. Now we prove thenecessity. By the assumption on the linear operator I , applying an argument similarto the proof of Lemma 2.12, we can find a family of functions g i ∈ K , | i | ≤ N , suchthat c If ( ξ ) = (cid:16) ˆ f ( ξ ) − X | i |≤ γ − d ∂ i ˆ f ( ) i ! ξ i (cid:17) Ω( ξ ) + X | i |≤ N ∂ i ˆ f ( ) i ! g i ( ξ )(2.37)for any Schwartz function f . This together with (2.32) and the translation-invarianceof the linear operator I implies that − X | i |≤ γ − d X j + k = i ∂ j ˆ f ( ) k ! j ! ( − i x ) k ξ i Ω( ξ ) + X | i |≤ N X j + k = i ∂ j ˆ f ( ) k ! j ! ( − i x ) k g i ( ξ )= e i x ξ (cid:16) − X | i |≤ γ − d ∂ i ˆ f ( ) i ! ξ i Ω( ξ ) + X | i |≤ N ∂ i ˆ f ( ) i ! g i ( ξ ) (cid:17) . As x ∈ R d in (2.38) is chosen arbitrarily, we conclude that − X | i |≤ γ − d ∂ i ˆ f ( ) i ! ξ i Ω( ξ ) + X | i |≤ N ∂ i ˆ f ( ) i ! g i ( ξ ) = 0 for all f ∈ S . Substituting the above equation into (2.37), we then obtain c If ( ξ ) = ˆ f ( ξ )Ω( ξ ) forall f ∈ S . This, together with the observation that ˆ f Ω ∈ K for all f ∈ S if andonly if γ < d , leads to the desired conclusion that I = J Ω and γ ∈ (0 , d ). Non-integrability in the spatial domain.
Let γ > γ − d Z + and Ω ∈ C ∞ ( R d \{ } ) be a nonzero homogeneous function of degree − γ . For anySchwartz function f , there exists a positive constant C by Theorem 2.1 such that | J Ω f ( x ) | ≤ C (1 + k x k ) γ − d for all x ∈ R d . Hence J Ω f ∈ L p , ≤ p ≤ ∞ , when γ < d (1 − /p ). In this subsection, we show that the above p -integrability propertyfor the generalized Riesz potential J Ω is no longer true when γ ≥ d (1 − /p ). Theorem 2.15.
Let ≤ p ≤ ∞ , < γ ∈ [ d (1 − /p ) , ∞ ) \ Z and Ω ∈ C ∞ ( R d \{ } ) be a nonzero homogeneous function of degree − γ . Then there exists a Schwartzfunction f such that J Ω f L p . Letting Ω( ξ ) = k ξ k − γ in Theorem 2.15 leads to the conclusion mentioned in theabstract: Corollary 2.16.
Let ≤ p ≤ ∞ and d (1 − /p ) ≤ γ Z + . Then I γ f is not p -integrable for some function f ∈ S .Proof of Theorem 2.15. Let the Schwartz functions φ and ψ i , i ∈ Z d + , be as in theproof of Lemma 2.13. We examine three cases to prove the theorem. Case I: d (1 − /p ) ≤ γ < min( d, d (1 − /p ) + 1). In this case, 1 ≤ p < ∞ and(2.38) J Ω ψ ( x ) = Z R d K ( x − y ) ψ ( y ) d y , by (2.2), where K is the inverse Fourier transform of Ω. By [7, Theorems 7.1.16and 7.1.18], K ∈ C ∞ ( R d \{ } ) is a homogeneous function of order γ − d ∈ ( − d, | ∂ i K ( x ) | ≤ C k x k γ − d −| i | for all i ∈ Z d + with | i | ≤ . Using (2.38) and (2.39), and noting that ψ ∈ S satisfies R R d ψ ( y ) d y = 1, weobtain that for all x ∈ R d with k x k ≥ | J Ω ψ ( x ) − K ( x ) | ≤ Z k y k≤k x k / | K ( x − y ) − K ( x ) || ψ ( y ) | d y + (cid:16) Z k x k / ≤k y k≤ k x k + Z k x k≤k y k (cid:17) | K ( x − y ) || ψ ( y ) | d y + | K ( x ) | Z k y k≥k x k / | ψ ( y ) | d y ≤ C (1 + k x k ) γ − d − . (2.40)We notice that R k x k≥ (1 + k x k ) ( γ − d − p d x < ∞ and R k x k≥ | K ( x ) | p d x = ∞ because K is a nonzero homogenous function of degree γ − d and d − p < ( d − γ ) p ≤ d . Theabove two observations together with the estimate in (2.40) prove that J Ω ψ L p ,the desired conclusion with f = ψ . Case II: d < γ < d (1 − /p ) + 1. In this case, d < p ≤ ∞ and J Ω ψ ( x ) = 1 d − γ X | j | =1 J Ω j ( ϕ j )( x ) + 1 d − γ X | i | =1 ( − x ) i J Ω i ψ ( x )(2.41)by taking k = 1 in (2.8), where Ω i ( ξ ) = ( iξ ) i Ω( ξ ) and ϕ i ( x ) = x i ψ ( x ). Let K i be the inverse Fourier transform of the function Ω i , | i | = 1. Noticing that Ω i is RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 19 homogeneous of degree − γ + 1 and that R R d ϕ i ( x ) d x = 0, we then apply similarargument to the one used in establishing (2.40) and obtain | J Ω i ( ϕ i )( x ) | + | J Ω i ψ ( x ) − K i ( x ) | ≤ C k x k γ − d − if k x k ≥ . Hence(2.42) Z k x k≥ (cid:12)(cid:12) J Ω ψ ( x ) − d − γ X | i | =1 ( − x ) i K i ( x ) (cid:12)(cid:12) p d x ≤ C Z k x k≥ k x k ( γ − d − p d x < ∞ if d < p < ∞ and(2.43) sup k x k≥ (cid:12)(cid:12) J Ω ψ ( x ) − d − γ X | i | =1 ( − i x ) i K i ( x ) (cid:12)(cid:12) ≤ C sup k x k≥ k x k γ − d − < ∞ if p = ∞ . Set K ( x ) := P | i | =1 ( − x ) i K i ( x ). Then K is homogeneous of degree γ − d by the assumption on Ω, and is not identically zero because h K, g i = Z R d Ω( ξ ) (cid:16) X | i | =1 ξ i ∂ i ˆ g ( ξ ) (cid:17) dξ = − Z R d (cid:16) X | i | =1 ∂ i ( ξ i Ω( ξ )) (cid:17) ˆ g ( ξ ) dξ = Z S d − Z ∞ (cid:0) d Ω( rξ ′ ) + r ddr Ω( rξ ′ ) (cid:1) ˆ g ( rξ ′ ) r d − drdσ ( ξ ′ )= ( d − γ ) Z R d Ω( ξ )ˆ g ( ξ ) dξ g ∈ S ∞ . Thus R k x k≥ | K ( x ) | p d x = + ∞ when d < p < ∞ , and K ( x ) isunbounded on R d \ B ( ,
1) when p = ∞ . This together with (2.42) and (2.43)proves that J Ω ψ L p and hence the desired conclusion with f = ψ . Case III: γ ≥ d (1 − /p ) + 1 . Let k be the integer such that d (1 − /p ) ≤ γ − k < d (1 − /p ) + 1, and set Ω j ( ξ ) = ( iξ ) j Ω( ξ ) , | j | = k . Noting that J Ω ψ j ( x ) = J Ω j ψ ( x ) / j ! and Ω j is homogeneous of degree − γ + k , we have obtained from theconclusions in the first two cases that J Ω ψ j L p . Hence the desired conclusionfollows by letting f = ψ j with | j | = k .2.7. Non-integrability in the Fourier domain. If γ < d , it follows from (2.2)that for Schwartz functions f and g , h J Ω f, g i can be expressed as a weighted integralof ˆ g :(2.44) h J Ω f, g i = Z R d h ( ξ )ˆ g ( ξ ) dξ, where h ( ξ ) = (2 π ) − d Ω( − ξ ) ˆ f ( − ξ ) ∈ K . In this subsection, we show that theabove reformulation (2.44) to define h J Ω f, g i via a weighted integral of ˆ g cannot be extended to γ > d . Theorem 2.17.
Let γ ∈ ( d, ∞ ) \ Z , Ω ∈ C ∞ ( R d \{ } ) be a nonzero homogeneousfunction of degree − γ , and let J Ω be defined by (2.1) . Then there exists a Schwartzfunction f such that the Fourier transform of J Ω f does not belong to K .Proof. Let φ and ψ be the Schwartz functions in the proof of Lemma 2.13, andlet g ∈ S ∞ be so chosen that its Fourier transform ˆ g is supported in B ( ,
1) and satisfies R R d Ω( ξ )ˆ g ( − ξ ) dξ = 1. Now we prove that [ J Ω ψ K . Suppose on thecontrary that [ J Ω ψ ∈ K . Then h J Ω ψ , n − d g ( · /n ) i = (2 π ) − d Γ( d − γ )Γ( d + k − γ ) Z S d − Z ∞ ǫ r k + d − γ − Ω( ξ ′ ) (cid:16) − ddr (cid:17) k (cid:16) b ψ ( rξ ′ )ˆ g ( − rnξ ′ ) (cid:17) drdσ ( ξ ′ )= (2 π ) − d Z R d ˆ g ( − nξ )Ω( ξ ) dξ = (2 π ) − d n γ − d Z R d Ω( ξ )ˆ g ( − ξ ) dξ → + ∞ as n → ∞ (2.45)by (2.1) and (2.5). On the other hand, |h J Ω ψ , n − d g ( · /n ) i| = (2 π ) − d (cid:12)(cid:12)(cid:12) Z R d [ J Ω ψ ( ξ )ˆ g ( − nξ ) dξ (cid:12)(cid:12)(cid:12) ≤ (2 π ) − d k ˆ g k ∞ Z | ξ |≤ /n | [ J Ω ψ ( ξ ) | dξ → n → ∞ , (2.46)where we have used the hypothesis that [ J Ω ψ ∈ K to obtain the limit. The limitsin (2.45) and (2.46) contradict each other, and hence the Fourier transform J Ω ψ does not belong to K .2.8. Proof of Theorem 1.1.
Observe that J Ω = I γ when Ω( ξ ) = k ξ k − γ and γ >
0, and that(2.47) J Ω = ( −△ ) − γ/ if Ω( ξ ) = k ξ k − γ and γ < . Then the necessity holds by Theorem 2.11, while the sufficiency follows from Corol-lary 2.3, Theorem 2.7, and Corollary 2.9.3.
Integrable Riesz Potentials
In Section 2, we have shown that the various attempts for defining a proper (inte-grable) Riesz potential that is translation-invariant are doomed to failure for γ > d .We now proceed by providing a fix which is possible if we drop the translation-invariance requirement.Let 1 ≤ p ≤ ∞ , γ ∈ R , and Ω ∈ C ∞ ( R d \{ } ) be a homogeneous function ofdegree − γ . We define the linear operator U Ω ,p from S to S ′ with the help of theFourier transform by(3.1) F ( U Ω ,p f )( ξ ) = (cid:16) ˆ f ( ξ ) − X | i |≤ γ − d (1 − /p ) ∂ i ˆ f ( ) i ! ξ i (cid:17) Ω( ξ ) , f ∈ S . We call the linear operator U Ω ,p a p -integrable Riesz potential associated with thehomogenous function Ω, or integrable Riesz potential for brevity, as(3.2) U Ω ,p = I γ,p if Ω( ξ ) = k ξ k − γ . Define(3.3) U ∗ Ω ,p f ( x ) = (2 π ) − d Z R d (cid:16) e i h x ,ξ i − X | i |≤ γ − d + d/p ( i x ) i ξ i i ! (cid:17) Ω( − ξ ) ˆ f ( ξ ) dξ, f ∈ S . RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 21
Then U ∗ Ω ,p is the adjoint operator of the integrable Riesz potenrial U Ω ,p :(3.4) h U Ω ,p f, g i = h f, U ∗ Ω ,p g i for all f, g ∈ S . If γ satisfies 0 < γ < d (1 − /p ), then(3.5) U Ω ,p f = J Ω f for all f ∈ S . Hence in this case, it follows from Theorem 2.7 that U Ω ,p is dilation-invariantand translation-invariant, and a continuous extension of the linear operator i Ω on the closed subspace S ∞ to the whole space S . Moreover U Ω ,p f ∈ L p and F ( U Ω ,p f ) ∈ L q , ≤ q ≤ p/ ( p − f by Theorem 2.1 andthe following estimate: |F ( U Ω ,p f )( ξ ) | ≤ C k ξ k − γ (1 + k ξ k ) γ − d − for all ξ ∈ R d . So from now on, we implicitly assume that γ ≥ d (1 − /p ), except when mentionedotherwise.In the sequel, we investigate with the properties of the p -integrable Riesz poten-tial U Ω ,p associated with a homogenous function Ω, such as dilation-invariance andtranslation-variance (Theorem 3.1), L p/ ( p − -integrability in the Fourier domain(Corollary 3.2), L p -integrability in the spatial domain (Theorem 3.5 and Corollary3.6), composition and left-inverse property (Theorem 3.3 and Corollary 3.4), theuniqueness of dilation-invariant extension of the linear operator i Ω from the closedsubspace S ∞ to the whole space S with additional integrability in the spatial do-main and in the Fourier domain (Theorems 3.7 and 3.8). The above properties ofthe p -integrable Riesz potential associated with a homogenous function will be usedto prove Theorem 1.2 in the last subsection.3.1. Dilation-invariance, translation-variance and integrability in the Fourierdomain.Theorem 3.1.
Let ≤ p ≤ ∞ , γ ≥ d (1 − /p ) , k be the integral part of γ − d (1 − /p ) , Ω ∈ C ∞ ( R d \{ } ) be a nonzero homogeneous function of degree − γ , and let U Ω ,p be defined as in (3.1) . Then the following statements hold. (i) U Ω ,p is dilation-invariant. (ii) U Ω ,p is not translation-invariant. (iii) If sup x ∈ R d | f ( x ) | (1 + k x k ) k + d +1+ ǫ < ∞ for some ǫ > , then there existsa positive constant C independent on f such that (3.6) |F ( U Ω ,p f )( ξ ) | ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d +1+ ǫ (cid:17) k ξ k k − γ +1 (1 + k ξ k ) − for all ξ ∈ R d . (iv) U Ω ,p is a continuous linear operator from S to S ′ , and an extension of theoperator i Ω on the subspace S ∞ to the whole space S . As a consequence of Theorem 3.1, we have the following result about the L p/ ( p − -integrability of the Fourier transform of U Ω ,p f for f ∈ S . Corollary 3.2.
Let ≤ p ≤ ∞ and γ ≥ d (1 − /p ) satisfy either p = 1 or γ − d (1 − /p ) Z + and < p ≤ ∞ , k be the integral part of γ − d (1 − /p ) , Ω ∈ C ∞ ( R d \{ } ) be a homogeneous function of degree − γ , and let U Ω ,p be definedas in (3.1) . Then the Fourier transform of U Ω ,p f belongs to L p/ ( p − for any f ∈ S . Proof of Theorem 3.1. (i)
Given any t > f ∈ S , F ( U Ω ,p ( δ t f ))( ξ ) = t − d (cid:16) ˆ f (cid:0) ξt (cid:1) − X | i |≤ γ − d + d/p ∂ i ˆ f ( ) i ! (cid:0) ξt (cid:1) i (cid:17) Ω( ξ ) = t − d − γ F ( U Ω ,p f ) (cid:0) ξt (cid:1) . This proves the dilation-invariance of the linear operator U Ω ,p . (ii) Suppose, on the contrary, that U Ω ,p is translation-invariant. Then(3.7) Ω( ξ ) X | i |≤ γ − d + d/p ∂ i d τ x f ( ) i ! ξ i = Ω( ξ ) e − i h x ,ξ i X | i |≤ γ − d + d/p ∂ i ˆ f ( ) i ! ξ i , ξ ∈ R d for all x ∈ R d and f ∈ S . Note that the left-hand side of equation (3.7) is apolynomial in x by (2.32) while its right hand side is a trigonometric function of x . Hence both sides must be identically zero, which implies that(3.8) Ω( ξ ) X | i |≤ γ − d + d/p ∂ i ˆ f ( ) i ! ξ i = 0 , ξ ∈ R d for all f ∈ S . Replacing f in the above equation by the function ψ in (2.29) andusing (2.30) and the assumption γ ≥ d (1 − /p ) leads to a contradiction. (iii) By the assumption on the homogeneous function Ω, | Ω( ξ ) | ≤ C k ξ k − γ .Then for ξ ∈ R d with k ξ k ≥ |F ( U Ω ,p f )( ξ ) | ≤ C (cid:16) k ˆ f k ∞ + X | i |≤ k k ∂ i ˆ f k ∞ k ξ k | i | (cid:17) k ξ k − γ ≤ C (cid:16) X | i |≤ k +1 k ∂ i ˆ f k ∞ (cid:17) k ξ k k − γ by (3.1), and for ξ ∈ R d with k ξ k ≤ |F ( U Ω ,p f )( ξ ) | ≤ C (cid:16) X | i |≤ k +1 k ∂ i ˆ f k ∞ (cid:17) k ξ k k − γ +1 by the Taylor’s expansion to the function ˆ f ( ξ ) at the origin. Combining the abovetwo estimates gives(3.9) |F ( U Ω ,p f )( ξ ) | ≤ C (cid:16) X | i |≤ k +1 k ∂ i ˆ f k ∞ (cid:17) k ξ k k − γ +1 (1 + k ξ k ) − , ξ ∈ R d . Note that(3.10) k ∂ i ˆ f k ∞ ≤ C Z R d | f ( x ) || x | | i | d x ≤ C sup z ∈ R d | f ( z ) | (1 + | z | ) k + d +1+ ǫ for all i ∈ Z d + with | i | ≤ k + 1. Then the desired estimate (3.6) follows from (3.9)and (3.10). (iv) By (3.1) and the first conclusion of this theorem, the Fourier transform of U Ω ,p f is continuous on R d \{ } , and satisfies Z R d |F ( U Ω ,p f )( ξ ) | (1 + k ξ k ) γ − k − d − dξ ≤ C sup z ∈ R d | f ( x ) | (1 + k x k ) k + d +2 . Hence U Ω ,p is a continuous linear operator from S to S ′ . For any f ∈ S ∞ , ∂ i ˆ f ( ) = 0for all i ∈ Z d + . Then F ( U Ω ,p f ) = F ( i Ω f ) for all f ∈ S ∞ . This shows that RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 23 U Ω ,p , ≤ p ≤ ∞ , is a continuous extension of the linear operator i Ω from thesubspace S ∞ ⊂ S to the whole space S .3.2. Composition and left-inverse of the fractional Laplacian.
Direct cal-culation leads to X | i |≤ γ − d (1 − /p ) ∂ i ( ξ k ˆ f ( ξ )) | ξ = i ! ξ i = X | j |≤ γ −| k |− d (1 − /p ) ∂ j ˆ f ( ) j ! ξ j + k , k ∈ Z d + for any γ ∈ R , ≤ p ≤ ∞ and f ∈ S . This together with (3.1) implies that(3.11) U Ω ,p ( ∂ k f ) = U Ω k ,p f, for all f ∈ S and k ∈ Z d + , where Ω k ( ξ ) = ( iξ ) k Ω( ξ ) for k ∈ Z d + . In general, we have the following result aboutcomposition of integrable Riesz potentials. Theorem 3.3.
Let ≤ p ≤ ∞ , real numbers γ , γ satisfy γ ≥ d (1 − /p ) and − γ is larger than the integral part of γ − d (1 − /p ) , and let Ω , Ω ∈ C ∞ ( R d \{ } ) behomogenous of degree − γ and − γ respectively. Then (3.12) U Ω ,p ( J Ω f ) = J Ω Ω f for all f ∈ S . As a consequence of Theorems 2.8 and 3.3, we have the following result aboutthe left-inverse of the fractional Laplacian ( −△ ) γ/ . Corollary 3.4.
Let ≤ p ≤ ∞ and γ > satisfy either < p ≤ ∞ or p = 1 and γ Z + , and the linear operator I γ,p be defined as in (1.12) . Then I γ,p isa left-inverse of the fractional Laplacian ( −△ ) γ/ , i.e., I γ,p ( −△ ) γ/ f = f for all f ∈ S . Proof of Theorem 3.3.
Let k be the integral part of γ − d (1 − /p ). Then − γ >k by the assumption. Then F ( J Ω f )( ξ ) = Ω ( ξ ) ˆ f ( ξ ) and ∂ i ( F ( J Ω f )( ξ )) | ξ = = 0for any i ∈ Z + with | i | ≤ k and any Schwartz function f . This implies that F ( U Ω ,p ( J Ω f ))( ξ ) is equal to (cid:16) [ J Ω f ( ξ ) − X | i |≤ γ − d (1 − /p ) ∂ i ( F ( J Ω f )( ξ )) | ξ = i ! ξ i (cid:17) Ω ( ξ ) , which is the same as F ( J Ω Ω f )( ξ ). Hence the equation (3.12) is established.3.3. L p -integrability in the spatial domain. If γ ∈ (0 , d (1 − /p )), then itfollows from (3.1) and Theorem 2.1 that | U Ω ,p f ( x ) | ≤ C (1 + k x k ) γ − d , x ∈ R d (hence U Ω ,p f ∈ L p ) for any Schwartz function f . In this subsection, we provide asimilar estimate for U Ω ,p f when γ ≥ d (1 − /p ). Theorem 3.5.
Let < ǫ < , ≤ p ≤ ∞ , γ ∈ [ d (1 − /p ) , ∞ ) \ Z , k be the integralpart of γ − d (1 − /p ) , and Ω ∈ C ∞ ( R d \{ } ) be a homogeneous function of degree − γ . If (3.13) | f ( x ) | ≤ C (1 + k x k ) − ( k +1+ d + ǫ ) , x ∈ R d then | U Ω ,p f ( x ) | ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k +1+ d + ǫ (cid:17) ×k x k min( γ − k − d, (1 + k x k ) max( γ − k − d, − (3.14) for all x ∈ R d , and | U Ω ,p f ( x ) − U Ω ,p f ( x ′ ) | ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k +1+ d + ǫ (cid:17) k x − x ′ k δ ×k x k min( γ − k − d − δ, (1 + k x k ) max( γ − k − d − δ, − (3.15) for all x , x ′ ∈ R d with k x − x ′ k ≤ k x k / , where δ < min( | γ − k − d | , ǫ ) . As an easy consequence of Theorem 3.5, we have
Corollary 3.6.
Let ≤ p ≤ ∞ , γ ≥ d (1 − /p ) , and Ω ∈ C ∞ ( R d \{ } ) be ahomogeneous function of degree γ . If both γ and γ − d (1 − /p ) are not nonnegativeintegers, then U Ω ,p f is H¨older continuous on R d \{ } and belong to L p for anySchwartz function f .Proof of Theorem 3.5. We investigate three cases to establish the estimates in (3.14)and (3.15).
Case I: k + 1 − γ <
0. Set h ξ ( t ) = ˆ f ( tξ ). Applying Taylor’s expansion to thefunction h ξ givesˆ f ( ξ ) = h ξ (1) = k X s =0 h ( s ) (0) s ! + 1 k ! Z h ( k +1) ξ ( t )(1 − t ) k dt = X | i |≤ k ∂ i ˆ f ( ) i ! ξ i + ( k + 1) X | j | = k +1 ξ j j ! Z ∂ j ˆ f ( tξ )(1 − t ) k dt. (3.16)Hence(3.17) (cid:16) ˆ f ( ξ ) − X | i |≤ k ∂ i ˆ f ( ) i ! ξ i (cid:17) Ω( ξ ) = X | j | = k +1 j ! Ω j ( ξ ) b g j ( ξ ) , where Ω j ( ξ ) = ( iξ ) j Ω( ξ ) and(3.18) g j ( x ) = ( k + 1) Z (1 − t ) k ( − x /t ) j f ( x /t ) t − d dt ∈ L , | j | = k + 1 . Taking inverse Fourier transform at both sides of the equation (3.17) yields(3.19) U Ω ,p f ( x ) = X | j | = k +1 j ! Z R d K j ( x − y ) g j ( y ) d y . where K j , | j | = k + 1, is the inverse Fourier transform of Ω j . Therefore | U Ω ,p f ( x ) | ≤ C Z Z R d k x − y k γ − d − k − k y /t k k +1 | f ( y /t ) | t − d d y dt = C Z Z R d k x − t y k γ − d − k − k y k k +1 | f ( y ) | d y dt ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k +1+ d + ǫ (cid:17) Z ( t + k x k ) γ − d − k − dt ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k +1+ d + ǫ (cid:17) ×k x k min( γ − d − k , (1 + k x k ) max( γ − d − k , − , (3.20) RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 25 where the first inequality holds because K j ∈ C ∞ ( R d \{ } ) is homogeneous of degree γ − d − k − ∈ ( − d,
0) [7, Theorems 7.1.16 and 7.1.18], and the second inequalityfollows from (2.10). Similarly, | U Ω ,p f ( x ) − U Ω ,p f ( x ′ ) |≤ C X | j | = k +1 Z k x − y k≥ k x − x ′ k k x − x ′ k δ k x − y k γ − d − k − − δ | g j ( y ) | d y + C X | j | = k +1 Z k x − y k≤ k x − x ′ k k x − y k γ − d − k − | g j ( y ) | d y + C X | j | = k +1 Z k x − y k≤ k x − x ′ k k x ′ − y k γ − d − k − | g j ( y ) | d y ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k +1+ d + ǫ (cid:17) k x − x ′ k δ ×k x k min( γ − d − k − δ, (1 + k x k ) max( γ − d − k − δ, − (3.21)for all x , x ′ ∈ R d with k x − x ′ k ≤ k x k /
4, where δ < min( ǫ, | γ − k − d | ). Thenthe desired estimate (3.14) and (3.15) follow from (3.20) and (3.21) for the case k + 1 − γ < Case II: k + 1 − γ > and k ≥ . Applying Taylor’s expansion to thefunction h ξ ( t ) = ˆ f ( tξ ), we haveˆ f ( ξ ) − X | i |≤ k ∂ i ˆ f ( ) i ! ξ i = k X | j | = k ξ j j ! Z (cid:0) ∂ j ˆ f ( tξ ) − ∂ j ˆ f ( ) (cid:1) (1 − t ) k − dt. Multiplying by Ω( ξ ) both sides of the above equation and then taking the inverseFourier transform, we obtain(3.22) U Ω ,p f ( x ) = X | j | = k j ! (cid:16) Z R d K j ( x − y ) g j ( y ) d y − K j ( x ) Z R d g j ( y ) d y (cid:17) , where(3.23) g j ( x ) = k Z (1 − t ) k − ( − x /t ) j f ( x /t ) t − d dt ∈ L , | j | = k . Recalling that K j ∈ C ∞ ( R d \{ } ) , | j | = k are homogeneous of degree γ − d − k ∈ ( − d, | ∂ i K j ( x ) | ≤ C k x k γ − d − k −| j | , | i | ≤ . Combining (2.10), (3.22), (3.23) and (3.24), we get | U Ω ,p f ( x ) | ≤ C X | j | = k Z Z R d | K j ( x − t y ) − K j ( x ) |k y k k | f ( y ) | d y ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d +1+ ǫ (cid:17) × n Z Z k y k≤k x k / t k y kk x k γ − d − k − (1 + k y k ) − d − − ǫ d y dt +(1 + k x k ) − Z Z k y k≥k x k / k x − t y k γ − d − k (1 + k y k ) − d − ǫ d y dt + k x k γ − d − k Z Z k y k≥k x k / (1 + k y k ) − d − − ǫ d y dt o ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d +1+ ǫ (cid:17) ×k x k min( γ − k − d, (1 + k x k ) max( γ − k − d, − , (3.25)and | U Ω ,p f ( x ) − U Ω ,p f ( x ′ ) |≤ C X | j | = k Z (cid:16) Z k t y k≤k x k / + Z k t y k≥ k x k + Z k x k / ≤k t y k≤ k x k (cid:17) | K j ( x − t y ) − K j ( x ) − K j ( x ′ − t y ) + K j ( x ′ ) |k y k k | f ( y ) | d y ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d +1+ ǫ (cid:17) X | j | = k n k x − x ′ k δ × Z Z k t y k≤k x k / t k y kk x k γ − d − k − − δ (1 + k y k ) − d − − ǫ d y dt + k x − x ′ k δ × Z Z t k y k≥ k x k (cid:0) k x k γ − k − d − δ + k y k γ − k − d − δ (cid:1) (1 + k y k ) − d − − ǫ d y dt + Z Z k x k / ≤k t y k≤ k x k (cid:16) | K j ( x − t y ) − K j ( x ′ − t y ) | + | K j ( x ) − K j ( x ) | (cid:17) (1 + k x k /t ) − d − − ǫ d y dt o ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d +1+ ǫ (cid:17) k x − x ′ k δ k x k γ − k − d − δ (1 + k x k ) − . (3.26)Then the desired estimates (3.14) and (3.15) are proved in the case that k +1 − γ > k ≥ Case III: k + 1 − γ > and k = 0 . In this case, γ ∈ (0 ,
1) and(3.27) U Ω ,p f ( x ) = Z R d (cid:0) K ( x − y ) − K ( x ) (cid:1) f ( y ) d y RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 27 where K is the inverse Fourier transform of Ω( ξ ). Then, by applying the argumentused in establishing (3.25), we have | U Ω ,p f ( x ) | ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) d +1+ ǫ (cid:17) × n Z k y k≤k x k / t k y kk x k γ − d − (1 + k y k ) − d − − ǫ d y +(1 + k x k ) − Z k y k≥k x k / k x − y k γ − d (1 + k y k ) − d − ǫ d y + k x k γ − d Z k y k≥k x k / (1 + k y k ) − d − − ǫ d y o ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) d +1+ ǫ (cid:17) k x k γ − d (1 + k x k ) − , (3.28)and | U Ω ,p f ( x ) − U Ω ,p f ( x ′ ) |≤ (cid:16) Z k y k≤k x k / + Z k y k≥ k x k + Z k x k / ≤k y k≤ k x k (cid:17) | K ( x − y ) − K ( x ) − K ( x ′ − y ) + K ( x ′ ) || f ( y ) | d y ≤ C (cid:16) sup z ∈ R d | f ( z ) | (1 + k z k ) k + d +1+ ǫ (cid:17) k x − x ′ k δ k x k γ − d − δ (1 + k x k ) − , (3.29)which yields the desired estimates (3.14) and (3.15) for k + 1 − γ > k =0.3.4. Unique dilation-invariant extension of the linear operator i Ω withadditional integrability in the spatial domain. We now show that U Ω ,p is theonly dilation-invariant extension of the linear operator i Ω from the subspace S ∞ tothe whole space S such that its image is contained in L p . Theorem 3.7.
Let ≤ p ≤ ∞ , γ > have the property that both γ and γ − d (1 − /p ) are not nonnegative integers, Ω ∈ C ∞ ( R d \{ } ) be a nonzero homogeneousfunction of degree − γ , and the linear map I from S to S ′ be a homogeneous ex-tension of the linear operator i Ω on S ∞ . Then If belongs to L p for any Schwartzfunction f if and only if I = U Ω ,p .Proof. The sufficiency follows from (3.1) and Theorems 1.1 and 2.1 for γ < d (1 − /p ), and from (3.1), Theorem 3.1 and Corollary 3.6 for γ ≥ d (1 − /p ). Now thenecessity. By the assumption on the linear operator I from S to S ′ , similar to theargument used in Lemma 2.12, we can find an integer N and tempered distributions H i , | i | ≤ N , such that(3.30) If = U Ω ,p f + X | i |≤ N ∂ i ˆ f ( ) i ! H i for all f ∈ S . Replacing f in (3.30) by ψ j in (2.29) and using (2.30) gives that H j / j ! = Iψ j − U Ω ,p ψ j . Hence(3.31) H j ∈ L p by Corollary 3.6 and the assumption on the linear map I . By (3.30), Theorem 3.1and the assumption on the linear operator I , ( I − U Ω ,p )( δ t f ) = t − γ δ t (( I − U Ω ,p ) f )for all f ∈ S . Hence H j is homogeneous of order γ − d − | j | by Lemma 2.13. Thistogether with (3.31) implies that H j = 0 for all j ∈ Z d + with | j | ≤ N . The desiredconclusion I = U Ω ,p then follows.3.5. Unique dilation-invariant extension of the linear operator i Ω withadditional integrability in the Fourier domain. In this subsection, we char-acterize all those dilation-invariant extensions I of the linear operator i Ω on thesubspace S ∞ to the whole space S such that c If is q -integrable for any Schwartzfunction f . Theorem 3.8.
Let ≤ q ≤ ∞ , γ ∈ [ d/q, ∞ ) \ Z and Ω ∈ C ∞ ( R d \{ } ) be a nonzerohomogeneous function of degree − γ , and the linear map I from S to S ′ be a dilation-invariant extension of the linear operator i Ω on S ∞ . Then the following statementshold. (i) If ≤ q < ∞ , then the Fourier transform of If belongs to L q for anySchwartz function f if and only if γ − d/q Z + and I = U Ω ,q/ ( q − . (ii) If q = ∞ and γ Z + , then the Fourier transform of If belongs to L ∞ forany Schwartz function f if and only if I = U Ω , . (iii) If q = ∞ and γ ∈ Z + , then the Fourier transform of If belongs to L ∞ forany Schwartz function f if and only if (3.32) c If ( ξ ) = \ U Ω , f ( ξ ) + X | i | = − γ ∂ i ˆ f ( ) i ! g i ( ξ ) for some bounded homogeneous functions g i , | i | = − γ , of degree .Proof. (i) The sufficiency follows from Theorem 3.1 and Corollary 3.2. Now weprove the necessity. As every q -integrable function belong to K , similar to theargument used in the proof of Lemma 2.12, we can find functions g i ∈ K , | i | ≤ N ,such that c If ( ξ ) = F ( U Ω ,q/ ( q − f )( ξ ) + X | i |≤ N ∂ i ˆ f ( ) i ! g i ( ξ ) . (3.33)Let ψ j , j ∈ Z d + be defined as in (2.29). Replacing f by ψ j with | j | ≤ N and using(2.30) gives d Iψ j ( ξ ) = (cid:16) b ψ j ( ξ ) − X | i |≤− γ − d/q ∂ i ˆ ψ j ( ) i ! ξ i (cid:17) Ω( ξ ) + g j ( ξ )= ( ξ j j ! ( φ ( ξ ) − ξ ) + g j ( ξ ) if | j | ≤ γ − d/q, ξ j j ! φ ( ξ )Ω( ξ ) + g j ( ξ ) if | j | > γ − d/q. (3.34)Note that ξ j j ! ( φ ( ξ ) − ξ ) ∈ L q when | j | < γ − d/q , and ξ j j ! φ ( ξ )Ω( ξ ) ∈ L p when | j | > γ − d/q . This, together with (3.34) and the assumption that d Iψ j ∈ L q , provesthat(3.35) g j ∈ L q for all j ∈ Z d + with γ − d/q = | j | ≤ N. RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 29
By the homogeneous property of the linear map I , the functions g i , | i | ≤ N , arehomogeneous of degree − γ + | i | , i.e.,(3.36) g i ( tξ ) = t − γ + | i | g i ( ξ ) , for all t > . Combining (3.35) and (3.36) proves that g j = 0 for all j ∈ Z d + with γ − d/q = | j | ≤ N , and the desired conclusion c If ( ξ ) = F ( U Ω ,q/ ( q − f )( ξ ) for all f ∈ S when γ − d/q Z + .Now it suffices to prove that γ − d/q Z + . Suppose on the contrary that γ − d/q ∈ Z + . Then 1 < q < ∞ as γ Z . By (3.34) and the assumption on thelinear map I , we have Z ξ supp φ | g j ( ξ ) − ξ j Ω( ξ ) / j ! | q dξ = Z ξ supp φ | d Iψ j ( ξ ) | q dξ < ∞ for all j ∈ Z d + with | j | = γ − d/q . This, together with (3.36) and the fact that thesupport supp φ of the function φ is a bounded set, implies that g j ( ξ ) − ξ j Ω( ξ ) / j ! = 0for all j ∈ Z d + with | j | = γ − d/q . By substituting the above equality for g j into(3.34) we obtain(3.37) d Iψ j ( ξ ) = φ ( ξ ) ξ j Ω( ξ ) / j !for all j ∈ Z d + with | j | = γ − d/q . This leads to a contradiction, as d Iψ j ( ξ ) ∈ L q by theassumption on the linear map I , and φ ( ξ ) ξ j Ω( ξ ) / j ! L q by direction computation. (ii) and (iii) The necessity is true by (3.32) and Theorem 3.1, while the suffi-ciency follows from (3.33) – (3.36).3.6.
Proof of Theorem 1.2.
The conclusions in Theorem 1.2 follow easily from(2.47), (3.2), Theorem 3.7 and Corollary 3.4.4.
Sparse Stochastic Processes
In this section, we will prove Theorem 1.3 and fully characterize the generalizedrandom process P γ w , which is a solution of the stochastic partial differential equa-tion (1.3). In particular, we provide its characteristic functional and its pointwiseevaluation.4.1. Proof of Theorem 1.3.
To prove Theorem 1.3, we recall the Levy continu-ity theorem, and a fundamental theorem about the characteristic functional of ageneralized random process.
Lemma 4.1. ([5])
Let ξ k , k ≥ , be a sequence of random variables whose charac-teristic functions are denoted by µ k ( t ) . If lim k →∞ µ k ( t ) = µ ∞ ( t ) for some contin-uous function µ ∞ ( t ) on the real line, then ξ k converges to a random variable ξ ∞ indistribution whose characteristic function E ( e − itξ ∞ ) is µ ∞ ( t ) . In the study of generalized random processes, the characteristic functional playsa similar role to the characteristic function of a random variable [6]. The idea isto formally specify a generalized random process Φ by its characteristic functional Z Φ given by(4.1) Z Φ ( f ) := E ( e − i Φ( f ) ) = Z R e − ix dP ( x ) , f ∈ D , where P ( x ) denotes the probability that Φ( f ) < x . For instance, we can show ([14])that the characteristic functional Z w of the white Poisson noise (1.13) is given by(4.2) Z w ( f ) = exp (cid:16) λ Z R d Z R (cid:0) e − iaf ( x ) − (cid:1) dP ( a ) d x (cid:17) , f ∈ D . The characteristic functional Z Φ of a generalized random process Φ is a functionalfrom D to C that is continuous and positive-definite, and satisfies Z Φ (0) = 1. Herethe continuity of a functional L from D to C means that lim k →∞ L ( f k ) = L ( f ) if f k ∈ D tends to f ∈ D in the topology of the space D , while a functional L from D to C is said to be positive-definite if(4.3) n X j,k =1 L ( f j − f k ) c j ¯ c k ≥ f , . . . , f n ∈ D and any complex numbers c , . . . , c n . The remarkable aspectof the theory of generalized random processes is that specification of Z Φ is sufficientto define a process in a consistent and unambiguous way. This is stated in thefundamental Minlos-Bochner theorem. Theorem 4.2. ([6])
Let L be a positive-definite continuous functional on D suchthat L (0) = 1 . Then there exists a generalized random process Φ whose char-acteristic functional is L . Moreover for any f , . . . , f n ∈ D , we may take thepositive measure P ( x , . . . , x n ) as the distribution function of the random variable Φ( f ) , . . . , Φ( f n ) , where the Fourier transform of the positive measure P ( x , . . . , x n ) is L ( y f + · · · + y n f n ) , i.e., L ( y f + . . . + y n f n ) = Z R n exp( − i ( x y + . . . + x n y n )) dP ( x , . . . , x n ) . We are now ready to prove Theorem 1.3.
Proof of Theorem 1.3.
Let N ≥ ϕ be a C ∞ function supported in B ( , B ( , f ∈ D , define a sequence of randomvariables Φ γ,N ( f ) associated with f by(4.4) Φ γ,N ( f ) := X k a k ϕ ( x k /N ) I γ, f ( x k ) , where the a k ’s are i.i.d. random variables with probability distribution P ( a ), andwhere the x k ’s are random point locations in R n which are mutually independentand follow a spatial Poisson distribution with Poisson parameter λ >
0. We willshow that Φ γ,N , N ≥
1, define a sequence of generalized random processes, whoselimit P γ w ( f ) := P k a k I γ, ( f )( x k ) is a solution of the stochastic partial differentialequation (1.3).As ϕ is a continuous function supported on B ( , γ,N ( f ) = X x k ∈ B ( , N ) a k ϕ ( x k /N ) I γ, f ( x k ) . Recall that I γ, f is continuous on R d \{ } by Corollary 3.6. Then the summationof the right-hand side of (4.5) is well-defined whenever there are finitely many x k in B ( , N ) with none of them belonging to B ( , ǫ ) , ǫ >
0. Note that the probability
RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 31 that at least one of x k lies in the small neighbor B ( , ǫ ) is equal to ∞ X n =1 e − λ | B ( ,ǫ ) | ( λ | B ( , ǫ ) | ) n n ! = 1 − e − λ | B ( ,ǫ ) | → ǫ → . We then conclude that Φ γ,N ( f ) is well-defined and Φ γ,N ( f ) < ∞ with probabilityone.Denote the characteristic function of the random variable Φ γ,N ( f ) by E γ,N,f ( t ): E γ,N,f ( t ) = E ( e − it Φ γ,N ( f ) ) = E ( e − i Φ γ,N ( tf ) ) . Applying the same technique as in [12, Appendix B], we can show that(4.6) E γ,N,f ( t ) = exp (cid:16) Z R d Z R (cid:0) e − iatϕ ( x /N ) I γ, f ( x ) − (cid:1) dP ( a ) d x (cid:17) . Moreover, the functional E γ,N,f ( t ) is continuous about t by the dominated conver-gence theorem, because (cid:12)(cid:12)(cid:12) e − iatϕ ( x /N ) I γ, f ( x ) − (cid:12)(cid:12)(cid:12) ≤ | a || t || I γ, f ( x ) | and Z R d Z R | a || I γ, f ( x ) | dP ( a ) d x = (cid:16) Z R | a | dP ( a ) (cid:17) × (cid:16) Z R d | I γ, f ( x ) | d x (cid:17) < ∞ by Corollary 3.6 and the assumption on the distribution P .Clearly the random variable Φ γ,N ( f ) is linear about f ∈ D ; i.e.,(4.7) Φ γ,N ( αf + βg ) = α Φ γ,N ( f ) + β Φ γ,N ( g ) for all f, g ∈ D and α, β ∈ R . For any sequence of functions f k in D that converges to f ∞ in the topology of D , itfollows from Theorem 3.5 and Corollary 3.6 that lim k →∞ k I γ, f k − I γ, f ∞ k = 0.Therefore (cid:12)(cid:12)(cid:12) Z R d Z R (cid:0) e − iatϕ ( x /N ) I γ, f k ( x ) − (cid:1) dP ( a ) d x − Z R d Z R (cid:0) e − iatϕ ( x /N ) I γ, f ∞ ( x ) − (cid:1) dP ( a ) d x (cid:12)(cid:12)(cid:12) ≤ | t | (cid:16) Z R | a | dP ( a ) (cid:17)(cid:16) Z R d ϕ ( x /N ) | I γ, f k ( x ) − I γ, f ∞ ( x ) | d x (cid:17) → k → ∞ , (4.8)which implies that the characteristic function of Φ γ,N ( f k ) converges to the con-tinuous characteristic function of Φ γ,N ( f ∞ ). Hence the random variable Φ γ,N ( f k )converges to Φ γ,N ( f ∞ ) by Lemma 4.1, which in turn implies that Φ γ,N is continuouson D .Set(4.9) L γ,N ( f ) = E γ,N,f (1) . For any sequence c l , ≤ l ≤ n , of complex numbers and f l , ≤ l ≤ n , of functionsin D , X ≤ l,l ′ ≤ n L γ,N ( f l − f l ′ ) c l c l ′ = E (cid:16) n X l,l ′ =1 e − i Φ γ,N ( f l − f l ′ ) c l c l ′ (cid:17) = E (cid:16)(cid:12)(cid:12)(cid:12) n X l =1 c l e − i Φ γ,N ( f l ) (cid:12)(cid:12)(cid:12) (cid:17) ≥ , (4.10)which implies that L γ,N is positive-definite. By Theorem 4.2, we conclude thatΦ γ,N defines a generalized random process with characteristic functional L γ,N .Now we consider the limit of the above family of generalized random processesΦ γ,N , N ≥
1. By Corollary 3.6, I γ, f is integrable for all f ∈ D . Then(4.11) lim N → + ∞ E γ,N,f ( t ) = exp (cid:16) Z R d Z R ( e − iatI γ, f ( x ) − dP ( a ) d x (cid:17) =: E γ,f ( t ) . Clearly E γ,f (0) = 1 and E γ,f ( t ) is continuous as I γ, ( f ) is integrable. Therefore byLemma 4.1, Φ γ,N ( f ) converges to a random variable, which is denoted by P γ ( f ) := P k a k I γ, f ( x k ), in distribution.As I γ, f is a continuous map from D to L , then lim k →∞ k I γ, f k − I γ, f ∞ k = 0whenever f k converges to f in D . Hence (cid:12)(cid:12)(cid:12) Z R d Z R (cid:0) e − iatI γ, f k ( x ) − (cid:1) dP ( a ) d x − Z R d Z R (cid:0) e − iatI γ, f ∞ ( x ) − (cid:1) dP ( a ) d x (cid:12)(cid:12)(cid:12) ≤ | t | (cid:16) Z R | a | dP ( a ) (cid:17)(cid:16) Z R d | I γ, f k ( x ) − I γ, f ∞ ( x ) | d x (cid:17) → k → ∞ , (4.12)which implies that the characteristic function of P γ ( f k ) converges to the charac-teristic function of P γ ( f ∞ ) (which is also continuous), and hence P γ ( f k ) convergesto P γ ( f ∞ ) in distribution by Lemma 4.1. From the above argument, we see that P γ ( f ) is continuous about f ∈ D .Define L γ ( f ) = E γ,f (1). From (4.10) and (4.11), we see that(4.13) X ≤ l,l ′ ≤ n L γ ( f l − f l ′ ) c i c i ′ = lim N →∞ X ≤ l,l ′ ≤ n L γ,N ( f l − f l ′ ) c i c l ′ ≥ c l , ≤ l ≤ n , of complex numbers and f l , ≤ l ≤ n , of functionsin D . Therefore by Theorem 4.2, P γ w defines a generalized random process withits characteristic functional given by(4.14) Z P γ w ( f ) = exp (cid:16) Z R d Z R ( e − iaI γ, f ( x ) − dP ( a ) d x (cid:17) . Pointwise evaluation.
In this section, we consider the pointwise character-ization of the generalized random process P γ w . RACTIONAL LAPLACIAN AND SPARSE STOCHASTIC PROCESSES 33
Theorem 4.3.
Let γ, λ, P ( a ) , P γ w be as in Theorem 1.3, and I γ, be defined as in (1.12) . Then (4.15) P γ w ( y ) := lim N →∞ P γ w ( g N, y ) is a random variable for every y ∈ R d whose characteristic function is given by (4.16) E ( e − itP γ w ( y ) ) = exp (cid:16) λ Z R Z R (cid:0) e − iatH y ( x ) − (cid:1) d x dP ( a ) (cid:17) , t ∈ R , where g ∈ D satisfies R R d g ( x ) d x = 1 , g N, y ( x ) = N d g ( N ( x − y )) , and (4.17) d H y ( ξ ) = (cid:16) e i h y ,ξ i − X | i |≤ γ ( i y ) i ξ i i ! (cid:17) k ξ k − γ . An interpretation is that the random variable P γ w ( y ) in (4.15) and its charac-teristic function E ( e − itP γ w ( y ) ) in (4.16) correspond formally to setting f = δ ( ·− y )(the delta distribution) in (1.18) and (1.19), respectively.To prove Theorem 4.3, we need a technical lemma. Lemma 4.4.
Let γ be a positive non-integer number, g ∈ D satisfy R R d g ( x ) d x = 1 ,and H y be defined in (4.17) . Then (4.18) lim N →∞ k I γ, g N, y − H y k = 0 for all y ∈ R d , where g N, y ( x ) = N d g ( N ( x − y )) .Proof. Let K j be the inverse Fourier transform of ( iξ ) j k ξ k − γ and k be the integralpart of the positive non-integer number γ . Then from the argument in the proof ofTheorem 3.5,(4.19) H y ( x ) = ( P | j | = k k j ! R ( K j ( x − t y ) − K j ( x ))( − y ) j (1 − t ) k − dt if k ≥ K ( x − y ) − K ( x ) if k = 0 . Therefore for y = 0, k I γ, g N, y − H y k ≤ C X | j | = k Z R d Z Z R d | ( K j ( x − t y ) − K j ( x )) y j − ( K j ( x − t y ) − K j ( x )) y j || g N, y ( y ) | d y dtd x ≤ C X | j | = k Z R d Z Z R d | K j ( x − t y ) − K j ( x − t y ) |k y k k | g N, y ( y ) | d y dtd x + C X | j | = k Z R d Z Z R d | K j ( x − t y ) − K j ( x ) || y j − y j || g N, y ( y ) | d y dtd x ≤ C Z Z R d ( t k y − y k ) γ − k ( k y k k + k y − y k k ) | g N, y ( y ) | d y dt + C Z Z R d ( t k y k ) γ − k ( k y k k − k y − y k + k y − y k k ) | g N, y ( y ) | d y dt → N → ∞ if k ≥
1, and k I γ, g N, y − H y k ≤ Z R d Z R d | K ( x − y ) − K ( x − y ) || g N, y ( y ) d y d x ≤ Z R d (cid:16) Z k x − y k≥ k y − y k | K ( x − y ) − K ( x − y ) | d x + Z k x − y k≤ k y − y k | K ( x − y ) | + | K ( x − y ) | d x (cid:17) | g N, y ( y ) | d y ≤ CN d Z R d k y − y k γ | g ( N ( y − y )) | d y = CN − γ Z R d k z k γ | g ( z ) | d z → N → , if k = 0. This shows that (4.18) for y = .The limit in (4.18) for y = can be proved by using a similar argument, thedetail of which are omitted here. Proof of Theorem 4.3.
By Lemma 4.4 and the dominated convergence theorem,(4.20)lim N →∞ Z R d Z R ( e − iatI γ, g N, y ( x ) − dP ( a ) d x = Z R d Z R ( e − iatH y ( x ) − dP ( a ) d x for all t ∈ R . Moreover as H y is integrable from Corollary 3.6 and Lemma 4.4, thefunction R R d R R ( e − iatI γ, H y ( x ) − dP ( a ) d x is continuous about t . Therefore (4.15)and (4.16) follows from Lemma 4.1. Acknowledgement.
This work was done when the first named author was visitingEcole Polytechnique Federale de Lausanne on his sabbatical leave. He would like tothank Professors Michael Unser and Martin Vetterli for the hospitality and fruitfuldiscussions.
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Department of Mathematics, University of Central Florida, Orlando, FL32816, USA
E-mail address : [email protected] (Michael Unser) Biomedical Imaging Group, ´Ecole Polytechnique F´ed´erale de Lau-sanne, Lausanne 1015, Switzerland
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