Invariant Parabolic equations and Markov process on Adéles
aa r X i v : . [ m a t h . A P ] M a y INVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS ONAD `ELES
VICTOR A. AGUILAR–ARTEAGA ∗ AND SAMUEL ESTALA–ARIAS ∗∗ Abstract.
In this article a class of additive invariant positive selfadjoint pseudodifferential un-bounded operators on L ( A f ), where A f is the ring of finite ad´eles of the rational numbers, is con-sidered to state a Cauchy problem of parabolic–type equations. These operators come from a setof additive invariant non-Archimedean metrics on A f . The fundamental solutions of these parabolicequations determines normal transition functions of Markov process on A f . Using the fractionalLaplacian on the Archimedean place, R , a class of parabolic–type equations on the complete ad`elering, A , is obtained. Introduction
The theory of stochastic processes on locally compact groups and their relationship to pseudodif-ferential equations have become an intense subject of study from some decades ago. In particular,the research of pseudodifferential operators over non–Archimedean spaces, such as the field Q p of p –adic numbers or more general local fields, has been deeply explored by several authors (see e.g.,[1, 3, 15, 16, 17, 26, 28] and the references therein). Pursuing a generalisation, there have beenmany attempts to formalise on both aspects of the theory and several general frameworks have beenproposed.This article deals with the study of a certain class of parabolic-type pseudodifferential equationsand its associated Markov stochastic processes on the complete ad`ele group A of the rational numbers Q . This locally compact topological ring can be factorised as A = R × A f , where A f is the totallydisconnected part of A and R its connected component at the identity. According to a theorem ofStruble, A f and A are metrizable with an additive invariant metric all whose spheres are bounded (see[23], including [19, 13]). In [25], an ultrametric on A f is the departing point to study a parabolic–typeequation on A f and its extension to A , considering the fractional Laplacian on R . More recently, aclass of invariant ultrametrics on the finite ad´ele ring A f has been introduced ([6]). These ultrametricsare the only additive invariant metrics whose balls centred at zero are compact and open subgroupsof A f . If the radius of any ball agree with its Haar measure, the ultrametric is called regular . Theseultrametrics have a subclass here called symmetric. Any symmetric regular ultrametric is given bya strictly increasing sequence of natural numbers, beginning with one, totally order by division andcofinal with the natural numbers. In this article we study a class of parabolic–type equation andits associated Markov processes of A f related to the afore mentioned symmetric regular ultrametric. Date : April 24, 2019.2010
Mathematics Subject Classification.
Primary: 35Kxx, 60Jxx Secondary: 35K05, 35K08, 60J25.
Key words and phrases. ring of adeles, ultrametrics, pseudodifferential equations, heat kernels, Markov processes.
Similar to [25], considering the Archimedean fractional Laplacian on R , we extend the mentionedresults to the ring of ad`eles.Given a symmetric regular ultrametric d A f on A f and a real number α > D α on L ( A f ) and to study the Cauchy problem ofa parabolic–type equation on A f . Nonetheless, the classical techniques of Fourier Analysis and ageometrical point of view coming from the regular condition on the ultrametrics give us a boundthat allows to prove regular considerations on the Markov process.Given 0 < β ≤ D β , in L ( R ), a positive selfadjoint pseudodiffer-ential operator D α,β = D α + D β on L ( A ), is defined. By construction, any of these operators areinvariant under translations. The following theorem encloses the results of this writing. Theorem 3.3: If f is any complex valued square integrable function on Dom( D α,β ), then theCauchy problem ∂u ( x,t ) ∂t + D α,β u ( x, t ) = 0 , x ∈ A , t > ,u ( x, t ) = f ( x )has a solution u ( x, t ) determined by the convolution of f with the heat kernel Z ( x, t ). Moreover, Z ( x, t ) is the transition density of a time and space homogeneous Markov process which is bounded,right–continuous and has no discontinuities other than jumps.The exposition is organised as follows. Section 1 presents preliminary results on non–Arquimedeanand Harmonic analysis on the finite ad`ele group. In section 2 the pseudodifferential operator D α defined on L ( A f ) is introduced, and the corresponding Cauchy problem related to the homogeneousheat equations is treated. Finally, in Section 3, a Cauchy problem for the pseudodifferential operator D α,β is studied. 1. The finite ad`ele ring of Q This section introduces the ring of finite ad`eles, A f , as a completion of the rational numbers withrespect to additive invariant ultrametrics. For a detailed description of these results we quote [2]and [6]. Complete information on the classical definition of the ad`ele rings can be found in [27], [21],[18], [12].1.0.1. Ultrametrics on A f . Let N = { , , · · · } denote the set of natural numbers. Let ψ ( n ) beimplicitly defined by any strictly increasing sequence of natural numbers { e ψ ( n ) } ∞ n =0 , which is totallyordered by division and cofinal with the natural numbers, and with e ψ (0) = 1 . The function ψ ( n ) =log( e ψ ( n ) ) can be defined to any integer number n , as ψ ( n ) = n | n | ψ ( | n | ) if n = 0 , n = 0 . The notation chosen here is inspired by the second Chebyshev function but it should not be confused with it, see[4].
NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 3
For any integer n define Λ( n ) implicitly by the equation e Λ( n ) = e ψ ( n ) e ψ ( n − . Then, for any integers n > m , there is a relation ψ ( n ) − ψ ( m ) = n X k = m +1 Λ( k ) or equivalently e ψ ( n ) /e ψ ( m ) = n Y k = m +1 e Λ( k ) . The collection { e ψ ( n ) Z ⊂ Q } n ∈ Z is a neighbourhood base of zero for an additive invariant topologyon Q formed by open and closed subgroups. Additionally, it satisfies the properties: \ n ∈ Z e ψ ( n ) Z = 0 , and [ n ∈ Z e ψ ( n ) Z = Q . For any element x ∈ Q the adelic order of x is given by:ord( x ) := max { n : x ∈ e ψ ( n ) Z } if x = 0 , ∞ if x = 0 . This function satisfies the following properties: • ord A f ( x ) ∈ Z ∪ {∞} and ord A f ( x ) = ∞ if and only if x = 0. • ord A f ( x + y ) ≥ min { ord A f ( x ) , ord A f ( y ) } . The ring of finite ad`eles A f is defined as the completion of Q with respect to the non–Archimedeanultrametric, d : Q × Q −→ R + ∪ { } , given by d ( x, y ) = e − ψ (ord( x − y )) . Every non–zero finite ad`ele x ∈ A f has a unique series representation of the form x = ∞ X l = γ x l e ψ ( l ) , ( x l ∈ { , , . . . , e Λ( l +1) − } )with x γ = 0 and γ = ord( x ) ∈ Z . This series is convergent in the ultrametric of A f and the numbers x l appearing in the representation of x are unique. The value γ , with γ (0) = + ∞ is the finite adelicorder of x .The fractional part of a finite ad`ele x ∈ A f is defined by { x } := − X k = γ ( x ) a k e ψ ( k ) if γ ( x ) < , γ ( x ) ≥ . The ultrametric d ( x, y ) makes the ring A f a second countable locally compact totally disconnectedtopological ring. The ring of adelic integers is the unit ball b Z = { x ∈ A f : k x k ≤ } which is the maximal compact and open subring of A f as well. Denote by dx the Haar measure ofthe topological Abelian group ( A f , +) normalised such that the Haar measure of b Z is equal to one. This definition does not depend on the initial filtration.
V.A. AGUILAR–ARTEAGA AND S. ESTALA–ARIAS
Note that the ultrametric d takes values in the set { e ψ ( n ) } n ∈ Z ∪ { } and the balls centred at zero B n are the sets { e ψ ( n ) b Z ⊂ A f } n ∈ Z , that is, B n := B (0 , e ψ ( n ) ) = e − ψ ( n ) b Z . We denote the sphere centred at zero and radius e ψ ( n ) as S n , i.e. S n := S (0 , e ψ ( n ) ) = B n \ B n − . The norm induced by this ultrametric is given by k x k = e − ψ (ord( x )) and k x k = e ψ ( n ) if and only if x ∈ S n .A function φ : A f −→ C is locally constant if for any x ∈ A f , there exists an integer ℓ ( x ) ∈ Z suchthat φ ( y ) = φ ( x ) , for all y ∈ B ℓ ( x ) ( x ) , where B ℓ ( x ) ( x ) is the closed ball with centre at x and radius e ψ ( ℓ ( x )) . Let D ( A f ) denote the C –vectorspace of all locally constant functions with compact support on A f . The vector space D ( A f ) is called Bruhat–Schwartz space of A f and an element φ ∈ D ( A f ) a Bruhat–Schwartz function (or simply a testfunction ) on A f .If φ belongs to D ( A f ) and φ ( x ) = 0 for some x ∈ A f , there exists a largest ℓ = ℓ ( φ ) ∈ Z , which iscalled the parameter of constancy of φ , such that, for any x ∈ A f , we have φ ( x + y ) = φ ( x ) , for all y ∈ B ℓ . Denote by D ℓk ( A f ) the finite dimensional vector space consisting of functions whose parameter ofconstancy is greater than or equal to ℓ and whose support is contained in B k .A sequence ( f m ) m ≥ in D ( A f ) is a Cauchy sequence if there exist k, ℓ ∈ Z and M > f m ∈ D ℓk ( A f ) if m ≥ M and ( f m ) m ≥ M is a Cauchy sequence in D ℓk ( A f ). That is, D ( A f ) = lim −→ ℓ ≤ k D ℓk ( A f ) . With this topology the space D ( A f ) is a complete locally convex topological vector over C . It is alsoa nuclear space because it is the inductive limit of the countable family of finite dimensional vectorspaces {D ℓk ( A f ) } .For each compact set K ⊂ A f , let D ( K ) ⊂ D ( A f ) be the subspace of test functions whose supportis contained in K . The space D ( K ) is dense in C( K ), the space of complex–valued continuousfunctions on K .An additive character of the field A f is defined as a continuous function χ : A f −→ C such that χ ( x + y ) = χ ( x ) χ ( y ) and | χ ( x ) | = 1, for all x, y ∈ A f . The function χ ( x ) = exp(2 πi { x } ) defines acanonical additive character of A f , which is trivial on b Z and not trivial outside b Z , and all characters of A f are given by χ ξ ( x ) = χ ( ξx ), for some ξ ∈ A f . The Fourier transform of a test function φ ∈ D ( A f )is given by the formula F φ ( ξ ) = b φ ( ξ ) = Z A f φ ( x ) χ ( ξx ) dx, ( ξ ∈ A f ) . NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 5
The Fourier transform is a continuous linear isomorphism of the space D ( A f ) onto itself and thefollowing inversion formula holds: φ ( x ) = Z A f b φ ( ξ ) χ ( − xξ ) dξ (cid:0) φ ∈ D ( A f ) (cid:1) . Additionally, the Parseval – Steklov equality reads as: Z A f φ ( x ) ψ ( x ) dx = Z A f b φ ( ξ ) b ψ ( ξ ) dξ, (cid:0) φ, ψ ∈ D ( A f ) (cid:1) . Last but not least, the Hilbert space L ( A f ) is a separable Hilbert space and the extended Fouriertransform F : L ( A f ) −→ L ( A f ) is an isometry of Hilbert spaces. Moreover, the Fourier inversionformula and the Parseval – Steklov identity hold on L ( A f ). Remark 1.1.
The Haar measure of any ball is equal to its radius: Z y + B n dξ = Z B n dξ = Z e − ψ ( n ) b Z dξ = e ψ ( n ) ( y ∈ A f , n ∈ Z ) , and the area of a sphere is given by Z y + S n dξ = Z S n dξ = e ψ ( n ) − e ψ ( n − ( y ∈ A f , n ∈ Z ) . Moreover, for any n ∈ Z the following formulae hold:(1) Z B n χ ( − ξx ) dx = e ψ ( n ) if k ξ k ≤ e − ψ ( n ) , if k ξ k > e − ψ ( n ) . (2) Z S n χ ( − ξx ) dx = e ψ ( n ) − e ψ ( n − if k ξ k ≤ e − ψ ( n ) , − e ψ ( n − if k ξ k = e − ψ ( n − , if k ξ k ≥ e − ψ ( n − . Parabolic-type equations on A f This section introduces a positive selfadjoint pseudodifferential unbounded operator D α on L ( A f ),the Hilbert space of square integrable functions on A f , and solves the abstract Cauchy problem for thehomogeneous heat equation on L ( A f ) related to D α . The properties of general evolution equationson Banach spaces can be found in [11], [7] and [20]. The reader can consult these and more topicsin the excellent books [14], [3], [26] and [28].2.1. Pseudodifferential operators on A f . For any α >
0, consider the pseudodifferential operator D α : Dom( D α ) ⊂ L ( A f ) −→ L ( A f ) defined by the formula D α φ ( x ) = F − ξ → x [ k ξ k α F x → ξ [ f ]] , for any φ in the domain Dom( D α ) := n f ∈ L ( A f ) : k ξ k α b f ( ξ ) ∈ L ( A f ) o , V.A. AGUILAR–ARTEAGA AND S. ESTALA–ARIAS
This operator is a pseudodifferential operator with symbol k ξ k α . It can be seen that the unboundedoperator D α , with domain Dom( D α ), is a positive selfadjoint operator which is diagonalized by the(unitary) Fourier transform. In other words, the following diagram commutes:(1) L ( A f ) F −−−→ L ( A f ) D α y y m α L ( A f ) F −−−→ L ( A f ) , where m α : L ( A f ) −→ L ( A f ) is the multiplicative operator given by f ( ξ ) ξ k α f ( ξ ), with(dense) domain Dom( m α ) := (cid:8) f ∈ L ( A f ) : k ξ k α f ( ξ ) ∈ L ( A f ) (cid:9) . As a result, several properties of D α , depending only on the inner product of L ( A f ), can be trans-lated into analogue properties of the multiplicative operator m α . In particular, the characteristicequation D α f = λf with f ∈ L ( A f ) \ { } can be solved by applying the Fourier transform. In fact,if λ ∈ { e αψ ( n ) } n ∈ Z , the characteristic function 1 S n , of the sphere S n , is a solution of the characteristicequation, ( k ξ k α − λ ) b f ( ξ ) = 0, of the multiplicative operator m α . Otherwise, if λ / ∈ { e αψ ( n ) } n ∈ Z , thefunction k ξ k α − λ is bounded from below and λ is in the resolvent set of m α . Since the Fourier trans-form is unitary, the point spectrum of D α is the set { e αψ ( n ) } n ∈ Z , with corresponding eigenfunctions {F − (∆ S n ) } n ∈ Z . Finally, { } forms part of the spectrum as a limit point. Each eigenspace is infinitedimensional and there exists a well defined wavelet base which is also made of eigenfunctions. Remark 2.1.
The operator D α is derived from the chosen double sequence (cid:0) e ψ ( n ) (cid:1) n ∈ b Z . Any operator D α is a finite adelic analogue of the Vladimirov operator on Q p . A Cauchy problem on L ( A f ) . For f ( x ) ∈ Dom( D α ) ⊂ L ( A f ), consider the abstract Cauchyproblem(2) ∂u ( x,t ) ∂t + D α u ( x, t ) = 0 , x ∈ A f , t ≥ u ( x, t ) = f ( x ) . This problem will be pointed as abstract Cauchy problem (2). Notice that for each invariant pseu-dodifferential operator D α , the abstract Cauchy problem above is a finite adelic counterpart of theArchimedean abstract Cauchy problem for the homogeneous heat equation.The abstract Cauchy problem (2) is considered in the sense of the Hilbert space L ( A f ), that is tosay, a function u : A f × [0 , ∞ ) −→ C is called a solution if:a. u : [0 , ∞ ) −→ L ( A f ) is a continuously differentiable function,b. u ( x, t ) ∈ Dom( D α ), for all t ≥ u ( x, t ) is satisfies the initial value problem (2).The abstract Cauchy problem (2) is well–posed and its solution is well understood from the theoryof semigroups of linear operators over Banach spaces. This solution is described in the followingsection. NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 7
Semigroup of operators.
From the Hille–Yoshida Theorem, to the positive selfadjoint oper-ator D α , there corresponds a strongly continuous contraction semigroup S ( t ) = exp( − tD α ) : L ( A f ) −→ L ( A f ) ( t ≥ , with infinitesimal generator − D α . It follows that { S ( t ) } t ≥ has the following properties: • For any t ≥ S ( t ) is a bounded operator with operator norm less or equal to one. • The application t S ( t ) is strongly continuous for t ≥ • S (0) is the identity operator in L ( A f ), i.e. S (0)( f ) = f , for all f ∈ L ( A f ), • It has the semigroup property: S ( t ) ◦ S ( s ) = S ( t + s ). • If f ∈ Dom( − D α ), then S ( t ) f ∈ Dom( − D α ) for all t ≥
0, the L derivative ddt S ( t ) f exists, itis continuous for t ≥
0, and is given by ddt S ( t ) f (cid:12)(cid:12)(cid:12) t = t +0 = − D α S ( t ) f = − S ( t ) D α f ( t ≥ . All this means that S ( t ) f is a solution of the Cauchy problem (6) with initial condition f ∈ Dom( D α ).On the other hand, for f ( ξ ) ∈ Dom( m α ) ⊂ L ( A f ), consider the abstract Cauchy problem(3) ∂u ( ξ,t ) ∂t + m α u ( ξ, t ) = 0 , ξ ∈ A f , t ≥ u ( ξ, t ) = f ( ξ ) . The solution of this problem is given by the strongly continuous contraction semigroup exp( − tm α ) : L ( A f ) −→ L ( A f ) given by f ( ξ ) f ( ξ ) exp( − t k ξ k α ) , which is the semigroup that corresponds to the positive selfadjoint multiplicative operator m α , underthe Hille–Yoshida Theorem and whose infinitesimal generator is equal to − m α . From the fact thatthe Fourier transform is an isometry on L ( A f ) and converts the abstract Cauchy problem (2) into(3), the commutative diagram (1), and corresponding definitions of the infinitesimal generators of S ( t ) and exp( − tm α ), the following diagram commutes L ( A f ) F −−−→ L ( A f ) S ( t ) y y exp( − tm α ) L ( A f ) F −−−→ L ( A f ) . A heat kernel.
In order to describe the theoretical solution given by the Hille–Yosida Theoremwe introduce the heat kernel: Z ( x, t ) = F − (cid:0) exp( − t k ξ k α ) (cid:1) = Z A f χ ( − xξ ) exp( − t k ξ k α ) dξ. The first estimate of the heat kernel is given in the following :
Lemma 2.2.
For any t > , α > , and x ∈ A f , Z ( x, t ) is well defined. Furthermore, for any t > and α > , the heat kernel Z ( x, t ) satisfies the inequality | Z ( x, t ) | ≤ Ct − /α , ( x ∈ A f ) , V.A. AGUILAR–ARTEAGA AND S. ESTALA–ARIAS where C is a constant depending on α .Proof. Since the Haar measure of any ball is equal to its radius, we obtain | Z ( x, t ) | ≤ Z A f exp( − t k ξ k α ) dξ < Z ∞ exp( − ts α ) ds = t − /α Γ(1 /α + 1) , where Γ denotes the Archimedean gamma function. Therefore, Z ( x, t ) is well defined and the secondassertion holds with C = Γ(1 /α + 1). (cid:3) Proposition 2.3.
The heat kernel Z ( x, t ) is a positive function for all x and t > . In addition Z ( x, t ) = X n ∈ Z e ψ ( n ) ≤k x k − e ψ ( n ) (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9) . Proof.
From Remark 1.1, if k x k = e − ψ ( m ) , then Z ( x, t ) = ∞ X n = −∞ Z S n χ ( − xξ ) exp( − t k ξ k α ) dξ = ∞ X n = −∞ exp( − te αψ ( n ) ) Z S n χ ( − xξ ) dξ = m +1 X n = −∞ exp( − te αψ ( n ) ) Z S n χ ( − xξ ) dξ = − exp( − te αψ ( m +1) ) e ψ ( m ) + m X n = −∞ exp( − te αψ ( n ) )( e ψ ( n ) − e ψ ( n − )= m X n = −∞ e ψ ( n ) (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9) = X n ∈ Z e ψ ( n ) ≤k x k − e ψ ( n ) (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9) . This implies that Z ( x, t ) is a positive function for all x and t > (cid:3) Remark 2.4.
It is important to notice that the expression of the heat kernel in Proposition 2.3 doesnot depend on the algebraic structure of A f . As a matter of fact, Z ( x, t ) depends only on the valuesof the double sequence ( e ψ ( n ) ) n ∈ Z and the ultrametric structure defined by this sequence on A f . Corollary 2.5.
The heat kernel is the distribution of a probability measure on A f , i.e. Z ( x, t ) ≥ and Z A f Z ( x, t ) dx = 1 , for all t > . NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 9
Proof.
From Proposition 2.3 it follows that Z A f Z ( x, t ) dx = ∞ X l = −∞ Z S l Z ( x, t ) dx = ∞ X l = −∞ Z ( e ψ ( l ) , t ) (cid:0) e ψ ( l ) − e ψ ( l − (cid:1) = ∞ X l = −∞ X n ∈ Z e ψ ( n ) ≤ e − ψ ( l ) e ψ ( n ) (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9) (cid:0) e ψ ( l ) − e ψ ( l − (cid:1) = ∞ X n = −∞ (cid:0) e ψ ( n ) (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9)(cid:1) − n X l = −∞ (cid:0) e ψ ( l ) − e ψ ( l − (cid:1)! = ∞ X n = −∞ (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9) = 1 . (cid:3) Lemma 2.6.
For any t > , α > and x ∈ A f , the heat kernel Z ( x, t ) is positive and satisfies theinequality Z ( x, t ) ≤ k x k − (cid:0) − exp( − te αψ ( m +1) ) (cid:1) , (4) where k x k = e − ψ ( m ) .Proof. This follows from the inequality Z ( x, t ) ≤ k x k − X n ∈ Z e ψ ( n ) ≤k x k − (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9) ≤ k x k − (cid:0) − exp( − te αψ ( m +1) ) (cid:1) (cid:3) Proposition 2.7.
The heat kernel satisfies the following properties: • It is the distribution of a probability measure on A f , i.e. Z ( x, t ) ≥ and Z A f Z ( x, t ) dx = 1 , for all t > . • It converges to the Dirac distribution as t tends to zero: lim t → Z A f Z ( x, t ) f ( x ) dx = f (0) , for all f ∈ D ( A f ) . • It has the Markovian property: Z ( x, t + s ) = Z A f Z ( x − y, t ) Z ( y, s ) dy. Proof.
From Corollary 2.5, Z ( x, t ) is in L ( A f ) for any t > Z A f Z ( x, t ) dx = 1 . Using this equality, the fact that f ∈ D ( A f ) is a locally constant function with compact support andLemma 2.6, we conclude that lim t → Z A f Z ( x, t ) (cid:0) f ( x ) − f (0) (cid:1) dx = 0 . The Markovian property follows from the Fourier inversion formula and the related property of theexponential function. (cid:3)
Remark 2.8.
It is worth mentioning that the heat kernel associated to the isotropic Laplacian of theultrametric space ( A f , d ) , with the Haar measure of A f as speed measure, and distribution function e /r , is equal to (see [5] for the definitions): X n ∈ Z e ψ ( n ) ≤k x k − e ψ ( n ) (cid:8) exp( − te αψ ( n − ) − exp( − te αψ ( n ) ) (cid:9) . This kernel differes from Z ( x, t ) only by a term. However this one is very important when consideringbounds . . The solution of the heat equation.
Given t > T ( t ) : L ( A f ) −→ L ( A f ) by the convolution with the heat kernel T ( t ) f ( x ) = Z ( x, t ) ∗ f ( x ) , ( f ∈ L ( A f )) , and let T (0) be the identity operator on L ( A f ). From Proposition 2.7 and Young’s inequality thefamily of operators { T ( t ) } t ≥ is a strongly continuous contraction semigroup.The main theorem of the diffusion equation on the ring A f is the following. Theorem 2.9.
Let α > and let S ( t ) be the C –semigroup generated by the operator − D α . Theoperators S ( t ) and T ( t ) agree for each t ≥ . In other words, for f ∈ Dom( D α ) and for t > thesolution of the abstract Cauchy problem (2) is given by u ( x, t ) = Z ( x, t ) ∗ f ( x ) .Proof. For f ∈ L ( A f ) ∩ L ( A f ), the convolution u ( x, t ) = Z ( x, t ) ∗ f ( x ) is in L ( A f ) ∩ L ( A f ) because Z ( x, t ) is integrable for t >
0. Then, the Fourier transform F x ξ u ( x, t ) is equal toˆ f ( ξ ) exp( − t k ξ k α ) . From the commutative diagram above and the last equation, S ( t )( f ) = T ( t )( f ). Since L ( A f ) ∩ L ( A f ) is dense in L ( A f ), S ( t )( f ) = T ( t ), for each t ≥
0. As a consequence, the function u ( x, t ) = Z ( x, t ) ∗ f ( x ) is a solution of the Cauchy problem for any f ∈ Dom( D α ). (cid:3) NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 11
Markov process on A f . In this section the fundamental solution of the heat equation, Z ( x, t ),is shown to be the transition density function of a Markov process on A f . For general informationon the theory of Markov process the reader can consult [24] and the classical writing [10].Let B denote the Borel σ –algebra of A f and for B ∈ B write 1 B for the characterisctic or indicatorfunction of B . Define p ( t, x, y ) := Z ( x − y, t ) ( t > , x, y ∈ A f )and P ( t, x, B ) = R B p ( t, x, y ) dy if t > , x ∈ A f , B ∈ B , B ( x ) if t = 0 . From Theorem 2.7, it follows that p ( t, x, y ) is a normal transition density and P ( t, x, B ) is a normalMarkov transition function on A f which corresponds to a Markov process on A f (see [10, Section2.1], for further details).In order to protray the properties of the path of the associated Markov process we first state thefollowing: Lemma 2.10.
Let k be any integer, then Z A f \ B k Z ( x, t ) ≤ − exp( − te αψ ( − k ) ) . Proof.
Similar to Corollary 2.5, we have the following Z A f \ B k Z ( x, t ) dx = ∞ X l = k +1 Z S l Z ( x, t ) dx = ∞ X l = k +1 Z ( e ψ ( l ) , t )( e ψ ( l ) − e ψ ( l − )= ∞ X l = k +1 X n ∈ Z e ψ ( n ) ≤ e − ψ ( l ) e ψ ( n ) (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9) ( e ψ ( l ) − e ψ ( l − )= − ( k +1) X n = −∞ (cid:0) e ψ ( n ) (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9)(cid:1) − n X l = k +1 (cid:0) e ψ ( l ) − e ψ ( l − (cid:1)! ≤ − ( k +1) X n = −∞ (cid:8) exp( − te αψ ( n ) ) − exp( − te αψ ( n +1) ) (cid:9) = 1 − exp( − te αψ ( − k ) ) . (cid:3) Proposition 2.11.
The transition function P ( t, y, B ) satisfies the following two conditions:a. For each s ≥ and compact subset B of A f lim x →∞ sup t ≤ s P ( t, x, B ) = 0 ( Condition LB ) . b. For each k > and compact subset B of A f lim t → sup x ∈ B P ( t, x, A f \ B k ( x )) = 0 ( Condition MB ) . Proof.
Let d ( x ) := dist ( x, B ) = e ψ ( − m x ) where m x ∈ Z . From Lemma 2.6 it follows that Z ( x − y, t ) ≤ [ d ( x )] − (cid:0) − exp( − se αψ ( m x +1) ) (cid:1) for any y ∈ B and t ≤ s . Since B is compact and α is positive, d ( x ) − −→ e αψ ( m x +1) −→
0, as x → ∞ . Hence P ( t, x, B ) ≤ [ d ( x )] − (cid:0) − exp( − se αψ ( m x +1) ) (cid:1) µ ( B ) −→ x → ∞ . This implies Condition L ( B ).Presently, we establish Condition M ( B ): for y ∈ A f \ B k ( x ), we have k x − y k > e ψ ( k ) . Therefore P ( t, x, A f \ B k ( x )) = Z A f \ B k ( x ) Z ( x − y, t ) dy = Z A f \ B k (0) Z ( y, t ) dy. From Lemma 2.10, Z A f \ B k Z ( y, t ) dy ≤ − exp( − te αψ ( − k ) ) −→ , t → + . Since P ( t, x, B k ( x )) is invariant under additive traslations, the last equation implies Condition M ( B ). (cid:3) Theorem 2.12.
The heat kernel Z ( x, t ) is the transition density of a time and space homogeneousMarkov process which is bounded, right–continuous and has no discontinuities other than jumps.Proof. The result follows from Proposition 2.11 and the fact that A f is a second countable and locallycompact ultrametric space (see [10, Theorem 3.6]). (cid:3) Cauchy problem for parabolic type equations on A In this section an abstract Cauchy problem on L ( A ) is presented. First, we recollect severalproperties of the ring of ad`eles A . The abstract Cauchy problem on L ( A ) is studied by consideringthe fractional Laplacian on the Archimedean completion, R , and the pseudodifferential operator on L ( A f ), studied in the previous section.3.1. The ring of ad`eles A . In the present section, the ring of ad`eles A of Q is described as theproduct of its Archimedean part with its non–Archimedean component. We first consider the locallycompact and complete Archimedean field of real numbers R .3.1.1. The Archimedean place.
Recall that the real numbers R is the unique Archimedean completionof the rational numbers. As a locally compact Abelian group, R , is autodual with pairing functiongiven by χ ∞ ( ξ ∞ x ∞ ), where χ ∞ ( x ∞ ) = e − πix ∞ is the canonical character on R . In addition, itis a commutative Lie group. The Schwartz space of R , which we denote here by D ( R ), consistsof functions ϕ ∞ : R −→ C which are infinitely differentiable and rapidly decreasing. D ( R ) has a NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 13 countable family of seminorms which makes it a nuclear Fr´echet space. Let dx ∞ denote the usualHaar measure on R . The Fourier transform F ∞ [ ϕ ∞ ]( ξ ∞ ) = Z R ϕ ∞ ( x ∞ ) χ ∞ ( ξ ∞ x ∞ ) dx ∞ is an isomorphism from D ( R ) onto itself. Moreover, the Fourier inversion formula and the Parseval–Steklov identities hold on D ( R ). Furthermore, L ( R ) is a separable Hilbert space, the Fourier trans-form is an isometry on L ( R ), and the Fourier inversion formula and the Parseval–Steklov identityhold on L ( R ). Definition 3.1.
The ad`ele ring A of Q is defined as A = R × A f . With the product topology, A is a second countable locally compact Abelian topological ring. If µ ∞ is the Haar measure on R and µ f denotes the Haar measure on A f , a Haar measure on A is givenby the product measure µ = µ ∞ × µ f . Recall that, if χ ∞ and χ f are the canonical characters on R and A f , respectively, then χ = ( χ ∞ , χ f ) defines a canonical character on A . A is a selfdual group inthe sense of Pontryagin and we have a paring χ ∞ ( x ∞ ξ ∞ ).3.1.2. Bruhat-Schwartz space.
For any ϕ ∞ ∈ D ( R ) and ϕ f ∈ D ( A f ), we have a function ϕ on A given by ϕ ( x ) = ϕ ∞ ( x ∞ ) ϕ f ( x f )for any ad`ele x = ( x ∞ , x f ). These functions are continuous on A and the linear vector space generatedby these functions is linearly isomorphic to the algebraic tensor product D ( R ) ⊗ D ( A f ). In thefollowing, we identify these spaces and write ϕ = ϕ ∞ ⊗ ϕ f .Since A is a locally compact Abelian topological group the Bruhat–Schwartz space D ( A ) hasnatural topology described as follows. First, recall that D lk ( A f ) denotes the set functions withsupport on B k ⊂ A f and parameter of constancy l . We have the algebraic and topological tensorproduct of a Fr´echet space and finite dimensional space, given by D ( R ) ⊗ D lk ( A f )which represents a well defined class of functions on A . These topological vector spaces are nuclearFr´echet, since D ( R ) is nuclear Fr´echet and D lk ( A f ) has finite dimension. We have D ( A ) = lim −→ l ≤ k D ( R ) ⊗ D lk ( A f ) . The space of
Bruhat–Schwartz functions on A is the algebraic and topological tensor product of nuclearspace vector spaces D ( R ) and D ( A f ), i.e. D ( A ) = D ( R ) ⊗ D ( A f ) . The Fourier transform on A . The Fourier transform on D ( A ) is defined as F [ ϕ ]( ξ ) = Z A ϕ ( x ) χ ( ξx ) dx, for any ξ ∈ A . It is well–defined on D ( A ) and for any function of the form ϕ = ϕ ∞ ⊗ ϕ f it is givenby F [ ϕ ]( ξ ) = F ∞ [ ϕ ∞ ]( ξ ∞ ) ⊗ F f ( ϕ f )( ξ f ) ( ξ = ( ξ ∞ , ξ f ) ∈ A )where F ∞ and F f are the Fourier transforms on D ( R ) and D ( A f ), respectively. In other words, wehave F A = F R ⊗ F A f .The Fourier transform F : D ( A ) −→ D ( A ) is a linear and continuous isomorphism. The inversionformula on D ( A ) reads as F − [ ϕ ]( ξ ) = Z A b ϕ ( − ξ ) χ ( ξx ) dξ, ( ξ ∈ A ) , and Parseval–Steklov equality as Z A ϕ ( x ) ψ ( x ) dx = Z A b ϕ ( ξ ) b ψ ( ξ ) dξ. The space of square integrable functions L ( A ) on A is a separable Hilbert space since it is the Hilberttensor product space L ( A ) ∼ = L ( R ) ⊗ L ( A f ). The Fourier transform F : L ( A ) −→ L ( A ) is anisometry. The Fourier inversion formula and the Parseval-Steklov identity hold.3.2. A Cauchy Problem on L ( A ) . In this paragraph we consider a parabolic type equation onthe complete ring of adeles.3.2.1.
Archimedean heat kernel.
Let us recall the theory of the fractional heat kernel on the real line.For a complete review of this topic the reader may consult [9] and the references therein. For any0 < β ≤
2, the fractional Laplatian D β ∞ : Dom( A ) ⊂ L ( R ) −→ L ( R ) is given by D β ∞ φ ( x ∞ ) = F − ξ ∞ → x ∞ h | ξ | β ∞ F x ∞ → ξ ∞ [ f ] i , for any φ in the domain Dom( D β ∞ ) := n f ∈ L ( R ) : | ξ | β ∞ b f ∈ L ( R ) o . Similar to the case of the finite ad`ele ring, the operator D β R φ ( x ∞ ) is diagonalized by the unitaryFourier transform: if m β ∞ denotes the multiplicative operator on L ( R ) given by f ( ξ ) ξ | β ∞ f ( ξ ),with domain Dom( m β ∞ ) := n f ∈ L ( R ) : | ξ | β ∞ b f ( ξ ) ∈ L ( R ) o , then the following diagram commutes:(5) L ( R ) F −−−→ L ( R ) D β ∞ y y m β ∞ L ( R ) F −−−→ L ( R ) . The pseudodifferential equation(6) ∂u ( x ∞ ,t ) ∂t + D β ∞ u ( x ∞ , t ) = 0 , x ∞ ∈ R , t ≥ u ( x, t ) = f ( x ) , f ∈ Dom( D β ∞ ) NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 15 is an abstract Cauchy problem whose solution is given by the convolution of f with the Archimedeanheat kernel : Z ∞ ( x ∞ , t ) = Z R χ ∞ ( ξ ∞ x ∞ ) e − t | ξ ∞ | β dξ ∞ ( t > . For 0 < β ≤
2, the following bound holds | Z ∞ ( x ∞ , t ) | ≤ Ct /β t /β + x ∞ (for t > , x ∞ ∈ R ) . Due to this bound, the Archimedean heat kernel satifies several properties: it is the distributionof a probability measure on R ; it converges to the Dirac delta distribution as t tends to zero, andit satisfies the Markovian property. Therefore the Archimedean heat kernel is the transition densityof a time and space homogeneous Markov process which is bounded, right–continuous and has nodiscontinuities other than jumps (see [9, Section 2]).Moreover, the formula S ∞ ( f ) = f ( x ∞ ) ∗ Z ∞ ( x ∞ , t ) ( f ∈ L ( R ))defines a strongly continuous contraction semigroup with the unbounded operator (cid:0) D β ∞ , Dom( D β ∞ ) (cid:1) as infinitesimal generator.Furthermore, there is a commutative diagram(7) L ( R ) F −−−→ L ( R ) exp( − tD β ∞ ) y y exp( − tm β ∞ ) L ( R ) F −−−→ L ( R ) . where exp( − tm β ∞ ) is the C –semigroup of contractions whose infinitesimal generator corresponds tothe operator − m β ∞ , under the Hille–Yoshida Theorem.3.2.2. Tensor product of operators.
Let us briefly recall the definition of tensor product of operatorson the Hilbert space L ( A ) = L ( A f ) ⊗ L ( R ) (see [22, Chapter VIII] for complete detail).Given two (unbounded) closable operators ( A, Dom( A )) and ( B, Dom( B )) on L ( A f ) and L ( R ),respectively, the algebraic tensor productDom( A ) ⊗ Dom( B ) = ( X finite λ i φ if ⊗ φ i ∞ : φ if ∈ Dom( A ) , φ i ∞ ∈ Dom( B ) ) ⊂ L ( A )is dense in L ( A ), and the operator A ⊗ B given by A ⊗ B ( φ f ⊗ φ ∞ ) = A ( φ f ) ⊗ B ( φ ∞ ) , for φ f ⊗ φ ∞ ∈ Dom( A ) ⊗ Dom( A ), is closable.The tensor product of A and B is the closure of the operator A ⊗ B defined on the algebraictensor product Dom( A ) ⊗ Dom( B ). We denote the closed operator by A ⊗ B and its domain byDom( A ⊗ B ). Furthermore, if A and B are selfadjoint, their tensor product A ⊗ B is essentiallyselfadjoint and the spectrum σ ( A ⊗ B ) of A ⊗ B is the closure in C of σ ( A ) σ ( B ), where σ ( A ) and σ ( B ) are the corresponding spectrum of A and B . On the other hand, if A and B are bounded operators, their tensor product A ⊗ B is boundedwith operator norm k A ⊗ B k L ( A ) = k A k L ( A f ) k B k L ( R ) . Now, let us recall the definition of the sum of unbounded operators on the Hilbert space L ( A ) = L ( A f ) ⊗ L ( R ) given by A + B = A ⊗ I + I ⊗ B . Once more, the algebraic tensor productDom( A ) ⊗ Dom( B ) ⊂ L ( A ) is dense in L ( A ) and the operator A + B = A ⊗ I + I ⊗ B given by( A + B )( φ f ⊗ φ ∞ ) = A ( φ f ) ⊗ φ ∞ + φ f ⊗ B ( φ ∞ ) , with φ f ⊗ φ ∞ ∈ Dom( A ) ⊗ Dom( B ) is essentially selfadjoint. The sum of A and B is the closure ofthe operator A + B defined on Dom ( A ) ⊗ Dom ( B ). We denote by Dom( A + B ) the domain of thethis closed unbounded operator and with abuse of notation we denote this unbounded operator by A + B . The spectrum of σ ( A + B ) of A + B is the closure in C of σ ( A ) + σ ( D β ), where σ ( A ) and σ ( B ) are the corresponding spectrum of A and B , respectively.3.2.3. Pseudodifferential operators on A . First, notice that the multiplicative operator [ m α,β : L ( A ) −→ L ( A ), given by f ( ξ ) ( k ξ f k α + | ξ ∞ | β ) f ( ξ ), with (dense) domainDom( [ m α,β ) := n f ∈ L ( A ) : (cid:0) k ξ f k α + | ξ ∞ | β (cid:1) b f ( ξ ) ∈ L ( A ) o is selfadjoint and coincides with m α,β = m α + m β = m α ⊗ I + I ⊗ m β on the set Dom( m α ) ⊗ Dom( m β ) ⊂ L ( A ). Since m α,β is essentially selfadjoint on the domain Dom( m α ) ⊗ Dom( m β ) it follows that m α,β = [ m α,β .For any 0 < α and 0 < β ≤
2, consider the pseudodifferential operator [ D α,β : Dom( [ D α,β ) ⊂ L ( A ) −→ L ( A ) defined by the formula [ D α,β φ ( x ) = F − ξ → x [ m α,β F x → ξ [ φ ]] , for any φ in the domain Dom( [ D α,β ) := n f ∈ L ( A ) : m α,β ( b φ ) ∈ L ( A ) o . This unbounded operator is a positive selfadjoint operator which is diagonalized by the (unitary)Fourier transform F , i.e. the following diagram commutes:(8) L ( A ) F −−−→ L ( A ) \ D α,β y y m α,β = \ m α,β L ( A ) F −−−→ L ( A ) , Therefore, the operator D α,β = D α + D β = D α ⊗ I + I ⊗ D β , which is essentially selfadjoint overthe domain Dom( D α ) ⊗ Dom( D β ) ⊂ L ( A ), is equal to the operator [ D α,β . NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 17
A heat equation on A . For f ( x ) ∈ Dom( D α,β ) ⊂ L ( A ), consider the abstract Cauchy problem(9) ∂u ( x,t ) ∂t + D α,β u ( x, t ) = 0 , x ∈ A , t ≥ u ( x, t ) = f ( x ) . As mentioned above, a function u : A × [0 , ∞ ) −→ C is called a solution of the abstract Cauchyproblem (9) in the Hilbert space L ( A ), if:a. u : [0 , ∞ ) −→ L ( A ) is a continuously differentiable function on the sense of Hilbert spaces,b. u ( x, t ) ∈ Dom( D α,β ), for all t ≥ u ( x, t ) is a solution of the initial value problem.Furthermore, this abstract Cauchy problem is well posed and its solution is given by a stronglycontinuous contraction semigroup. From the Hille–Yoshida theorem, the unbounded operator − D α,β is the infinitesimal generator of a strongly continuous contraction semigroup S A ( t ) = exp( − tD α,β ).Additionally, to the unbounded operator − m α,β there corresponds a strongly continuous contractionsemigroup exp( − tm α,β ) with m α,β as infinitesimal generator.From an argument as in 2.3, there is a commutative diagram:(10) L ( A ) F −−−→ L ( A ) exp( − tD α,β ) y y exp( − tm α,β ) L ( A ) F −−−→ L ( A ) . In order to describe the solution of problem (9), for fixed α > < β ≤
2, we define the adelic heat kernel as Z A ( x, t ) = Z A χ ( − ξx ) e − t ( k ξ f k α + | ξ ∞ | β ) dξ ( t > , x, ξ ∈ A ) , where ξ = ( ξ f , ξ ∞ ). That is to say, Z A ( x, t ) = F − A ( e − t ( k ξ f k α + | ξ ∞ | β ) )= F − ∞ ( e − t | ξ ∞ | β ) F − f ( e − t k ξ f k α )= Z f ( x f , t ) ⊗ Z ∞ ( x ∞ , t ) . Proposition 3.2.
The adelic heat kernel, Z A ( x, t ) , satisfies the following properties: • It is the distribution of a probability measure on A f , i.e. Z A ( x, t ) ≥ and Z A Z A ( x, t ) dx = 1 , for all t > . • It converges to the Dirac distribution as t tends to zero: lim t → Z A Z A ( x, t ) f ( x ) dx = f (0) , for all f ∈ D ( A ) . • It has the Markovian property: Z ( x, t + s ) = Z A Z A ( x − y, t ) Z ( y, s ) dy. Proof.
From the equality Z A ( x, t ) = Z A f ( x f , t ) ⊗ Z R ( x ∞ , t ) it follows that Z A ( x, t ) is in L ( A ) for any t >
0, and also Z A Z A ( x, t ) dx = 1 . Using the corresponding properties of the Arquimedian heat kernel and the finite adelic heat kernel,for f ∈ D ( A ), we have lim t → Z A Z A ( x, t ) (cid:0) f ( x ) − f (0) (cid:1) dx = 0 . The Markovian property follows from the Fourier inversion formula and the related property of theexponential function. (cid:3)
Now, for any f ∈ L ( A ), define T A ( t )( f )( x ) = Z A ( x, t ) ∗ f ( x ) t > ,f ( x ) t = 0 . From Proposition 3.2 and Young’s inequality it follows that { T A ( t ) } t ≥ is a strongly continuouscontraction semigroup. On the other hand, from definition, it follows that S A ( t )( φ f ⊗ φ ∞ ) = (cid:0) Z A f ( x f , t ) ∗ φ f (cid:1) ⊗ ( Z R ( x ∞ , t ) ∗ φ ∞ ) . Theorem 3.3. If f is any complex valued square integrable function on Dom( D α,β ) , then the Cauchyproblem ∂u ( x,t ) ∂t + D α,β u ( x, t ) = 0 , x ∈ A , t > ,u ( x, t ) = f ( x ) has a classical solution u ( x, t ) determined by the convolution of f with the heat kernel Z A ( x, t ) .Moreover, Z A ( x, t ) is the transition density of a time and space homogeneous Markov process whichis bounded, right-continuous and has no discontinuities other than jumps.Proof. Similar to Section 2, for f ∈ L ( A ) ∩ L ( A ), since the adelic heat kernel is absolute integral,the convolution Z A ( x, t ) ∗ f ( x ) is in L ( A ) ∩ L ( A ) and F x → ξ ( Z A ( x, t ) ∗ f ( x )) = ˆ f ( ξ ) exp( − t k ξ f k α + | ξ ∞ | β ) . Therefore T A ( t ) = S A ( t ) coincides on a dense set of L ( A ). The properties of the Markov processfollow because, the product of two Markov process which satisfy conditions MB and LB also satisfiesthose conditions (see [28, Section 4.9]). (cid:3) Remark 3.4.
A slightly different proof of the Theorem above can be given as follows. Notice thatthe expression e S A ( t ) = S A f ( t ) ⊗ S R ( t ) gives a strongly continuous contraction semigroup on L ( A ) which satisfies (cid:13)(cid:13)(cid:13) e S A ( t ) (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13) S A f ( t ) (cid:13)(cid:13) k S R ( t ) k . By the Leibniz rule, the infinitesimal generator of e S A ( t ) is D α,β . NVARIANT PARABOLIC EQUATIONS AND MARKOV PROCESS AD`ELES 19
Acknowledgements
The authors would like to thank Manuel Cruz-L´opez, Sergii Torba and Wilson A. Zu˜niga–Galindofor very useful discussions. Work was partially supported by CONACYT–FORDECYT grant number265667.
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E-mail address : [email protected] ∗∗ Instituto de Matem´aticas, UNAM.
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