Generalised Fractional Evolution Equations of Caputo Type
aa r X i v : . [ m a t h . A P ] J un Generalised Fractional Evolution Equations of Caputo Type
M. E. Hern´andez-Hern´andez, V. N. Kolokoltsov, L. Toniazzi
Abstract.
This paper is devoted to the study of generalised time-fractional evolutionequations involving Caputo type derivatives. Using analytical methods and probabilisticarguments we obtain well-posedness results and stochastic representations for the so-lutions. These results encompass known linear and non-linear equations from classicalfractional partial differential equations such as the time-space-fractional diffusion equa-tion, as well as their far reaching extensions.Meaning is given to a probabilistic generalisation of Mittag-Leffler functions. Introduction
The main purpose of this article is to prove well-posedness and stochastic representationfor the solutions of the following evolution equations − t D ( ν ) a +∗ u ( t, x ) = − Au ( t, x ) − g ( t, x ) , ( t, x ) ∈ ( a, b ] × R d ,u ( a, x ) = φ a ( x ) , x ∈ R d , (1)and − t D ( ν ) a +∗ u ( t, x ) = − Au ( t, x ) − f ( t, x, u ( t, x )) , ( t, x ) ∈ ( a, b ] × R d ,u ( a, x ) = φ a ( x ) , x ∈ R d , (2)where − t D ( ν ) a +∗ is a generalised differential operator of Caputo type of order less than 1 act-ing on the time variable t ∈ [ a, b ] (as introduced in [25]), A is the (infinitesimal) generatorof a Feller semigroup on C ∞ ( R d ) acting on the variable x ∈ R d , φ a belongs to the domainof the generator A (denoted by Dom ( A ) ), g ∶ [ a, b ] × R d → R is a bounded measurablefunction, and f ∶ [ a, b ] × R d × R → R is a non-linear function satisfying a certain Lipschitzcondition.Since Caputo derivatives of order β ∈ ( , ) are special cases of the operators − t D ( ν ) a +∗ ,the evolution equations in (1)-(2) include as particular cases a variety of equations stud-ied in the theory of fractional partial differential equations (FPDE’s). The latter equa-tions have been successfully used for describing diffusions in disordered media, also called anomalous diffusions , which include both subdiffusions and superdiffusions . Subdiffusionphenomena are usually related to time-FPDE’s, whereas superdiffusions are related tospace-FPDE’s. We refer, e.g., to [6], [7], [20], [29], [30], [19], [1], [23], [28] [33], [36], [43][22] (and references cited therein) for an account of historical notes, theory and appli-cations of fractional calculus, as well as different analytical and numerical methods toaddress both fractional ordinary differential equations (FODE’s) and fractional partialdifferential equations. Mathematics Subject Classification.
Key words and phrases.
Fractional evolution equation, Generalised derivatives of Caputo type, Mittag-Leffler functions, Feller process, β -stable subordinator, Stopping time, Boundary point. In the classical fractional setting, special cases of equation (1) include fractional Cauchyproblems , that is initial value problems of the form − t D βa +∗ u ( t, x ) = − Au ( t, x ) , ( t, x ) ∈ [ a, b ] × R d ,u ( a, x ) = φ a ( x ) , x ∈ R d , β ∈ ( , ) , (3)where t D βa +∗ stands for the Caputo derivative of order β (acting on the variable t ). Equa-tions of the type in (3) have been actively studied in the literature. Amongst the standardanalytical approaches to solve FPDE’s, the Laplace-Fourier transform method plays animportant role (see, e.g., [9], [11], [19], [36], [37], and references therein). From a prob-abilistic point of view, interesting connections have been found between the solution oftime-FPDE’s and the transition densities of time-changed Markov processes (see for ex-ample [2], [4], [16], [23], [24], [33], [34]). For instance, a very standard example of theequation (3), first studied by Schneider and Wyss [39] and Kochubei [21] (see also [6],[29], [33] and references therein), is given by the time- fractional diffusion equation , where − A = − ∆, ∆ being the Laplace operator. The work in [3] provides strong solutions for A being the generator of a Feller process. The work in [27] provides strong solutions for A being the generator of a Pearson diffusion on an interval. In these cases the fundamentalsolution (or Green function) corresponds to the probability density of a self-similar non-Markovian stochastic process, given by the time-changed transition probability functionof the diffusion associated with A by the hitting time of a β -stable subordinator.An example of equation (3) (with a potential), was studied in [12], wherein the authorsdetermined the fundamental solution of the non-homogeneous Cauchy problem associatedwith the second-order differential operator with variable coefficients given by A = d ∑ i,j a ij ( x ) ∂ ∂x i ∂x j + d ∑ j = b j ( x ) ∂∂x j + c ( x ) . The well-posedness of the (abstract) Cauchy problem (3) for A being a closed operator ina Banach space was studied in [5]. Moreover, evolution equations of the type (3) arise, forexample, as the limiting evolution of an uncoupled and properly scaled continuous timerandom walk (CTRW) with the waiting times in the domain of attraction of β − stablelaws . This probabilistic model and some of its extensions have been widely studied (see,e.g., [33], [38], [24], and references therein). The authors in [26] addressed the regularityof the non-homogeneous time-space fractional linear equation t D β +∗ u ( t, x ) = − c ( − ∆ ) α / u ( t, x ) + g ( t, x ) , x ∈ R d , t ≥ ,u ( , x ) = φ ( x ) , x ∈ R d as well as the well-posedness for the fractional Hamilton-Jacobi-Bellman (HJB) type equa-tion t D β +∗ u ( t, x ) = − c ( − ∆ ) α / u ( t, x ) + H ( t, x, ∇ u ( t, x )) , x ∈ R d , t ≥ ,u ( , x ) = φ ( x ) , x ∈ R d , for β ∈ ( , ) , α ∈ ( , ] and a positive constant c > − t D ( ν ) a −∗ ).We will first show the well-posedness of problem (1) (for two notions of solution) and the ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 3 stochastic representation for both notions of solution (see Theorem 4.20). The stochasticrepresentation for the solution u , will be given by u ( t, x ) = E [ φ a ( X x,A ( τ ( ν ) a ( t ))) + ∫ τ ( ν ) a ( t ) g ( X t, ( ν ) a +∗ ( s ) , X x,A ( s )) ds ] , (4)where { X t, ( ν ) a +∗ ( s )} s ≥ is the decreasing [ a, b ] -valued stochastic process generated by − t D ( ν ) a +∗ started at t ∈ [ a, b ] , { X x,A ( s )} s ≥ is the stochastic process generated by A started at x ∈ R d , τ ( ν ) a ( t ) is the first time { X t, ( ν ) a +∗ ( s )} s ≥ hits { a } . Note that the stochastic representation(4) features the (time-changed) process { X x,A ( τ ( ν ) a ( t ))} t ≥ .For A bounded and a stronger assumption on the function ν (see assumption (H1b)), wewill give the series representation to the solution of problem (1) u ( t, x ) = ∞ ∑ n = (( AI ( ν ) a + ) n φ a )( t, x ) + ∞ ∑ n = (( AI ( ν ) a + ) n I ( ν ) a + g )( t, x ) , (5)where I ( ν ) a + is the potential operator of the semigroup generated by the (generalised) RLfractional operator − t D ( ν ) a + (see Theorem 4.24). The series in (5) provides a generalisationof a certain class of Mittag-Leffler functions. To see this take A = λ , λ ∈ R , a = − t D ( ν ) a +∗ = − t D βa +∗ , the Caputo derivative of order β ∈ ( , ) , then I ( ν ) a + = I βa + , the RLfractional integral of order β , and u ( t, x ) = φ a ( x ) E β ( λt β ) + ∫ t g ( t − y, x ) βt β − ddy E β ( λy β ) dy, where E β ( z ) ∶= ( ∑ ∞ n = z n Γ ( βn + ) ) (see [9, Theorem 7.2] for example). By approximating thegenerator of a Feller process A with bounded operators (namely the Yosida approxiamtion)we show the convergence of the series representation (5) to the stochastic stochastic rep-resentation (4) for the operator A (see Theorem 4.27).As for the non-linear problem (2), we study the well-posedness following a similar strategyto the one used for the non-linear equation studied by the authors in [18]. Namely, bymeans of the the integral representation ( mild form ) of the solution to the linear problem(1), we reduce the analysis of (2) to a fixed point problem for a suitable linear operator(see Theorem 5.3). Let us mention that, even though in this work we do not includethe HJB type case, our results for the generalised non-linear equation (2) can be used toextend the well-posedness for the corresponding equations of HJB type.The results concerning the series representations (5) of the solutions to the linear evolu-tion equation (1) and the well-posedness of the non-linear evolution equation (2) rely onthe bounds in Theorem 3.4. Theorem 3.4 is a consequence of assumption (H1b), whichimplies that for every t, y ∈ [ a, b ] , s ∈ R + , P [ X t, ( ν ) a +∗ ( s ) ≥ y ] ≤ P [ X t,β + ( s ) ≥ y ] where X t,β + issome inverted β -stable subordinator of order β ∈ ( , ) .Let us briefly describe the two notions of solution used in this work for problem (1).We call u ∈ C ∞ ([ a, b ] × R d ) a solution in the domain of the generator for problem (1)with g ∈ C ∞ ([ a, b ] × R d ) , φ a ∈ Dom ( A ) , if u satisfies the two equalities in (1) and u ∈ Dom ( − D ( ν ) a +∗ + A ) , the domain of the generator − D ( ν ) a +∗ + A .This notion of solution is quite natural from the point of view of semigroup theory. To see ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 4 this consider a strongly continuous semigroup { T s } s ≥ acting on a Banach space B , let G be its generator and Dom ( G ) the domain of G . Suppose now that the potential operator ( − G ) − is bounded on B , then ( − G ) − ∶ B → Dom ( G ) is a bijection and G ( − G ) − g = − g (see [10, Theorem 1.1’]). By viewing problem (1) as a Dirichlet problem of the form Gu ( t, x ) = − g ( t, x ) , in ( a, b ] × R d , u ( a, x ) = φ a ( x ) on { a } × R d , for G = ( − t D ( ν ) a + + A ) , where − D ( ν ) a + is the generalised Riemann-Liouville (RL) fractionalderivative, φ a =
0, we will see that ( − G ) − is bounded. From the RL case we extend thedefinition to the Caputo case. Of course such definition of solution does not allow tochoose the boundary condition φ a , as u ( a, ⋅ ) is determined by the choice of g ∈ B .The second notion of solution overcomes this issue. Roughly speaking, a function u ∈ B ([ a, b ] × R d ) is said to be a generalised solution to problem (1) if u is the point-wiselimit of a certain sequence of solutions in the domain of the generator. The stochasticrepresentation of solutions in the domain of the generator allows us to pass to the limitand obtain well-posedness along with the stochastic representation (4) of the generalisedsolution.All results of this work concerning solutions in the domain of the generator hold true(with no change in the proofs) if we substitute R d with the closure of an open subset of R d , call it X , and we let A be the generator of a Feller semigroup on C ∞ ( X ) .The paper is organized as follows. Section 2 sets standard notation and gives a quickreview about generalised Caputo type operators of order less than 1. Section 3 introducesthe generalised RL integral operator I ( ν ) a + . Section 4 focuses on the well-posedness resultsfor the equation (1) along with providing stochastic and series representations for thesolutions. Section 5 deals with the well-posedness of the non-linear equation (2).2. Preliminaries
Notation.
Let N and R d be the set of positive integers and the d -dimensional Eu-clidean space, d ∈ N , respectively.For any subset A ⊂ R d , we define the standard sets of functions B ( A ) ∶ = { f ∶ A → R ∶ f is bounded and Borel measurable } ,C ( A ) ∶ = { f ∈ B ( A ) ∶ f is continuous } ,C ∞ ( A ) ∶ = { f ∈ C ( A ) ∶ f vanishes at infinity } . All these spaces are equipped with the usual sup-norm ∥ ⋅ ∥ , making them Banach spaces.For an open set A ⊂ R d we define C k ( A ) ∶ = { f ∈ C ( A ) ∶ D γ f ∈ C ( A ) , ∀ ∣ γ ∣ ≤ k } ,C k ∞ ( ¯ A ) ∶ = { f ∈ C ∞ ( A ) ∶ D γ f ∈ C ∞ ( A ) & D γ f is uniformly continuous on A, ∀ ∣ γ ∣ ≤ k } ,C ∞ ( ¯ A ) ∶ = ∩ ∞ k = C k ( ¯ A ) and C ∞∞ ( ¯ A ) ∶ = ∩ ∞ k = C k ∞ ( ¯ A ) , where γ is a multi-index, D γ the associated integer-order derivative operator, ¯ A denotesthe closure of A , and for the last three spaces of continuous functions we identify thefunctions on A along with their partial derivatives with their unique continuous extensionto ¯ A . If A is compact we write C k ( ¯ A ) = C k ∞ ( ¯ A ) .Two special spaces of continuous function will be of interest to us, namely C a ([ a, b ]) ∶ = { f ∈ C ([ a, b ]) ∶ f ( a ) = } , and C a, ∞ ([ a, b ] × X ) ∶ = { f ∈ C ∞ ([ a, b ] × X ) ∶ f ( a, x ) = ∀ x ∈ X } , ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 5 for X ⊂ R d , both equipped with the supremum norm, turning them into Banach spaces.When we write ∥ f ∥ for some real-valued function f ∶ X → R we mean the supremum normof f over its domain. If L is a linear operator acting on a subset of a Banach space B toa Banach space ˜ B , we denote by Dom ( L ) the domain of L . If L is bounded we denoteits operator norm by ∥ L ∥ .Notation Γ ( z ) and B ( α, β ) stands for the Gamma and the Beta function, respectively.For all α, β >
0, the Beta function is defined by B ( α, β ) ∶ = ∫ u α − ( − u ) β − du. We shall use the following rather standard identitiesΓ ( z + ) = z Γ ( z ) , B ( α, β ) = Γ ( α ) Γ ( β ) Γ ( α + β ) , (6)and the inequality Γ ( na ) > ( n − ) ! a ( n − ) ( Γ ( a )) n , (7)for n ∈ N and a >
0. Letters P and E are reserved for the probability and the mathe-matical expectation, respectively. We will use the lower case letter s as the time variablewhen indexing stochastic processes or semigroups (the letter t will generally be used todenote the starting point of a process on [ a, b ] ). For a stochastic process { X z ( s )} s ≥ the superscript z means that the process starts at z . The notation E [ f ( X z ( s ))] and E [ f ( X ( s )) ∣ X ( ) = z ] are used interchangeably.For a topological space X we write B ( X ) to denote its Borel σ -algebra. All the stochasticprocesses { X z ( s )} s ≥ considered in this paper are assumed to be defined on some com-plete filtered probability space ( Ω , F , { F s } s ≥ , P ) such that σ ( X z ( s )) ⊂ F s for each s ≥ σ ( X z ( s )) is the smallest σ -algebra generated by X z ( s ) . The notation a.e. standsfor almost everywhere with respect to Lebesgue measure.2.2. Feller processes.
Let { T s } s ≥ be a strongly continuous semigroup of linear boundedoperators on a Banach space ( B, ∥ ⋅ ∥ B ) , i.e., T s ∶ B → B s ∈ R + , T s + t = T s T t ∀ s, t ∈ R + , T = I the identity operator and lim s → ∥ T s f − f ∥ B = f ∈ B . Its (infinitesimal)generator L is defined as the (possibly unbounded) operator L ∶ Dom ( L ) ⊂ B → B givenby the strong limit Lf ∶ = lim s ↓ T s f − fs , f ∈ Dom ( L ) , (8)where the domain of the generator Dom ( L ) consists of those functions f ∈ B for whichthe limit in (8) exists in the norm sense. We denote the resolvent operator for λ ≥ ( λ − L ) − . Sometimes we use the notation e Ls = T s for a semigroup { T s } s ≥ with generator L . If L is a closed operator and D ⊂ Dom ( L ) is a subspace of B , then D is called a core for L if L is the closure in the graph norm of the restriction of L to D . If D ⊂ B and T s D ⊂ D for all s ≥
0, then D is said to be invariant (under the semigroup).We say that a (time homogeneous) Markov process Z = ( Z ( s )) s ≥ taking values in E ⊂ R d is a Feller process (see, e.g., [24, Section 3.6]) if its semigroup { T s } s ≥ , defined by T s f ( z ) ∶ = E [ f ( Z ( s )) ∣ Z ( ) = z ] , s ≥ , z ∈ E, f ∈ B ( E ) , gives rise to a Feller semigroup when reduced to C ∞ ( E ) , i.e., it is a strongly continuoussemigroup on C ∞ ( E ) and it is formed by positive linear contractions (0 ≤ T s f ≤ ≤ f ≤ C ∞ ( E ) to B ( E ) bythe same notation. ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 6
Generalised fractional operators of Caputo type.
This section provides thebasics on generalised fractional operators as introduced in [25], along with some propertiesand related definitions.Let ν ∶ R × ( R + /{ }) → R + be a non-negative function of two variables. The followingcondition will be always assumed when dealing with generalised fractional operators. (H0) The function ν ( t, r ) is continuous as a function of two variables andcontinuously differentiable in the first variable. Furthermore,sup t ∫ ∞ min { , r } ν ( t, r ) dr < ∞ , sup t ∫ ∞ min { , r }∣ ∂∂t ν ( t, r )∣ dr < ∞ , and lim δ → sup t ∫ < r ≤ δ rν ( t, r ) dr = . Remark 2.1.
The second bound in (H0) is both a natural assumption for concrete ex-amples (see, e.g., the L´evy kernel (12)) and a natural assumption for the proof of [25,Theorem 4.1]. The last bound in (H0) is a tightness assumption also used in the proof of[25, Theorem 4.1].
Definition 2.2.
Let a, b ∈ R , a < b . For any function ν satisfying condition (H0), theoperator − D ( ν ) a +∗ , defined by − D ( ν ) a +∗ f ( t ) ∶ = ∫ t − a ( f ( t − r ) − f ( t )) ν ( t, r ) dr + ( f ( a ) − f ( t )) ∫ ∞ t − a ν ( t, r ) dr, (9) t ∈ ( a, b ] , is called the generalised Caputo type operator .The operator − D ( ν ) a + , defined by − D ( ν ) a +∗ f ( t ) ∶ = ∫ t − a ( f ( t − r ) − f ( t )) ν ( t, r ) dr − f ( t ) ∫ ∞ t − a ν ( t, r ) dr, (10) t ∈ ( a, b ] , is called the generalised RL type operator . Remark 2.3.
Note that the operator (9) is well-defined at least on C ∞ ([ a, ∞ )) and thatthe operator (10) is well-defined at least on C ∞ ([ a, ∞ )) ∩ { f ( a ) = } .The sign − in the notation − D ( ν ) a +∗ is introduced to comply with the standard notation offractional derivatives.The subscript t will be added to operators (9) and (10) by denoting them as − t D ( ν ) a +∗ and − t D ( ν ) a + , respectively, if we want to emphasise the variable they act on.2.3.1. Special cases: the Caputo derivatives of order β ∈ ( , ) . The classical fractionalCaputo derivatives are particular cases of the operator (9). Namely, on regular enoughfunctions f ,if ν ( t, r ) = − ( − β ) r + β , β ∈ ( , ) , then − D ( ν ) a +∗ f ( t ) = − D βa +∗ f ( t ) , (11)where D βa +∗ stands for the Caputo derivative of order β ∈ ( , ) . Hence, D βa +∗ f ( t ) = ( − β ) ∫ t − a f ( t − r ) − f ( t ) r + β dr − f ( a ) − f ( t ) Γ ( − β )( t − a ) β , β ∈ ( , ) . For β ∈ ( , ) and smooth enough functions f , the expression in (9) coincides with thestandard analytical definition ( Riemann-Liouville approach ) which is given in terms ofthe
Riemann-Liouville fractional integral operator and the standard differential operatorof integer order (see, e.g., [9], [36], [37] and references therein).
ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 7
Other particular cases include the fractional derivatives of variable order − D ( ν ) a +∗ ≡ − D β ( t ) a +∗ ,which are obtained by taking ν as the function ν ( t, r ) = − ( − β ( t )) r + β ( t ) (12)with a suitable function β ∶ R → ( , ) (see [17]). Even more generally, these operatorsinclude the generalised distributed order fractional derivatives : − D ( ν ) a +∗ f ( t ) = − ∫ ∞−∞ ω ( s, t ) D β ( s,t ) a +∗ f ( t ) µ ( ds ) , (13)where ω ∶ R × [ a, b ] → R + is a differentiable function in the second variable such that ν ( t, r ) = − ∫ ∞−∞ ω ( s, t ) µ ( ds ) Γ ( − β ( s, t )) r + β ( s,t ) is a function satisfying condition (H0). In the classical fractional framework, particularcases of (13) have been studied for example in [31], [15]. Let us mention that temperedL´evy kernels of the form ν ( t, r ) = − e − λr Γ ( − β ) r + β , β ∈ ( , ) , λ > , fall under the assumptions (H0). Tempered L´evy kernels are actively studied, see forexample [8], [41].2.3.2. Probabilistic interpretation and basic results.
For ν satisfying (H0), we define thefollowing processes: Definition 2.4. (i) We denote by X t, ( ν )+ ∶ = { X t, ( ν )+ ( s )} s ≥ the Feller process started at t ∈ [ a, b ] induced by the semigroup { T ( ν )+ s } s ≥ generated by the operator (14) (be-low) on the space C ∞ (( −∞ , b ]) with core C ∞ (( −∞ , b ]) ∩ { f ∶ − D ( ν )+ f ∈ C ∞ (( −∞ , b ])} (see [24, Theorem 5.1.1]).(ii) We denote by X t, ( ν ) a +∗ ∶ = { X t, ( ν ) a +∗ ( s )} s ≥ the Feller process started at t ∈ [ a, b ] inducedby the semigroup { T ( ν ) a +∗ s } s ≥ generated by the operator (9) on the space C ([ a, b ]) with core C ([ a, b ]) (see [25, Theorem 4.1]).(iii) We denote by X t, ( ν ) a + ∶ = { X t, ( ν ) a + ( s )} s ≥ the sub-Feller process started at t ∈ ( a, b ] induced by the semigroup { T ( ν ) a + s } s ≥ generated by the operator (10) on the space C a ([ a, b ]) with core C ([ a, b ]) ∩ C a ([ a, b ]) (this follows from a simple modificationof [25, Theorem 4.1]).The operator (9) was introduced in [25] as a probabilistic extension of the classical frac-tional derivatives when applied to sufficiently regular functions. It can be seen as thegenerator of an interrupted Feller processes. The generator of the decreasing Feller pro-cess X t, ( ν )+ is given by − D ( ν )+ f ( t ) = ∫ ∞ ( f ( t − r ) − f ( r )) ν ( t, r ) dr, (14)and the process X t, ( ν ) a +∗ with generator ( ) is obtained by absorbing at the point a theprocess X t, ( ν )+ on its first attempt to leave the interval ( a, b ] . The process X t, ( ν ) a + withgenerator ( ) is obtained by killing the process X t, ( ν )+ on its first attempt to leave theinterval ( a, b ] . ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 8
Remark 2.5.
Since we are interested in the solutions to differential equations on finitetime intervals, we only consider the operators − D ( ν ) a +∗ and − D ( ν ) a + acting on functions definedon the interval [ a, b ] instead of [ a, ∞ ) , as was done originally in [25, Theorem 4.1]. Remark 2.6. If a = −∞ , then the operator − D ( ν )−∞+∗ coincides with the operator − D ( ν )+ onfunctions vanishing at infinity. This operator can be seen as the left-sided generalisationof the Marchaud derivative [37, Formulas 5.57-5.58]. This operator is also known as the generator form of fractional derivatives [24], [33].Notation p ( ν )+ s ( r, E ) and p ( ν ) a +∗ s ( r, E ) denote the transition probabilities (with s beingthe time variable) for the processes X t, ( ν )+ and X t, ( ν ) a +∗ , respectively.We collect some results in the following Proposition 2.7. (i) The processes X t, ( ν )+ , X t, ( ν ) a + and X t, ( ν ) a +∗ are non-increasing andthe sets { X t, ( ν )+ ( s ) ∈ ( c, d )} , { X t, ( ν ) a + ( s ) ∈ ( c, d )} , { X t, ( ν ) a +∗ ( s ) ∈ ( c, d )} have the sameprobability, for every t ∈ ( a, b ] , a < c < d ≤ b , s ∈ R + . In particular p ( ν ) a +∗ s ( t, { a }) = p ( ν )+ s ( t, ( −∞ , a ]) , t ∈ ( a, b ] .(ii) The law of τ ( ν ) a ( t ) ∶ = inf { s ≥ ∶ X t, ( ν )+ ( s ) ≤ a } equals the law of the first exit timefrom the interval ( a, b ] of the processes X t, ( ν ) a +∗ for each t ∈ ( a, b ] (so that we willuse indistinctly the same notation τ ( ν ) a ( t ) ).(iii) The first exit time τ ( ν ) a ( t ) has finite expectation and E [ τ ( ν ) a ( t )] → t → a undereither the assumption (H1a) or (H1b): (H1a) : There exist ǫ > δ >
0, such that the function ν satisfies ν ( t, r ) ≥ δ > t and ∣ r ∣ < ǫ , or (H1b) : The function ν satisfies ν ( t, r ) ≥ Cr − − β for some constant C > β ∈ ( , ) . Proof.
Part (i) is proved in Appendix 6.2. Part (ii) is implied by (i). For part (iii) see [25,Theorem 4.1]. (cid:3)
Remark 2.8.
Note that (H1b) implies (H1a).For our notion of generalised solution we will assume (H2) : the measures p ( ν )+ s ( t, ⋅ ) and p As ( x, ⋅ ) are absolutely continuous withrespect to Lebesgue measure for each t ∈ [ a, b ] , x ∈ R d , t ∈ R + ,where A is the generator of a Feller process { X x,A ( s )} s ≥ on R d , x ∈ R d , and we denoteby p As ( x, ⋅ ) the law of X x,A ( s ) , s ≥ x ∈ R d .3. Generalised RL integral operator I ( ν ) a + We use the potential operator corresponding to the generator − D ( ν ) a + as in Definition 2.4-(iii) to define an integral operator on B ([ a, b ]) , which can be thought of as a generalisationof the RL integral operator I βa + of order β ∈ ( , ) (see, e.g., [9, Definition 2.1]). ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 9
Definition 3.1.
Let ν be a function satisfying assumption (H0) and (H1a). The operator I ( ν ) a + ∶ B ([ a, b ]) → B ([ a, b ]) defined by ( I ( ν ) a + f ) ( t ) ∶ = ∫ ( a,t ] f ( y ) ( ∫ ∞ p ( ν )+ s ( t, dy ) ds ) , t > a, and 0 for t = a , will be called the generalised RL fractional integral associated with ν .The generalised fractional integral I ( ν ) a + satisfies the following:(i) for the process X t, ( ν )+ we have I ( ν ) a + f ( t ) = E [ ∫ τ ( ν ) a ( t ) f ( X t, ( ν )+ ( s )) ds ] , which follows from Proposition 2.7-(i)-(ii).(ii) For each f ∈ B [ a, b ] , ∣ ( I ( ν ) a + f ) ( t )∣ ≤ ∥ f ∥ sup t ∈ [ a,b ] E [ τ ( ν ) a ( t )] . In particular, if f = (the constant function 1), then ( I ( ν ) a + ) ( t ) = ∫ ( a,t ] ∫ ∞ p ( ν ) + s ( t, dy ) ds = E [ τ ( ν ) a ( t )] . Remark 3.2.
The operator I ( ν ) a + can be thought of as the left inverse operator of the RLtype operator − D ( ν ) a + . Note that the RL type operator − D ( ν ) a + coincides with the Caputotype operator − D ( ν ) a +∗ on functions vanishing at a . Remark 3.3. If ν ( x, y ) is given by (11), then I ( ν ) a + coincides with the Riemann-Liouvilleintegral operator I βa + of order β ∈ ( , ) (see, e.g., [9, Chapter 2]). Let τ βa ( t ) be the firstexit time from the interval ( a, b ] of the inverted β − stable subordinator started at t ∈ ( a, b ] .If p β − s ( t, y ) denotes the transition densities of the β − stable subordinator, then p β − s ( t, y ) = s − / β ω β ( s − / β ( y − t ) ; 1 , ) , where ω β ( ⋅ ; σ, γ ) stands for the β -stable density with scaling parameter σ , skewness pa-rameter γ and zero location parameter (see, e.g., [24, Equation (7.2), page 311]). Let p β + s ( t, y ) denote the transition density of the respective inverted β − stable subordinator.Then ∫ ∞ p β + s ( t, y ) ds = ∫ ∞ s − / β ω β ( s − / β ( t − y ) ; 1 , ) ds = ( t − y ) β − ∫ ∞ u − / β ω β ( u − / β ; 1 , ) du = ( β ) ( t − y ) β − , (15)using the Mellin transform of the β − stable densities ω β ( z ; 1 , ) for the last equality (see,e.g., [44, Theorem 2.6.3, p. 117]). The previous yields the known results ∣ ( I βa + f ) ( t )∣ ≤ ( β + ) ∥ f ∥( b − a ) β , and ( I βa + ) ( t ) = ∫ ta ∫ ∞ p β + s ( t, y ) dsdy = E [ τ βa ( t )] = ( t − a ) β Γ ( β + ) . ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 10
Let I ( ν ) ,na + denote the n -fold iteration of the operator I ( ν ) a + , n ∈ N . For convention I ( ν ) , a + stands for the identity operator.The following result shall be important for the following sections. It provides an explicitbound for ∣ I ( ν ) a + f ( t )∣ under assumption (H1b). Theorem 3.4.
Let ν be a function satisfying assumptions (H0), (H1b). Then, for each f ∈ B ([ a, b ]) , ∣ ( I ( ν ) ,na + f ) ( t )∣ ≤ ∥ f ∥ t ( b − a ) nβ ( Γ ( β + )) n n − ∏ k = B ( kβ + , β ) , n ≥ , (16) where ∥ f ∥ t ∶ = sup y ≤ t ∣ f ( y )∣ . Moreover, the series ∞ ∑ n = ( I ( ν ) ,na + f ) ( t ) (17) converges uniformly on [ a, b ] . Proof.
By definition of the generalised fractional integral ∣ ( I ( ν ) a +∗ f ) ( t )∣ ≤ ∫ ∞ ( ∫ ( a,t ] ∣ f ( y )∣ p ( ν ) + s ( t, dy )) ds ≤ ∫ ∞ ( ∫ ( a,t ] sup z ≤ y ∣ f ( z )∣ p ( ν ) + s ( t, dy )) ds. Fix β ∈ ( , ) as in (H1b) and denote by { X t,β + ( s )} s ≥ the associated inverted β -stablesubordinator. By assumption (H1b) it follows from [42, Theorem 1.5] that P [ X t, ( ν ) + ( s ) > y ] ≤ P [ X t,βt ( s ) > y ] , t, y ∈ ( a, b ] , s ∈ R + . Therefore E [ g ( X t, ( ν ) a + ( s ))] = E [ g ( X t, ( ν ) + ( s ))] ≤ E [ g ( X t,β + ( s ))] for any non-decreasing function g ∈ C ∞ (( −∞ , b ]) such that g ( t ) = , ∀ x ≤ a , wherethe equality holds as a consequence of the proof of Proposition 2.7-(i). By a standardapproximation argument we obtain P [ X t, ( ν ) a + ( s ) > y ] ≤ P [ X t,β + ( s ) > y ] , t, y ∈ ( a, b ] , s ∈ R + . Another approximation argument yields E [ g ( X t, ( ν ) a + ( s ))] ≤ E [ g ( X t,β + ( s ))] , (18)for any non-decreasing bounded function g ∶ [ a, b ] → R . In particular (18) holds for thefunction g ( y ) = sup z ≤ y ∣ f ( z )∣ . Hence ∣ ( I ( ν ) a +∗ f ) ( t )∣ ≤ ∫ ∞ ( ∫ ( a,t ] ∣ f ( y )∣ p ( ν ) + s ( t, dy )) ds ≤ ∫ ∞ ∫ ta sup z ≤ y ∣ f ( z )∣ p β + s ( t, y ) dyds ≤ ∥ f ∥ t ∫ ∞ ∫ ta p β + s ( t, y ) dyds ≤ ( β + ) ∥ f ∥ t ( t − a ) β , (19)To prove the inequality (16) we proceed by induction. Case n = n −
1. Then, using standard identities for the Beta
ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 11 function, the inequality in (19) and the induction hypothesis ∣ ( I ( ν ) ,na + f ) ( t )∣ = ∣ I ( ν ) a + I ( ν ) ,n − a + f ( t )∣ ≤ ∫ ∞ ∫ ta sup z ≤ y ∣ I ( ν ) ,n − a + f ( z )∣ p β + s ( t, y ) dyds ≤ ∫ ∞ ∫ ta ∥ f ∥ y ( y − a ) ( n − ) β ( Γ ( β + )) n − n − ∏ k = B ( kβ + , β ) p β + s ( t, y ) dyds ≤ ∥ f ∥ t ( Γ ( β + )) n − n − ∏ k = B ( kβ + , β ) ∫ ∞ ∫ ta ( y − a ) ( n − ) β p β + s ( t, y ) dyds ≤ ∥ f ∥ t ( Γ ( β + )) n − n − ∏ k = B ( kβ + , β ) ∫ ta ( y − a ) ( n − ) β ( t − y ) β − ( β + ) dy = ∥ f ∥ t ( b − a ) nβ ( Γ ( β + )) n n − ∏ k = B ( kβ + , β ) , where the last inequality uses Fubini’s theorem and the equality in (15).To prove the convergence of (17) we use the identity (6) and the inequality (7) to obtainthat for each n ∈ N n − ∏ k = B ( kβ + , β ) = ( Γ ( β ) ) n nβ Γ ( nβ ) ≤ ( Γ ( β ) ) n nβ ( n − ) ! β ( n − ) ( Γ ( β ) ) n ≤ n ! β n . Hence, ∣ ( I ( ν ) ,na + f ) ( t ) ∣ ≤ ∥ f ∥ ( ( b − a ) β β Γ ( β + ) ) n n ! = ∶ M n . Since ∑ ∞ n = M n converges, Weierstrass M − test implies the uniform convergence of (17) on [ a, b ] , as required. (cid:3) Remark 3.5.
In the classical fractional setting, the n − fold RL integral I β,na + has an explicitexpression obtained from its semigroup property [9, Theorem 2.2] ( I β,na + f ) ( t ) = ( I nβa + f ) ( t ) . Hence, for f ( t ) = , ( I β,na + f ) ( t ) = ( nβ ) ∫ ta ( t − y ) nβ − dy = ( t − a ) nβ Γ ( nβ + ) . Generalised fractional evolution equation: Linear case
Using the theory of strongly continuous semigroups and the properties of the process X ( ν ) a +∗ (in particular Proposition 2.7-(iii)), we first prove the wellposedness and stochasticrepresentation for two notions of solution to the problem ( − t D ( ν ) a + + A ) u ( t, x ) = − g ( t, x ) , ( t, x ) ∈ ( a, b ] × R d ,u ( a, x ) = , x ∈ R d , (20)for g ∈ B ([ a, b ] × R d ) , A being the generator of a Feller semigroup on C ∞ ( R d ) , and ν satisfying assumptions (H0) and (H1a) (see Theorem 4.10).The series representation is obtained under the additional assumptions (H1b) and A bounded (see Theorem 4.15). We then show convergence of such series representationto the stochastic representation for A generator of a Feller semigroup (see Theorem4.16). ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 12
We use the following technical results whose proof is provided in Appendix 6.1.
Theorem 4.1.
Let G, ˜ G be generators of strongly continuous, uniformly bounded semi-groups T = { T s } s ≥ , ˜ T = { ˜ T s } s ≥ on C ∞ ( X ) , C ∞ ( ˜ X ) with domains D = Dom ( G ) , ˜ D = Dom ( ˜ G ) , respectively, where X, ˜ X are the closure of non-empty open subsets of R n and C ∞∞ ( X ) ⊂ D , C ∞∞ ( ˜ X ) ⊂ ˜ D , respectively.Define L ∶ = Linear span of D ˜ D, where D ˜ D ∶ = { g = f ˜ f ∶ f ∈ D, ˜ f ∈ ˜ D } .Then(i) the closure of G + ˜ G in C ∞ ( X × ˜ X ) on the set L generates a uniformly boundedstrongly continuous semigroup { Φ s } s ≥ on C ∞ ( X × ˜ X ) , with invariant core L , where Φ s ∶ = T s ˜ T s = ˜ T s T s s ∈ R + (where G and T s act on the X -variable, ˜ G and ˜ T s act onthe ˜ X -variable, s ∈ R + ).We denote by L the generator of { Φ s } s ≥ .(ii) If G, ˜ G are generators of Feller semigroups, then { Φ s } s ≥ is a Feller semigroup. If G, ˜ G are generators of sub-Feller semigroups, then { Φ s } s ≥ is a sub-Feller semi-group.(iii) The same statement in (i) holds for X = [ a, b ] , G acting on C a ([ a, b ]) and { Φ s } s ≥ acting on C a, ∞ ([ a, b ] × ˜ X ) instead of C ∞ ([ a, b ] × ˜ X ) . Remark 4.2.
Theorem 4.1 allows us to solve the resolvent equation Lu = λu + g, λ ∈ R + /{ } , g ∈ C ∞ ( X × ˜ X ) , but what we are particularly interested in is the case λ =
0, which requires more care asthe potential operator is not well-defined in general.The next Proposition will be used in Section 4.1.3.
Proposition 4.3.
Suppose that ˜ G is bounded. Then, under the assumptions of Theorem4.1, f ∈ Dom ( L ) implies that f ( ⋅ , ˜ x ) ∈ Dom ( G ) for each ˜ x ∈ ˜ X . In particular Lf = ( G + ˜ G ) f . Proof.
Let f ∈ Dom ( L ) . Since L is a core for the generator L , there exists { f n } n ∈ N ⊂ L such that f n → f and ( G + ˜ G ) f n = Lf n → Lf . As ˜ G is bounded ˜ Gf n → ˜ Gf and so { ˜ Gf n } n ∈ N is Cauchy in C ∞ ( X × ˜ X ) . For each ˜ x ∈ ˜ X f n ( ⋅ , ˜ x ) → f ( ⋅ , ˜ x ) in C ∞ ( X ) and f n ( ⋅ , ˜ x ) ∈ Dom ( G ) for each n ∈ N , by the definition of L . If we show that Gf n ( ⋅ , ˜ x ) isCauchy in C ∞ ( X ) we are done as G is a closed operator on C ∞ ( X ) . This follows fromthe inequality ∣( Gf n − Gf m )( x, ˜ x )∣ ≤ ∥ Lf n − Lf m ∥ + ∥ ˜ Gf n − ˜ Gf m ∥ , and by taking n and m large. (cid:3) We now identify two independent processes associated with the semigroups { T s } s ≥ and { ˜ T s } s ≥ from the process on X × ˜ X induced by the semigroup { Φ s } s ≥ in Theorem 4.1-(ii). Definition 4.4.
Let { Φ s } s ≥ be a Feller semigroup generated as in Theorem 4.1-(ii) anddenote by Y ( t, ˜ x ) ∶ = { Y ( t, ˜ x ) ( s )} s ≥ , ( t, ˜ x ) ∈ X × ˜ X the induced Feller process.For each ( t, ˜ x ) ∈ X × ˜ X , define the process X t ∶ = { X t ( s )} s ≥ and the process ˜ X ˜ x ∶ = ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 13 { ˜ X ˜ x ( s )} s ≥ to be the processes induced by the collection of probability measures on X and on ˜ X defined as P ( X t ( s ) ∈ B ) ∶ = Φ s ( B × ˜ X )( t, ˜ x ) , B ∈ B ( X ) , and P ( ˜ X ˜ x ( s ) ∈ ˜ B ) ∶ = Φ s ( X × ˜ B )( t, ˜ x ) , ˜ B ∈ B ( ˜ X ) , respectively. Define the stochastic process {( X t ( s ) , ˜ X ˜ x ( s ))} s ≥ on X × ˜ X by P ( X t ( s ) ∈ B, ˜ X ˜ x ( s ) ∈ ˜ B ) ∶ = Φ s ( B × ˜ B )( t, ˜ x ) , B ∈ B ( X ) , ˜ B ∈ B ( ˜ X ) . Corollary 4.5.
Let { Φ s } s ≥ be a Feller semigroup generated as in Theorem 4.1-(ii).Then Y ( t, ˜ x ) ( s ) = ( X t ( s ) , ˜ X ˜ x ( s )) , s ∈ R + , ( t, ˜ x ) ∈ X × ˜ X . Moreover the processes X t and˜ X ˜ x are independent and they equal the processes generated by G and ˜ G on C ∞ ( X ) and C ∞ ( ˜ X ) , respectively. Proof.
The first statement is straightforward. The latter two statements follow fromΦ s ( B × ˜ B )( t, ˜ x ) = P ( X t ( s ) ∈ B ) P ( ˜ X ˜ x ( s ) ∈ ˜ B ) , B ∈ B ( X ) , ˜ B ∈ B ( ˜ X ) , s ∈ R + . (cid:3) Linear evolution equation: RL Case.
Well-posedness and stochastic representation.
We drop the subscript t from theoperators − t D ( ν ) a +∗ and − t D ( ν ) a + .With respect to the notation in Theorem 4.1, from now on G = − D ( ν ) a + , D = Dom ( − D ( ν ) a + ) , C ∞ ( X ) = C a ([ a, b ]) , or (21) G = − D ( ν ) a +∗ , D = Dom ( − D ( ν ) a +∗ ) , C ∞ ( X ) = C ([ a, b ]) , and (22)˜ G = A, ˜ D = Dom ( A ) , C ∞ ( ˜ X ) = C ∞ ( R d ) , (23)where the triples (21) and (22) are the ones given in Definition 2.4-(i) and Definition2.4-(ii), respectively. The triple (23) is any such triple arising from a Feller semigroup on C ∞ ( R d ) with C ∞∞ ( R d ) ⊂ Dom ( A ) , and we denote the corresponding process by X x,A ∶ = { X x,A ( s )} s ≥ .We will show that the potential operator ( − L ) − of L = − D ( ν ) a + + A (as in Theorem 4.1-(iii))is bounded. We will use this fact to solve problem (20).Define the stopping times τ Ya (( t, x )) ∶ = inf s { s ≥ ∶ Y ( t,x ) ( s ) ∉ ( a, b ] × R d } , τ Xa ( t ) ∶ = inf s { s ≥ ∶ X t ( s ) ∉ ( a, b ]} , where Y ( t,x ) = { Y ( t,x ) ( s )} s ≥ and X t = { X t ( s )} s ≥ are defined as in Definition 4.4. Proposition 4.6.
The stopping times τ Ya (( t, x )) , τ Xa ( t ) and τ ( ν ) a ( t ) have the same dis-tribution, in particular E [ τ Ya (( t, x ))] = E [ τ ( ν ) a ( t )] < ∞ , (24)uniformly in ( t, x ) ∈ [ a, b ] × R d . Moreover τ Ya (( t, x )) is independent of { ˜ X x ( s )} s ≥ . ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 14
Proof.
By Corollary 4.5 the process X t has the same distribution of X t, ( ν ) a +∗ . In particular X t is non-increasing and P ( X t ( s ) > a ) = P ( X t, ( ν ) + ( s ) > a ) . Also { τ Ya (( t, x )) > s } = { X t ( s ) > a } ∩ { ˜ X x ( s ) ∈ R d } and by independence of X t ( s ) and˜ X x ( s ) (Corollary 4.5) we have P ( τ Ya (( t, x )) > s ) = P ({ X t ( s ) > a ) P ( ˜ X xs ∈ R d ) = P ( X t, ( ν ) + ( s ) > a ) = P ( τ ( ν ) a ( t ) > s ) . This proves that τ Ya (( t, x )) , τ Xa ( t ) and τ ( ν ) a ( t ) have the same distribution. In particularwe obtain the equality in (24).The inequality in (24) follows from Proposition 2.7-(iii).The last statement can be proved using the computations in this proof. (cid:3) From now on we will use the notation τ ( ν ) a ( t ) for the stopping time τ Ya (( t, x )) . In the nextproposition we obtain the boundedness and the stochastic representation for the potentialoperator ( − L ) − . Proposition 4.7.
Let Φ ∗ ∶ = { T ∗ s ˜ T s } s ≥ be the Feller semigroup obtained in Theorem 4.1-(ii) for the triples (22) and (23). Denote the generator of Φ ∗ by L ∗ .Let Φ ∶ = { T s ˜ T s } s ≥ be the semigroup obtained from in Theorem 4.1-(iii) for the triples(21) and (23). Denote the generator of Φ by L .Then ( − L ) − ∶ C a, ∞ ([ a, b ] × ˜ X ) → C a, ∞ ([ a, b ] × ˜ X ) is well-defined and it is bounded.Moreover the equality ( − L ∗ ) − g = ( − L ) − g holds if g ∈ C a, ∞ ([ a, b ] × ˜ X ) and we obtain thestochastic representation ( − L ) − g ( t, x ) = E ∫ τ ( ν ) a ( t ) e As g ( X t, ( ν ) + ( s ) , x ) ds. Proof.
For each ( t, x ) ∈ [ a, b ] × R d , s ∈ R + T ∗ s ˜ T s ( B × ˜ B )( t, x ) = T s ˜ T s ( B × ˜ B )( t, x ) (25)if a ∉ B , B ∈ B ( X ) , ˜ B ∈ B ( ˜ X ) from Proposition 2.7-(i) and Corollary 4.5. Let g ∈ C a, ∞ ([ a, b ] × R d ) , then ( − L ∗ ) − g ( t, x ) = ∫ ∞ T ∗ s ˜ T s g ( t, x ) ds = E ( ∫ τ ( ν ) a ( t ) + ∫ ∞ τ ( ν ) a ( t ) ) ˜ T s g ( X t, ( ν ) a +∗ ( s ) , x ) ds = E ∫ τ ( ν ) a ( t ) ˜ T s g ( X t, ( ν ) a +∗ ( s ) , x ) ds + , where we used Proposition 4.6. A similar computation using (25) yields ( − L ) − g ( t, x ) = E ∫ τ ( ν ) a ( t ) ˜ T s g ( X t, ( ν ) a + ( s ) , x ) ds. (26)That ( − L ) − ∶ C a, ∞ ([ a, b ] × ˜ X ) → C a, ∞ ([ a, b ] × ˜ X ) is well-defined and bounded follows fromProposition 2.7-(iii) with the representation (26), as ∣( − L ) − g ( t, x )∣ ≤ E ∫ τ ( ν ) a ( t ) ∣ ˜ T s g ( X t, ( ν ) + ( s ) , ⋅ )( x )∣ ds ≤ ∥ g ∥ E [ τ ( ν ) a ( t )] < ∞ , ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 15 and noting that E [ τ ( ν ) a ( t )] ≤ E [ τ ( ν ) a ( b )] < ∞ . (cid:3) We are now ready to prove the well-posedness of problem (20) for two notions of solutions(following [17]) and to obtain stochastic representations for such solutions.
Definition 4.8.
Let g ∈ C a, ∞ ([ a, b ] × R d ) . A function u ∈ C a, ∞ ([ a, b ] × R d ) is said to be a solution in the domain of the generator to problem (20) if u ∈ Dom ( L ) and u satisfies theequalities in (20), where L is the generator obtained in Theorem 4.1-(iii). Definition 4.9.
Let g ∈ B ([ a, b ] × R d ) . A function u ∈ B ([ a, b ] × R d ) is said to be a generalised solution to problem (20) if u = lim n → ∞ u n point-wise, where u n is the solution inthe domain of the generator to problem (20) with { g n } n ∈ N ⊂ C a, ∞ ([ a, b ] × R d ) , lim n → ∞ g n = g a.e. and sup n ∥ g n ∥ < ∞ . Theorem 4.10.
Let ν be a function satisfying conditions ( H ) , ( H a ) and let A be thegenerator of a Feller semigroup on C ∞ ( R d ) .(i) If g ∈ C a, ∞ ([ a, b ] × R d ) , then there exists a unique u ∈ C a, ∞ ([ a, b ] × R d ) solutionin the domain of the generator to problem (20). Moreover u admits the stochasticrepresentation u ( t, x ) = E [ ∫ τ ( ν ) a ( t ) e As g ( X t, ( ν ) + ( s ) , ⋅ )( x ) ds ] . (27) (ii) If g ∈ B ([ a, b ] × R d ) and (H2) holds, then there exists a unique u ∈ B ([ a, b ] × R d ) generalised solution to problem (20). Moreover u has the stochastic representationgiven in (27). Proof. (i) The potential operator ( − L ) − of the semigroup { Φ s } s ≥ is bounded by Proposition4.7. Hence by Theorem 1.1’ in [10] ( − L ) − ∶ C a, ∞ ([ a, b ] × R d ) → Dom ( L ) is abijection, and ( − L ) − g solves the equation L ( − L ) − g ( t, x ) = − g ( t, x ) , ( t, x ) ∈ [ a, b ] × R d , g ∈ C a, ∞ ([ a, b ] × R d ) , giving the existence and uniqueness of a solution in the domain of the generator.The stochastic representation follows from Proposition 4.7.(ii) Let g ∈ B ([ a, b ] × R d ) and take { g n } n ∈ N ⊂ C a, ∞ ([ a, b ] × R d ) such that g n → g a.e. as n → ∞ and sup n ∥ g n ∥ ∞ < ∞ (such sequence can be constructed using [14, Theorem7-(i)-(ii), Appendix C]). Note that condition (H2) and g ∈ C a, ∞ ([ a, b ] × R d ) implythat E g ( X t, ( ν ) a + ( s ) , X x,A ( s )) = E g ( X t, ( ν ) a +∗ ( s ) , X x,A ( s )) = ∫ ta ∫ R d g ( z, y ) p ( ν ) + s ( t, z ) p As ( x, y ) dydz. Then by Dominated Convergence Theorem (DCT) for each ( t, x ) ∈ [ a, b ] × R d F ( t,x ) ,n ( s ) ∶ = E [ e As g n ( X t, ( ν ) a + ( s ) , ⋅ )( x )] → E [ e As g ( X t, ( ν ) a + ( s ) , ⋅ )( x )] = ∶ F ( t,x ) ( s ) , as n → ∞ . Define G ( t,x ) ( s ) ∶ = sup n ∥ g n ∥ P ( τ ( ν ) a ( t ) > s ) . Thensup n ∣ F ( t,x ) ,n ( s )∣ ≤ G ( t,x ) ( s ) , ∫ ∞ G ( t,x ) ( s ) ds = sup n ∥ g n ∥ E [ τ ( ν ) a ( t )] < ∞ , ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 16 and by DCT we obtainlim n → ∞ E [ ∫ τ ( ν ) a ( t ) e As g n ( X t, ( ν ) + ( s ) , ⋅ )( x ) ds ] = E [ ∫ τ ( ν ) a ( t ) e As g ( X t, ( ν ) + ( s ) , ⋅ )( x ) ds ] , which gives existence of a generalised solution, independence of the approximatingsequence, hence uniqueness, and the claimed stochastic representation. (cid:3) Approximation by Yosida operators.
Lemma 4.11.
Let L λ ∶ = λL ( λ − L ) − be the Yosida approximation for the generator L of a Feller semigroup on C ∞ ( R d ) . Let g ∈ C ∞ ([ a, b ] × R d ) . Let u λ ∈ C a, ∞ ([ a, b ] × R d ) bethe generalised solution to problem (20) with A = L λ . Let u ∈ C a, ∞ ([ a, b ] × R d ) be thegeneralise solution to problem (20), with A = L .Then for each t ∈ [ a, b ] , u λ ( t, x ) → u ( t, x ) as λ → ∞ , uniformly in x ∈ R d . Proof.
By [13, Chapter 1, Proposition 2.7] we have that for each g ∈ C ∞ ([ a, b ] × R d ) , t ∈ [ a, b ] , ∥( e L λ s − e Ls ) g ( t, ⋅ )∥ R d → λ → ∞ , uniformly for s ≥ ∥ g ∥ as the dominating function. Then ∥ e L λ s g ( t, ⋅ )( x )∥ ≤ ∥ g ( ⋅ , x )∥ ≤ ∥ g ∥ which implies E [ ∫ τ ( ν ) a ( t ) ∣ e L λ s g ( X t, ( ν ) + ( s ) , ⋅ )( x )∣ ds ] ≤ ∥ g ∥ E [ τ ( ν ) a ( t )] < ∞ , and the result follows from the application of DCT. (cid:3) Series representation.
Under the additional assumptions A is bounded and ν satisfies assumption (H1b) , we give a series representation for the solution in the domain of the generator and thegeneralised solution to problem (20) obtained in Theorem 4.10.Once we have the series representation we will obtain convergence of a sequence of seriesrepresentations of solutions to the stochastic representation obtained in Theorem 4.10 for A the generator of a Feller semigroup on C ∞ ( R d ) (see Theorem 4.16 below).Let us give well-posedness and stochastic representation for the solution to the (FODE)problem − D ( ν ) a + u ( t ) = − g ( t ) , t ∈ ( a, b ] ,u ( a ) = , g ∈ B ([ a, b ]) . (28) Definition 4.12.
Let g ∈ C a ([ a, b ]) . A function u ∈ C a ([ a, b ]) is a solution in the domainof the generator for problem (28) if u ∈ Dom ( − D ( ν ) a + ) and u satisfies (28). Definition 4.13.
A function u ∈ B ([ a, b ]) is a generalised solution to problem (28) if u = lim n → ∞ u n point-wise, where u n is the solution in the domain of the generator toproblem (28) for g n ∈ C a ([ a, b ]) , n ∈ N , g n → g a.e. and sup n ∈ N ∥ g n ∥ < ∞ .The following is just a simpler version of Theorem 4.10. ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 17
Theorem 4.14.
Let ν be a function satisfying conditions (H0), (H1a). If g ∈ C a ([ a, b ]) there exists a unique solution in the domain of the generator u ∈ C a ([ a, b ]) to problem(28), and u has the representation u = I ( ν ) a + g .Under the additional assumption (H2), if g ∈ B ([ a, b ]) there exists a unique u ∈ B ([ a, b ]) generalised solution to problem (28), also with the representation u = I ( ν ) a + g . Theorem 4.15.
Let ν be a function satisfying assumption (H0), (H1b). Suppose that A is bounded.(1) If g ∈ C a, ∞ ([ a, b ] × R d ) the unique solution u ∈ C a, ∞ ([ a, b ] × R d ) in the domain ofthe generator to problem (20) has the series representation u ( t, x ) = ∞ ∑ n = (( I ( ν ) a + A ) n I ( ν ) a + g ) ( t, x ) , (29) where the convergence is in the sense of the norm of C a, ∞ ([ a, b ] × R d ) .(ii) If g ∈ B ([ a, b ] × R d ) and (H2) holds, the unique generalised solution u ∈ B ([ a, b ] × R d ) to problem (20) has the series representation given in (29). Proof.
Note that by Riesz-Representation Theorem ([24, Theorem 1.7.3]) A and I ( ν ) a + commute.(i) Let u ∈ C a, ∞ ([ a, b ] × R d ) be the solution in the domain of the generator to problem(20) obtained in Theorem 4.10. As A is bounded and u ∈ Dom ( L ) we obtain byProposition 4.3 that for each x ∈ R d , u ( ⋅ , x ) ∈ Dom ( − D ( ν ) a + ) , Lu ( ⋅ , x ) = ( − D ( ν ) a + + A ) u ( ⋅ , x ) . Hence u ( ⋅ , x ) solves − D ( ν ) a + u ( ⋅ , x ) = − ˜ g ( ⋅ , x ) , u ( a, x ) = g ( ⋅ , x ) ∶ = Au ( ⋅ , x ) + g ( ⋅ , x ) ∈ C a [ a, b ] , as Au ( a, ⋅ ) =
0. Hence, by Theorem4.14, u ( ⋅ , x ) is the unique solution in the domain of the generator to problem (30)and it has the representation u ( ⋅ , x ) = I ( ν ) a + ˜ g ( ⋅ , x ) .By induction, for each N ∈ N u ( t, x ) = N ∑ n = (( I ( ν ) a + A ) n I ( ν ) a + g ) ( t, x ) + (( I ( ν ) a + A ) N + u ) ( t, x ) . (31)Now observe that, a n ( t, x ) ∶ = (( I ( ν ) a + A ) n I ( ν ) a + g ) ( t, x ) ≤ ∣ ( I ( ν ) a + A ) n I ( ν ) a + g ) ( t, x )∣ ≤ ∥ g ∥∥ A ∥ n ∣ ( I ( ν ) ,n + a + ) ( t )∣ = ∶ b n ( t ) . Hence Theorem 3.4 implies the uniform convergence of ∑ ∞ n = b n ( t ) , which in turnimplies the uniform convergence of ∑ ∞ n = a n ( t, x ) . Moreover ∣ (( I ( ν ) a + A ) N + u ) ( t, x )∣ ≤ ∥ u ∥∥ A ∥ N ∣ I ( ν ) ,N + a + ( t, x )∣ → , N → ∞ , due to the uniform convergence of ∑ ∞ n = ∥ A ∥ n ( I ( ν ) ,na + ) ( t ) on [ a, b ] , again by The-orem 3.4. Then, letting N → ∞ in the equality (31) yields the result in (29).(ii) Consider a sequence { g n } n ∈ N ⊂ C a, ∞ ([ a, b ] × R d ) such that g n → g a.e. andsup n ∥ g n ∥ < ∞ . Fix ( t, x ) ∈ [ a, b ] × R d . By DCT we obtainlim n → ∞ ∞ ∑ m = F ( t,x ) ,n ( m ) = ∞ ∑ m = (( I ( ν ) a + A ) m I ( ν ) a + g ) ( t, x ) , (32) ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 18 where F ( t,x ) ,n ( m ) ∶ = ( I ( ν ) a + A ) m I ( ν ) a + g n . To see this observe that for every m ∈ N lim n → ∞ F ( t,x ) ,n ( m ) = (( I ( ν ) a + A ) m I ( ν ) a + g ) ( t, x ) , and ∣ F ( t,x ) ,n ( m )∣ ≤ F ( t,x ) ( m ) ∶ = sup n ∥ g n ∥∥ A ∥ m ( I ( ν ) a + ) m + ( t ) .By part (i) of this Theorem and part (ii) of Theorem 4.10 the limit on the left-hand-side of (32) equals the unique generalised solution to problem (20). (cid:3) Convergence of the series representation to the stochastic representation.
Theorem 4.16.
Let ν be a function satisfying assumptions (H0), (H1b). Let A λ bethe Yosida approximation for the generator of a Feller semigroup A on C ∞ ( R d ) . Let g ∈ C ∞ ([ a, b ] × R d ) .Then for each t ∈ [ a, b ] ∞ ∑ n = ( I ( ν ) a + A λ ) n I ( ν ) a + g ( t, x ) → E [ ∫ τ ( ν ) a ( t ) e As g ( X t, ( ν ) + ( s ) , ⋅ )( x ) ds ] , λ → ∞ , (33) uniformly in x ∈ R d . Proof.
The result follows from combining Lemma 4.11 with Theorem 4.15. (cid:3)
Linear evolution equation: Caputo case.
We now transfer the results for theRL generalised fractional operator − D ( ν ) a + to the Caputo generalised fractional operator − D ( ν ) a +∗ . We will indeed look at the problem ( − t D ( ν ) a +∗ + A ) u ( t, x ) = − g ( t, x ) , ( t, x ) ∈ ( a, b ] × R d ,u ( a, x ) = φ a ( x ) , x ∈ R d , (34)where g ∈ B ([ a, b ] × R d ) , φ a ∈ Dom ( A ) ⊂ C ∞ ( R d ) , A is the generator of a Feller semigroupon C ∞ ( R d ) with C ∞∞ ( R d ) ⊂ Dom ( A ) and ν is a function satisfying conditions (H0), (H1a).We again drop the subscript t in − t D ( ν ) a +∗ . Remark 4.17.
Note that if φ a ∈ Dom ( A ) then u satisfies ( − D ( ν ) a +∗ + A ) u ( t, x ) = − g ( t, x ) ,u ( a, x ) = φ a ( x ) , if and only if ˜ u = u − φ a satisfies ( − D ( ν ) a +∗ + A ) ˜ u ( t, x ) = − ( g + Aφ a )( t, x ) , ˜ u ( a, x ) = , using the fact that − D ( ν ) a +∗ c ( t, x ) = c constant in the t variable (whichis an immediate consequence of Definition 9). We indirectly use this fact to connectthe results obtained in last section about RL type evolution equations to Caputo typeevolution equations. ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 19
Well-posedness and stochastic representation.
Definition 4.18.
Let g ∈ C ([ a, b ] × R d ) , φ a ∈ Dom ( A ) such that Aφ a ( x ) = − g ( a, x ) ∀ x ∈ R d . A function u ∈ C ([ a, b ] × R d ) is a solution in the domain of the generator to problem (34) if u = ˜ u + φ a , where ˜ u is a solution in the domain of the generator for problem (20)with ˜ g = g + Aφ a ∈ C a, ∞ ([ a, b ] × R d ) . Definition 4.19.
Let g ∈ B ([ a, b ] × R d ) , φ a ∈ Dom ( A ) . A function u ∈ B ([ a, b ] × R d ) isa generalised solution for problem (34) if u = ˜ u + φ a , where ˜ u is a generalised solution toproblem (20) for ˜ g ∶ = g + Aφ ∈ B ([ a, b ] × R d ) . Theorem 4.20.
Assume that ν is a function that satisfies (H0) and (H1a).(i) If g ∈ C ∞ ([ a, b ] × R d ) and φ a ∈ Dom ( A ) such that Aφ a ( ⋅ ) = − g ( a, ⋅ ) , then thereexists a unique solution u ∈ C ∞ ([ a, b ] × R d ) in the domain of the generator toproblem (34) and u has the stochastic representation u ( t, x ) = E [ φ a ( X x,A ( τ ( ν ) a ( t )))] + E [ ∫ τ ( ν ) a ( t ) g ( X t, ( ν ) + ( s ) , X x,A ( s )) ds ] . (35) (ii) If g ∈ B ([ a, b ] × R d ) , φ a ∈ Dom ( A ) and (H2) holds, then there exists a unique u ∈ B ([ a, b ] × R d ) generalised solution for problem (34) and u has the stochasticrepresentation given by (35). Proof. (i) By the assumptions on g and φ a we have that ˜ g ∶ = g + Aφ a ( x ) ∈ C a, ∞ ([ a, b ] × R d ) ,and it follows from Theorem 4.10-(i) that a unique solution ˜ u in the domain of thegenerator to problem (20) exists.The above gives existence of a solution in the domain of the generator to problem(34) and uniqueness.By Theorem 4.10 ˜ u has the stochastic representation˜ u ( t, x ) = E ∫ τ ( ν ) a ( t ) ˜ g ( X t, ( ν ) + ( s ) , X x,A ( s )) ds = E ∫ τ ( ν ) a ( t ) g ( X t, ( ν ) + ( s ) , X x,A ( s )) ds + E ∫ τ ( ν ) a ( t ) Aφ a ( X x,A ( s )) ds. Consider u = ˜ u + φ a , then by Dynkin formula (see [24, Theorem 3.9.4]) we have theequality φ a ( x ) + E ∫ τ ( ν ) a ( t ) Aφ a ( X x,A ( s )) ds = E φ a ( X x,A ( τ ( ν ) a ( t ))) , and we obtain the stochastic representation in (35).(ii) As g + Aφ a ∈ B ([ a, b ] × R d ) existence and uniqueness follows immediately fromTheorem 4.10-(ii), and we have the stochastic representation (35) by the sameargument at the end of part (i) of this proof. (cid:3) Remark 4.21.
The solution in the domain of the generator u ∈ C ∞ ([ a, b ] × R d ) of Theorem4.20 solves problem (34), in the sense that L ∗ u ( t, x ) = L ˜ u ( t, x ) + Aφ a ( x ) = − g ( t, x ) − Aφ a ( x ) + Aφ a ( x ) = − g ( t, x ) , ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 20 and u ( a, x ) = ˜ u ( a, x ) + φ a ( x ) = φ a ( x ) , where we use the fact that u = ˜ u + φ a ∈ Dom ( L ∗ ) , L ∗ ˜ u = L ˜ u and L ∗ φ a = Aφ a . Here L ∗ is the generator obtained in Theorem 4.1-(ii) and L is the generator obtained in Theorem 4.1-(iii). For the equality L ∗ u = ( − D ( ν ) a +∗ + A ) u , it isin general necessary to prove smoothness properties of u . Remark 4.22.
As mentioned in the introduction, all results for the solution in the domainof the generator hold (with the same proofs) if A is the generator of a Feller semigroup ona bounded domain such that the respective conditions of Theorem 4.1 are satisfied. Toobtain the results for the generalised solution it is necessary to modify assumption (H2).Such stochastic representations have been obtained for example in the case of Pearsondiffusions ([27, Theorem 4.2]). Example 4.23.
In the standard Caputo case, i.e. − D ( ν ) a +∗ = − D βa +∗ , a =
0, the generalisedsolution u to problem (34) has the stochastic representation u ( t, x ) = ∫ R d φ ( y ) ( ∫ ∞ p As ( x, y ) tβ s − β − ω β ( ts − β ; 1 , ) ds ) dy + ∫ R d ∫ t g ( z, y ) ( ∫ ∞ ∫ ∞ ( s < r ) ϕ βt,s ( r, z ) p As ( x, y ) drds ) dzdy, (36)where ϕ βt,s ( r, z ) ∶ = ( s < r ) p β + s ( t, z ) ddr ∫ −∞ p β + r − s ( z, γ ) dγ = ( s < r ) s − β ω β (( t − z ) s − β ; 1 , ) zβ ( r − s ) − β − ω β ( z ( r − s ) − β ; 1 , ) is the joint density of ( τ β ( t ) , X t,β +∗ ( s )) (see [17, Proposition 4.2]), using the notation ofassumption (H2) and Remark 3.3. To obtain the last equality we used standard change ofvariables and identities for the stable densities ω β ( ⋅ ; ⋅ , ⋅ ) . In the homogeneous case ( g = Series representation.
Theorem 4.24.
Let ν be a function satisfying conditions (H0), (H1b). Let A be a boundedlinear operator on C ∞ ( R d ) .(i) If g ∈ C ∞ ([ a, b ] × R d ) , φ a ∈ Dom ( A ) , Aφ a ( ⋅ ) = − g ( a, ⋅ ) , then the unique solution u ∈ C ∞ ([ a, b ] × R d ) in the domain of the generator to problem (34) has the seriesrepresentation u ( t, x ) = ∞ ∑ n = A n φ a I ( ν ) ,na + ( t, x ) + ∞ ∑ n = ( I ( ν ) a + A ) n I ( ν ) a + g ( t, x ) . (37) (ii) If g ∈ B ([ a, b ] × R d ) , φ a ∈ Dom ( A ) , condition (H2) holds, then the unique gen-eralised solution u ∈ B ([ a, b ] × R d ) to problem (34) has the series representation(37). Proof. (i) Let u ∈ C ∞ ([ a, b ] × R d ) be the solution in the domain of the generator to problem(34). By Proposition 4.3, ˜ u ∶ = u − φ a ∈ Dom ( L ) ⊂ C a, ∞ ([ a, b ] × R d ) solves − D ( ν ) a +∗ ˜ u ( t, x ) = − A ˜ u ( t, x ) − ( g ( t, x ) + Aφ a ( x )) , ˜ u ( a, ⋅ ) = . (38) ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 21
By the assumptions of the Theorem ˜ g ∶ = g + Aφ a ∈ C a, ∞ ([ a, b ] × R d ) . Thereforeby Theorem 4.15-(i) ˜ u is the unique solution in the domain of the generator toproblem (38) and it has the series representation˜ u ( t, x ) = ∞ ∑ n = ( I ( ν ) a + A ) n I ( ν ) a + ˜ g ( t, x ) = ∞ ∑ n = ( I ( ν ) a + A ) n I ( ν ) a + g ( t, x ) + ∞ ∑ n = ( I ( ν ) a + A ) n I ( ν ) a + Aφ a ( t, x ) . (39)using the fact that both series in the right-hand side converge in C ∞ ([ a, b ] × R d ) by Theorem 3.4. Then u = ˜ u + φ a has the series representation given in (37).(ii) For g ∈ B ([ a, b ] × R d ) , let ˜ u be the unique generalised solution to problem (20) with˜ g = g + Aφ a . Then by Theorem 4.15-(ii) ˜ u has the representation (39), using thefact that both series in the right-hand side converge in B ([ a, b ] × R d ) by Theorem3.4. Then u = ˜ u + φ a has representation (37). (cid:3) Definition 4.25.
Let ν satisfy conditions (H0), (H1b) and let A be bounded. We call E ( ν ) ( A ( ⋅ ) I ( ν ) a + ) ∶ B ( R d ) → B ([ a, b ] × R d ) the generalised Mittag-Leffler function for A and ν , defined as φ a ↦ E ( ν ) ( Aφ a I ( ν ) a + )( t, x ) ∶ = ∞ ∑ n = A n φ a ( x ) I ( ν ) ,na + ( t ) , (40) ( t, x ) ∈ [ a, b ] × R d . Remark 4.26.
The function E ( ν ) ( A ( ⋅ ) I ( ν ) a + ) provides a probabilistic generalisation, for λ = A bounded operator, to the Mittag-Leffler function E β ( λ ( t − a ) β ) = ∞ ∑ n = λ n ( t − a ) βn Γ ( βn + ) = ∞ ∑ n = λ n φ a ( x ) I β,na + ( t ) , where β ∈ ( , ) , φ a ( ⋅ ) = Convergence of the series representation to the stochastic representation.
Theorem 4.27.
Let ν be a function satisfying (H0), (H1b), and assume that (H2) holds.Let A be the generator of a Feller semigroup on C ∞ ( R d ) and A λ its Yosida approximation.Fix g ∈ C ∞ ([ a, b ] × R d ) and φ a ∈ Dom ( A ) .Then for each t ≥ E ( ν ) ( Aφ a ( ν ) )( t, x ) → E φ a ( X x,A ( τ ( ν ) a ( t ))) , and ∞ ∑ n = ( I ( ν ) a + A λ ) n I ( ν ) a + g ( t, x ) → E ∫ τ ( ν ) a ( t ) e As g ( X t, ( ν ) + ( s ) , ⋅ )( x ) ds as λ → ∞ , uniformly in x ∈ R d . Proof.
Let u λ ∈ B ([ a, b ] × R d ) be the generalised solution for problem (34) for A = A λ .Let u ∈ B ([ a, b ] × R d ) be the generalised solution for problem (34) for A = A .By Theorem 4.20 u λ ( t, x ) = E φ a ( X x,A λ ( τ ( ν ) a ( t ))) + E ∫ τ ( ν ) a ( t ) e A λ s g ( X t, ( ν ) + ( s ) , ⋅ )( x ) ds, (41) ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 22 and u ( t, x ) = E φ a ( X x,A ( τ ( ν ) a ( t ))) + E ∫ τ ( ν ) a ( t ) e As g ( X t, ( ν ) + ( s ) , ⋅ )( x ) ds. (42)As a consequence of Theorem 4.15 and Theorem 4.10 the second term in (41) equals theseries representation (29) and by Theorem 4.16 it converges as required to the secondterm in (42).The considerations above along with Theorem 4.24 imply that the first term in (41) equalsthe first term on the right-hand side of (37). For the first term in (41) observe that by[13, Chapter 1, Proposition 2.7] e A λ s φ a ( x ) → e As φ a ( x ) , λ → ∞ , uniformly in x ∈ R d , for each s ≥
0. For each λ ≥ E φ a ( X x,A λ ( τ ( ν ) a ( t ))) = ∫ ∞ e A λ s φ a ( x ) µ τ ( ν ) a ( t ) ( ds ) , by independence of X x,A λ and τ ( ν ) a ( t ) (Corollary 4.6), where µ τ ( ν ) a ( t ) ( ds ) is the law of τ ( ν ) a ( t ) . Also ∣ e A λ s φ a ( x )∣ ≤ ∥ φ a ∥ ∀ λ > , and ∫ ∞ ∥ φ a ∥ µ τ ( ν ) a ( t ) ( ds ) ≤ ∥ φ a ∥ , and the result follows from the application of DCT. (cid:3) Remark 4.28.
Theorem 4.27 allows us to give meaning to a generalised Mittag-Lefflerfunction for A generator of a Feller semigroup on C ∞ ( R d ) .5. Generalised fractional evolution equation: Non-linear case
Let us now study the well-posedness for the non-linear equation given in (2). We introducea notion of solution and then we proceed as in [18] via fixed point arguments.
Definition 5.1.
Let ν be a function satisfying (H0), (H1b). A function u ∶ [ a, b ] × R d → R is said to be a generalised solution to the non-linear equation (2) if u is a generalisedsolution to the linear equation (1) with g ( t, x ) ∶ = f ( t, x, u ( t, x )) for all ( t, x ) ∈ [ a, b ] × R d . Lemma 5.2.
Let ν be a function satisfying conditions (H0), (H1b). Assume that A isthe generator of a Feller semigroup on C ∞ ( R d ) and φ a ∈ Dom ( A ) and that (H2) holds.Suppose that f ∶ [ a, b ] × R d × R → R is a bounded measurable function. Then, a function u ∈ C ([ a, b ] × R d ) is a generalised solution to equation (2) if, and only if, u solves thenon-linear integral equation u ( t, x ) = ∫ ∞ ( e As φ a )( x ) µ τ ( ν ) a ( t ) ( ds ) + E ∫ τ ( ν ) a ( t ) e As f ( X t, ( ν ) + ( s ) , ⋅ , u ( X t, ( ν ) + , ⋅ ))( x ) ds, (43)where µ τ ( ν ) a ( t ) is the law of τ ( ν ) a ( t ) . Proof.
By Definition 5.1, u ∈ C ([ a, b ] × R d ) is a generalised solution to (2) if and onlyif u is a generalised solution to the the linear equation (1) with g ( t, x ) ∶ = f ( t, x, u ( t, x )) .Note that if u ∈ C ([ a, b ] × R d ) , then g is a measurable and bounded function on [ a, b ] × R d .Hence Theorem 4.20-(ii) yields the integral equation (43), as required. (cid:3) Using Weissenger’s fixed point theorem we prove that the integral equation (43) possesses aunique solution (for a given boundary φ a ) under the following additional assumption: ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 23 (H3):
The function f ∶ [ a, b ] × R d × R → R is bounded and fulfils thefollowing Lipschitz condition with respect to the third variable: for all ( t, x, y ) , ( t, x, y ) ∈ [ a, b ] × R d × R , ∣ f ( t, x, y ) − f ( t, x, y )∣ < L f ∣ y − y ∣ , (44)for a constant L f > t and x ). Theorem 5.3.
Let [ a, b ] ⊂ R and φ a ∈ Dom ( A ) . Suppose that ν is a function satisfyingconditions (H0), (H1b). Suppose that (H2) holds and that f is a function satisfyingcondition (H3). Then problem (2) has a unique generalised solution u ∈ C ([ a, b ] × R d ) . Proof.
By Lemma 5.2, the existence of a unique generalised solution to (2) means theexistence of a unique solution to the integral equation (43). The latter equation can berewritten as a fixed point problem u ( t, x ) = ( Ψ u )( t, x ) for a suitable operator Ψ. Step a)
Definition of the operator Ψ. Denote by B φ a the closed convex subset of C ([ a, b ] × R d ) consisting of functions satisfying f ( a ) = φ a . This set is a metric space when endowed withthe metric induced by the norm on C ([ a, b ] × R d ) .Next, define the operator Ψ on B φ a by ( Ψ u )( t, x ) ∶ = ∫ ∞ ( e As φ a )( x ) µ τ ( ν ) a ( t ) ( ds ) + E ∫ τ ( ν ) a ( t ) e As f ( X t, ( ν ) + , ⋅ , u ( X t, ( ν ) + , ⋅ ))( x ) ds, t ∈ [ a, b ] . (45)Note that if u ∈ B φ a , then ( Ψ u )( ⋅ , x ) ∈ C [ a, b ] for each x ∈ R d and ( Ψ u )( t, ⋅ ) ∈ C ( R d ) for each t ∈ [ a, b ] . Further, ( Ψ u )( a, x ) = φ a ( x ) as µ τ ( ν ) a ( a ) ( ds ) = δ ( ds ) . Therefore,Ψ ∶ B φ a → B φ a . Step b)
Let Ψ n denote the n-fold iteration of the operator Ψ for n ≥ n ∈ N . Forconvention Ψ denotes the identity operator. Note that for n =
1, the Lipschitz conditionof f and the fact that e As is a contraction semigroup imply ∣ Ψ u − Ψ v ∣( t, x ) = ∣ E ∫ τ ( ν ) a ( t ) e As ( f ( X t, ( ν ) + , ⋅ , u ( X t, ( ν ) + , ⋅ )) − f ( X t, ( ν ) + , ⋅ , v ( X t, ( ν ) + , ⋅ ))( x ) ds ∣ ≤ E ∫ τ ( ν ) a ( t ) e As (∣ f ( X t, ( ν ) + , ⋅ , u ( X t, ( ν ) + , ⋅ )) − f ( X t, ( ν ) + , ⋅ , v ( X t, ( ν ) + , ⋅ )∣)( x ) ds ≤ L f ∥ u − v ∥ t I ( ν ) a + ( t ) , where ∥ u − v ∥ t ∶ = sup z ≤ t ∥ u ( z, ⋅ ) − v ( z, ⋅ )∥ , t ∈ [ a, b ] , and L f is the Lipschitz constant of the function f . Proceeding by induction we can provethat ∣ Ψ n u ( t, x ) − Ψ n v ( t, x )∣ ≤ ∥ u − v ∥ t L nf ( I ( ν ) ,na + ) ( t ) , n ≥ , where I ( ν ) ,na + is the n th fold iteration of the generalised fractional operator I ( ν ) a + . Moreover,by Theorem 3.4, we know that ∞ ∑ n = L nf ( I ( ν ) ,na + ) ( t ) ≤ ( L nf ( b − a ) β β Γ ( β + ) ) n n ! = ∶ α n . ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 24
Hence, ∥ Ψ n u − Ψ n v ∥ ≤ α n ∥ u − v ∥ , for every n ≥ u, v ∈ B φ a , where α n ≥ ∑ ∞ n = α n converges.Therefore, the Weissinger fixed point theorem [9, Theorem D.7] guarantees the existenceof a unique fixed point u ∗ ∈ B φ a to the integral equation (43), which in turn implies theexistence of a generalised solution to (2), as required. (cid:3) Appendix
Proof of Theorem 4.1. (i). It is easy to show that D ˜ D ⊂ C ∞ ( X × ˜ X ) , Φ t ∶ = T t ˜ T t isa well-defined continuous linear operator on C ∞ ( X × ˜ X ) , { Φ t } t ≥ is a uniformly boundedsemigroup. That T t ˜ T t = ˜ T t T t follows from Riesz-Markov representation Theorem ([24,Theorem 1.7.4]).That L is a dense subspace of C ∞ ( X × ˜ X ) follows from Stone-Weierstrass Theorem (forlocally compact spaces, see [40, 44A.I]) by taking as a sub-algebra the linear span of C ∞∞ ( X ) C ∞∞ ( ˜ X ) ⊂ L where C ∞ ( X ) C ∞ ( ˜ X ) ∶ = { f ∶ f = g ˜ g, C ∞∞ ( X ) , ˜ g ∈ C ∞∞ ( ˜ X )} , as it separates points and it does not vanish on X × ˜ X .Let f = ∑ Nn = λ n g n ˜ g n ∈ L . Then ∥ Φ t f − f ∥ ≤ N ∑ n = ∣ λ n ∣∥ T t g n ˜ T t ˜ g n − g ˜ g ∥ ≤ N ∑ n = ∣ λ n ∣(∥ T t g n ˜ T t ˜ g n − T g n ˜ g ∥ + ∥ T t g n ˜ g n − g ˜ g ∥) ≤ N ∑ n = ∣ λ n ∣(∥ T t ∥∥ g n ∥∥ ˜ T t ˜ g n − ˜ g ∥ + ∥ ˜ g n ∥∥ T t g n − g ∥) , which can be made arbitrarily small by choice of t small, using the strong continuity andthe uniform boundedness of ( T t ) and ( ˜ T t ) .As L is dense in C ∞ ( X × ˜ X ) , it follows that Φ t strongly continuous on C ∞ ( X × ˜ X ) .The semigroup { Φ t } t ≥ is invariant on L as T in invariant on D and ˜ T is invariant on ˜ D and Φ t f = N ∑ n = λ n T t g n ˜ T t ˜ g n , f ∈ L . We now show that L belongs to the domain of the generator of { Φ t } t ≥ .It is enough to show that D ˜ D belongs to the domain of the generator of Φ t as the domainof a generator is closed under linear combinations. ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 25
To do so we show that t − ( Φ t g ˜ g − g ˜ g ) converges to ˜ gAg + g ˜ A ˜ g as t →
0. Compute ∣ t − ( Φ t g ˜ g − g ˜ g ) − ˜ gAg − g ˜ A ˜ g ∣ ≤ ∣ t − ( T t g ˜ T t ˜ g − g ˜ g ) ± t − T t g ˜ g ± T t g ˜ A ˜ g − ˜ gAg − g ˜ A ˜ g ∣ ≤ ∣ t − ( T t g ˜ T t ˜ g − T t g ˜ g ) − T t g ˜ A ˜ g ∣ + ∣ t − ( T t g ˜ T t ˜ g − g ˜ g ) + t − T t g ˜ g + T t g ˜ A ˜ g − ˜ gAg − g ˜ A ˜ g ∣ ≤ ∥ T t g ∥ X ∥ t − ( ˜ T t ˜ g − ˜ g ) − ˜ A ˜ g ∥ ˜ X + ∣ − t − g ˜ g + t − T t g ˜ g + T t g ˜ A ˜ g − ˜ gAg − g ˜ A ˜ g ∣ ≤ ∥ g ∥ X ∥ t − ( ˜ T t ˜ g − ˜ g ) − ˜ A ˜ g ∥ ˜ X + ∥ ˜ g ∥ ˜ X ∥ t − ( T t g − g + ) − Ag ∥ X + ∥ ˜ A ˜ g ∥ ˜ X ∥ T t g − g ∥ X , which can be made arbitrarily small independently of ( x, ˜ x ) ∈ X × ˜ X by choosing t smallby strong continuity and the uniform boundedness of ( T t ) and ( ˜ T t ) (here the notation ∥ h ∥ Y means the supremum norm of the function h ∶ Y → R ).Therefore we have shown that L is a dense invariant subspace of Dom ( L ) and by [24,Proposition 1.9.1] L is a core for the generator of Φ t , and L = A + ˜ A on L .(ii). That the semigroup { Φ t } t ≥ is a Feller semigroup if { T t } t ≥ and { ˜ T t } t ≥ are Fellersemigroups follows easily. The same for the sub-Feller case.(iii). The case of C a, ∞ ([ a, b ] × ˜ X ) has the same proof as above apart from the statementabout the density of L . Briefly, to obtain the density of the respective set L , considerthe linear span of the product of smooth functions in C ∞ (( −∞ , b ] × ˜ X ) , apply Stone-Weierstrass as above, then use an isometric isomorphism between C a, ∞ (( a, b ] × ˜ X ) and C a, ∞ (( −∞ , b ] × ˜ X ) .6.2. Proof of Proposition 2.7-(i).
Fix h > bounded generators − D ( ν ) ,h + , − D ( ν ) ,ha + and − D ( ν ) ,ha +∗ defined as − D ( ν ) ,h + f ( x ) = ∫ ∞ h ( f ( x − y ) − f ( x )) ν ( x, y ) dy, − D ( ν ) ,ha + f ( x ) = ∫ max {( x − a ) ,h } h ( f ( x − y ) − f ( x )) ν ( x, y ) dy − f ( x ) ∫ ∞ max {( x − a ) ,h } ν ( x, y ) dy, − D ( ν ) ,ha +∗ f ( x ) = ∫ max {( x − a ) ,h } h ( f ( x − y ) − f ( x )) ν ( x, y ) dy + ( f ( a ) − f ( x )) ∫ ∞ max {( x − a ) ,h } ν ( x, y ) dy, acting on the spaces C ∞ (( −∞ , b ]) , C a ([ a, b ]) , C ([ a, b ]) , respectively. Then T ( ν ) + ,hs = ∞ ∑ n = t n n ! ( − D ( ν ) ,h + ) n , T ( ν ) a + ,hs = ∞ ∑ n = t n n ! ( − D ( ν ) ,ha + ) n , T ( ν ) a +∗ ,hs = ∞ ∑ n = t n n ! ( − D ( ν ) ,ha +∗ ) n , (46) s ∈ R + , are the respective semigroups. We first prove the second part of Proposition 2.7-(i).The key observation is that − D ( ν ) ,h + f ( t ) = − D ( ν ) ,ha + f ( t ) = − D ( ν ) ,ha +∗ f ( t ) , t > a, ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 26 if f ∈ { f ( x ) = ∀ x ≤ a } ∩ C ∞ (( −∞ , b ]) , and − D ( ν ) ,h + f ∈ { f ( x ) = ∀ x ≤ a } ∩ C ∞ (( −∞ , b ]) , − D ( ν ) ,ha + f ∈ { f ( x ) = ∀ x ≤ a } ∩ C a ([ a, b ]) , − D ( ν ) ,ha +∗ f ∈ { f ( x ) = ∀ x ≤ a } ∩ C a ([ a, b ]) . Hence for every n ∈ N , ( − D ( ν ) ,h + ) n f ( t ) = ( − D ( ν ) ,ha + ) n f ( t ) = ( − D ( ν ) ,ha +∗ ) n f ( t ) , t > a, (47)if f ∈ { f ( x ) = ∀ x ≤ a } .The identities in (47) imply that T ( ν ) + ,hs f ( t ) = T ( ν ) a + ,hs f ( t ) = T ( ν ) a +∗ ,hs f ( t ) , t > a, s ∈ R + , (48)if f ∈ { f ( x ) = ∀ x ≤ a } , as each of the semigroups is given by the exponentiation formulain (46). By the proofs of [24, Theorem 5.1.1] and [25, Theorem 4.1] T ( ν ) + ,hs f ( t ) → T ( ν ) + s f ( t ) , h → , (49)for each s ≥ t ∈ ( a, b ] , f ∈ C ∞ (( −∞ , b ]) , and T ( ν ) a + ,hs f ( t ) → T ( ν ) a + s f ( t ) , T ( ν ) a +∗ ,hs f ( t ) → T ( ν ) a +∗ s f ( t ) , h → , (50)for each s ≥ t ∈ ( a, b ] , f ∈ C ([ a, b ]) ∩ C a ([ a, b ]) .Hence, If f ∈ C (( −∞ , b ]) ∩ { f ( x ) = ∀ x ≤ a, f ′ ( a ) = } , then (48) holds and we also havethe convergence in (49) and (50).Now approximate point-wise from below the indicator function of any interval in ( a, b ] with functions in C ([ a, b ]) ∩ { f ( x ) = ∀ x ≤ a, f ′ ( a ) = } to obtain the second part ofProposition 2.7-(i).The first part of Proposition 2.7-(i) follows similarly after observing that T ( ν ) + ,hs f y ( t ) = T ( ν ) a + ,hs f y ( t ) = T ( ν ) a +∗ ,hs f y ( t ) = ∀ a < t ≤ y for any f ∈ { f ( x ) = ∀ x ≤ y } ∩ C ∞ (( −∞ , b ]) . In the last step we approximate the indicatorfunction ( ⋅ > y ) with functions in C (( −∞ , b ]) ∩ { f ( x ) = ∀ x ≤ y } , y > a to obtain thatfor every s ∈ R + = T ( ν ) + s ( ⋅ > y )( t ) = P [ X t, ( ν ) + ( s ) > y ] = P [ X t, ( ν ) a + ( s ) > y ] = P [ X t, ( ν ) a +∗ ( s ) > y ] , if t ≤ y . References [1] Anh, V. V., Leonenko, N. N. (2001),
Spectral analysis of fractional kinetic equations with randomdata . J. Statist. Phys. 104, no. 5-6, 1349-1387.[2] Baeumer, B., Kov´acs, M., Meerschaert, M. M., Schilling, R., Straka, P. (2016),
Reflected spectrallynegative stable processes and their governing equations . Transactions of the American MathematicalSociety, 368(1), 227-248.[3] Baeumer, B., Meerschaert, M. M. (2001),
Stochastic solutions for fractional Cauchy problems . Frac-tional Calculus and Applied Analysis 4.4: 481-500.[4] Baeumer, B., Kurita, S., Meerschaert, M. M. (2005),
Inhomogeneous fractional diffusion equations.
Fractional Calculus and Applied Analysis 8.4 (2005): 371-386.[5] Bazhlekova, E. (1998),
The abstract Cauchy problem for the fractional evolution equation.
FractionalCalculus and Applied Analysis 1.3: 255-270.[6] Bouchaud, J.P., Georges, A. (1990),
Anomalous diffusion in disordered media: statistical mechanism,models and physical applications , Physics Reports, , 127-293.
ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 27 [7] Carpinteri, A., Mainardi, F. (1997),
Fractals and Fractional Calculus in Continuum Mechanics ,CISM International Centre for Mechanical Sciences, Springer Verlag, Wien-New York.[8] Chakrabarty, A., Meerschaert, M. M. (2011),
Tempered stable laws as random walk limits . Statistics& Probability Letters, 81(8), 989-997.[9] Diethelm, K. (2010),
The Analysis of Fractional Differential Equations, An application-oriented ex-position using differential operators of Caputo Type , Lecture Notes in Mathematics, v. 2004, Springer.[10] Dynkin, E. B. (1965),
Markov processes , Vol. I, Springer-Verlag.[11] Edwards, J. T., Ford, N. J., Simpson, A. C. , (2001),
The Numerical Solutions of Linear Multi-termFractional Differential Equations: Systems of Equations , Journal of Computational and AppliedMathematics, , 401-418.[12] Eidelman, S. E., Kochubei, A. N. (2004),
Cauchy problem for fractional differential equations . Else-vier, Journal of differential equations, , pp. 211-255.[13] Ethier, S.N., Kurtz, T. G. (1986),
Markov processes. Characterization and Convergence .Wiley Seriesin Probability and Mathematical Statistics, New York Chicester, Wiley.[14] L. C. Evans,
Partial Differential Equations , Graduate Studies in mathematics, Vol 19, AmericanMathematical Society, 1997.[15] Gorenflo, R., Luchko, Y., Stojanovic, M. (2013),
Fundamental solution of a distributed order time-fractional diffusion-wave equation as probability density , Fractional Calculus and Applied Analysis,Volume , Number 2, pp. 297-316.[16] Gorenflo, R., Mainardi, F. (1998), Fractional calculus and stable probability distributions , Archive ofMechanics, (3), 377-388.[17] Hern´andez-Hern´andez, M.E., Kolokoltsov, V. N. (2015), On the probabilistic approach to the solutionof generalized fractional differential equations of Caputo and Riemann-Liouville type , Journal ofFractional Calculus and Applications, Vol. 7(1) Jan. 2016, pp. 147-175.[18] Hern´andez-Hern´andez, M.E., Kolokoltsov, V. N. (2015),
Probabilistic solutions to non-linear frac-tional differential equations of generalized Caputo and Riemann-Liouville type , submitted for publi-cation.[19] Kilbas, A. A., Srivastava, H. M., Trujillo, J. J. (2006),
Theory and Applications of Fractional Dif-ferential Equations , North-Holland Mathematics Studies, , Elsevier.[20] Klafter, J., I. M. Sokolov,
Anomalous Diffusion Spreads its Wings , Physics World , 29 August(2005).[21] Kochubei, A. N., (1990), Fractional-order diffusion , Differential Equations 26, 485-492.[22] Kochubei, A. N., Kondratiev, Y. (2017).
Fractional kinetic hierarchies and intermittency . Kinet.Relat. Models 10:3, 725 - 740.[23] Kolokoltsov, V. N. (2009),
Generalized continuous-time random walks (CTRW), Subordination byHitting times and fractional dynamics.
Theory Probab. Appl. Vol. 53, No. 4, pp. 549-609.[24] Kolokoltsov, V. N. (2011),
Markov processes, semigroups and generators.
DeGruyter Studies inMathematics, Book 38.[25] Kolokoltsov, V. N. (2015),
On fully mixed and multidimensional extensions of the Caputo andRiemann-Liouville derivatives, related Markov processes and fractional differential equations , Frac-tional Calculus and Applied Analysis, 18.4 (2015): 1039-1073.[26] Kolokoltsov, V. N., Veretennikova, M. (2014),
Well-posedness and regularity of the Cauchy problemfor non-linear fractional in time and space equations , Fractional Differential Calculus 4:1, 1-30.[27] Leonenko, N. N., Meerschaert, M. M., Sikorskii, A. (2013),
Fractional Pearson diffusions . J. Math.Anal. Appl. 403, no. 2, 532-546.[28] L¨orinczi, J´ozsef, Hiroshima, Fumio; Betz, Volker
Feynman-Kac-type theorems and Gibbs measures onpath space. With applications to rigorous quantum field theory . De Gruyter Studies in Mathematics,34. Walter de Gruyter & Co., Berlin, 2011. xii+505 pp.[29] Mainardi, F. (2001),
Fractional calculus: some basic problems in continuum and statistical mechanics ,http://arxiv.org/abs/1201.0863v1.[30] Mainardi, F. (2010),
Fractional Calculus and Waves in Linear Viscoelasticity. An introduction toMathematical Models , Imperial College Press.[31] Mainardi, F., Mura, A., Pagnini, G., Gorenflo, R. (2008),
Time-fractional diffusion of distributedorder , J. Vib. Control , pp. 1267-1290.[32] Meerschaert, M. M., Nane, E., Vellaisamy, P. (2009), Fractional Cauchy problems on bounded do-mains . The Annals of Probability, Vol. 37, No. 3.[33] Meerschaert, M.M., Sikorskii, A. (2012),
Stochastic Models for Fractional Calculus , De GruyterStudies in Mathematics, Book . ENERALISED FRACTIONAL EVOLUTION EQUATIONS OF CAPUTO TYPE 28 [34] Nonnenmacher, T. F. (1990),
Fractional integral and differential equations for a class of Levy-typeprobability densities , J. Phys. A: Math. Gen. .[35] E. Nane (2010), Fractional Cauchy problems on bounded domains: survey of recent results , FractionalDynamics and Control (2011): 185.[36] Podlubny, I. (1999),
Fractional differential equations. An introduction to fractional derivatives, frac-tional differential equations, to methods of their solution and some of their applications.
Mathematicsin Science and Engineering, v. 198. Academic Press, Inc., San Diego.[37] Samko, S. G., Kilbas, A. A., Marichev, O. I. (1993),
Fractional integrals and derivatives: theory andapplications , Gordon and Breach Science Publishers S. A.[38] Scalas, E. (2012),
A class of CTRW’s: compound fractional Poisson processes. In: Fractional dy-namics, World Sci. Publ., Hackensack, NJ, pp. 353-374 .[39] W. R. Schneider and W. Wyss,
Fractional diffusion and wave equations , J. Math. Phys. 30 (1989),134-144.[40] Willard, S.,
General Topology , Addison-Wesley Series in Mathematics, (1970).[41] Wy loma´nska, A. (2013),
The tempered stable process with infinitely divisible inverse subordinators .Journal of Statistical Mechanics: Theory and Experiment, 2013(10), P10011.[42] Zhang, Y., (2000),
Sufficient and necessary conditions for stochastic comparability of jump processes.
Acta Mathematica Sinica 16.1 : 99-102.[43] Zaslavsky, G. M. (2002)
Chaos, fractional kinetics, and anomalous transport , Physics Reports, ,pp. 461-580.[44] Zolotarev, V. M. (1986)
One-dimensional stable distributions . Translations of Mathematical mono-graphs, vol. 65, American Mathematical Society, 1986.
M. E. Hern´andez-Hern´andezDepartment of Statistics, University of Warwick, Coventry, United Kingdom.
E-mail address : [email protected]
V. N. KolokoltsovDepartment of Statistics, University of Warwick, Coventry, United Kingdom,And Associate member of Institute of Informatics Problems, FRC CSC RAS.
E-mail address : [email protected]
L. ToniazziDepartment of Mathematics, University of Warwick, Coventry, United Kingdom.
E-mail address ::