aa r X i v : . [ m a t h . P R ] F e b Cesaro Limits for Fractional Dynamics
Yuri Kondratiev
Department of Mathematics, University of Bielefeld,D-33615 Bielefeld, Germany,Dragomanov University, Kiev, UkraineEmail: [email protected]
José Luís da Silva ,CIMA, University of Madeira, Campus da Penteada,9020-105 Funchal, Portugal.Email: joses@staff.uma.ptMarch 1, 2021
Abstract
We study the asymptotic behavior of random time changes of dy-namical systems. As random time changes we propose three classeswhich exhibits different patterns of asymptotic decays. The subordi-nation principle may be applied to study the asymptotic behavior ofthe random time dynamical systems. It turns out that for the specialcase of stable subordinators explicit expressions for the subordinationare known and its asymptotic behavior are derived. For more gen-eral classes of random time changes explicit calculations are essentiallymore complicated and we reduce our study to the asymptotic behaviorof the corresponding Cesaro limit.
Keywords : Dynamical systems; random time change; inverse sub-ordinator; asymptotic behavior.
AMS Subject Classification 2010 : 37A50, 45M05, 35R11, 60G52. ontents In this paper we will deal with Markov processes or dynamical systems in R d .These processes or dynamics starting from x ∈ R d , denote by X x ( t ) , t ≥ ,have associated evolution equations on R d . In the Markov case we define forsuitable f : R d −→ R the function u ( t, x ) = E [ f ( X x ( t ))] which satisfied theKolmogorov equation ∂∂t u ( t, x ) = Lu ( t, x ) , where L is the generator of the Markov process.For a dynamical system we introduce u ( t, x ) = f ( X x ( t )) . Then thisfunction is the solution of the Liouville equation ∂∂t u ( t, x ) = Lu ( t, x ) , where now L is the Liouville operator for the dynamical system, see e.g.[KdS20].Let S ( t ) , t ≥ be a subordinator and E ( t ) , t ≥ denotes the inversesubordinator, that is, for each t ≥ , E ( t ) := inf { s > | S ( s ) > t } . This2andom process we consider as a random time and assume to be independentof X x ( t ) . Define a random process Y x by Y x ( t, ω ) := X x ( E ( t, ω )) . Then as above we may introduce u E ( t, x ) = E [ f ( Y x ( t ))] . For both Markov and dynamical system cases this function satisfies the evo-lution equations D Et u E ( t, x ) = Lu E ( t, x ) where L is the Kolmogorov or Liouville operator correspondingly. Here D Et is a generalized fractional time derivative corresponding to the inverse sub-ordinator E ( t ) , see Section 2 below for details, in particular the definitionin (2.14). The main relation which is true for both cases is the followingsubordination formula: u E ( t, x ) = Z ∞ u ( τ, x ) G t ( τ ) dτ, (1.1)where G t ( τ ) is the density of the inverse subordinator E ( t ) , see, e.g., [Toa15],[KdS20] and especially the book [MS12]. This formula which relates thesolutions of the evolution equations with usual and fractional derivativesplays an important role in the study of dynamics with random times. Notethat there exist such relations between random times, fractional equationsand subordination in the framework of physical models, see, e.g., [MTM08].The goal of this paper is to study and analyze the asymptotic behaviorof two elementary dynamical system after the random time change, namely u ( t, x ) = e − at , a > and u ( t, x ) = t n , n ≥ . Here the dynamical systemare considered as a deterministic Markov processes. For particular classes ofrandom times the subordination formula (1.1) is evaluated explicitly. This istrue, for example, in the case of inverse stable subordinators. For a generalinverse subordinator the properties of the density G t ( τ ) are unknown andthe evaluation of (1.1) is not possible. Actually, it is a long standing openproblem in the theory of stochastic processes.We propose an alternative approach to study the asymptotic behavior of u E ( t, x ) . More precisely, we consider Cesaro limits (the asymptotic of theCesaro mean of u E ( t, x ) , see (3.2) below) of u E ( t, x ) using the subordination3ormula representation (1.1) together with the Fe< ller–Karamata Tauberiantheorem, see Theorem 2.7. For many classes of random times this approachleads to a precise asymptotic behavior. In this paper we investigate threeclasses of random time change, denote by (C1), (C2), and (C3), see Section 2,which exhibits different patterns of decays of the Cesaro limit of u E ( t, x ) .We would like to emphasize that for particular classes of random times,namely inverse stable subordinators, the asymptotic of u E ( t, x ) which maybe computed explicitly, coincides with the Cesaro limit. For other classesof random times the Cesaro limit gives one possible characteristic of theasymptotic for u E ( t, x ) . To the best of our knowledge at the present timeno other information on the asymptotic of u E ( t, x ) is known for a generalsubordinator.The remaining of the paper is organized as follows. In Section 2 we in-troduce three classes ((C1), (C2), and (C3)) of subordinator processes whichserves as random times. These classes are given in terms of their local behav-ior of the Laplace exponent at λ = 0 . In addition, we state the main resultsof the paper. Section 3 is a preparation for the more general study of theasymptotic of the subordination in Section 4. More precisely, we investigatein detail the special case of the inverse stable subordinator where explicitexpressions are known. Hence, the expression for the subordination (1.1) isderived (for the two dynamical systems considered above) as well as theirCesaro limit. It turns out that both asymptotic for u E ( t, x ) (the explicitcalculations and Cesaro limit) are the same. Finally in Section 4 we studythe Cesaro limit for the general classes (C1), (C2), and (C3) of random timechanges. In this section we introduce three classes of subordinators which serves asrandom times processes. More precisely, the random times corresponds tothe inverse of subordinator processes whose Laplace exponent satisfies certainconditions, see below for details. The simplest example in class (C1) below,is the well known α -stable subordinators whose inverse processes are wellstudied in the literature, see for example [Bin71] or [Fel71].The classes of processes to be introduced which serve as random timeshave a connection with the concept of general fractional derivatives (see[Koc11] for details and applications to fractional differential equations) asso-4iated to an admissible kernels k ∈ L ( R + ) which is characterized in termsof their Laplace transforms K ( λ ) as λ → , see assumption (H) below. Let S = { S ( t ) , t ≥ } be a subordinator without drift starting at zero, thatis, an increasing Lévy process starting at zero, see [Ber96] for more details.The Laplace transform of S ( t ) , t ≥ is expressed in terms of a Bernsteinfunction Φ : [0 , ∞ ) −→ [0 , ∞ ) (also known as Laplace exponent) by E ( e − λS ( t ) ) = e − t Φ( λ ) , λ ≥ . The function Φ admits the Lévy-Khintchine representation Φ( λ ) = Z (0 , ∞ ) (1 − e − λτ ) d σ ( τ ) , (2.1)where the measure σ (called Lévy measure) has support in [0 , ∞ ) and fulfills Z (0 , ∞ ) (1 ∧ τ ) d σ ( τ ) < ∞ . (2.2)In what follows we assume that the Lévy measure σ satisfy σ (cid:0) (0 , ∞ ) (cid:1) = ∞ . (2.3)Using the Lévy measure σ we define the kernel k as follows k : (0 , ∞ ) −→ (0 , ∞ ) , t k ( t ) := σ (cid:0) ( t, ∞ ) (cid:1) . (2.4)Its Laplace transform is denoted by K , that is, for any λ ≥ one has K ( λ ) := Z ∞ e − λt k ( t ) d t. (2.5)The relation between the function K and the Laplace exponent Φ is given by Φ( λ ) = λ K ( λ ) , ∀ λ ≥ . (2.6)We make the following assumption on the Laplace exponent Φ( λ ) of thesubordinator S . 5 H) Φ is a complete Bernstein function (that is, the Lévy measure σ isabsolutely continuous with respect to the Lebesgue measure) and thefunctions K , Φ satisfy K ( λ ) → ∞ , as λ → K ( λ ) → , as λ → ∞ ; (2.7) Φ( λ ) → , as λ →
0; Φ( λ ) → ∞ , as λ → ∞ . (2.8) Example 2.1 ( α -stable subordinator) . A classical example of a subordinator S is the so-called α -stable process with index α ∈ (0 , . Specifically, asubordinator is α -stable if its Laplace exponent is Φ( λ ) = λ α = Z ∞ (1 − e − λτ ) ατ − − α Γ(1 − α ) d τ. In this case it follows that the Lévy measure is d σ α ( τ ) = α Γ(1 − α ) τ − (1+ α ) d τ .The corresponding kernel k α has the form k α ( t ) = g − α ( t ) := t − α Γ(1 − α ) , t ≥ and its Laplace transform is K α ( λ ) = λ α − , λ > . Example 2.2 (Sum of two stable subordinators) . Let < α < β < begiven and S α,β ( t ) , t ≥ the driftless subordinator with Laplace exponentgiven by Φ α,β ( λ ) = λ α + λ β . It is clear from Example 2.1 that the corresponding Lévy measure σ α,β is thesum of two Lévy measures, that is, d σ α,β ( τ ) = d σ α ( τ ) + d σ β ( τ ) = α Γ(1 − α ) τ − (1+ α ) d τ + β Γ(1 − β ) τ − (1+ β ) d τ. Then the associated kernel k α,β is k α,β ( t ) := g − α ( t ) + g − β ( t ) = t − α Γ(1 − α ) + t − β Γ(1 − β ) , t > and its Laplace transform is K α,β ( λ ) = K α ( λ ) + K β ( λ ) = λ α − + λ β − , λ > .Let E be the inverse process of the subordinator S , that is, E ( t ) := inf { s > | S ( s ) > t } = sup { s ≥ | S ( s ) ≤ t } . (2.9)6or any t ≥ we denote by G t ( τ ) , τ ≥ the marginal density of E ( t ) or,equivalently G t ( τ ) d τ = ∂∂τ P ( E ( t ) ≤ τ ) d τ = ∂∂τ P ( S ( τ ) ≥ t ) d τ = − ∂∂τ P ( S ( τ ) < t ) d τ. The density G t ( τ ) is the main object in our considerations below. There-fore, in what follows, we collect the most important properties of G t ( τ ) needed in the next sections. Remark . If S is the α -stable process, α ∈ (0 , , then the inverse process E ( t ) , has Laplace transform (cf. Prop. 1(a) in [Bin71] or [Fel71]) given by E ( e − λE ( t ) ) = Z ∞ e − λτ G t ( τ ) d τ = ∞ X n =0 ( − λt α ) n Γ( nα + 1) = E α ( − λt α ) , (2.10)where E α is the Mittag-Leffler function. It follows from the asymptotic be-havior of the function E α that E ( e − λE ( t ) ) ∼ Ct − α as t → ∞ . It is possibleto find explicitly the density G t ( τ ) in this case using the completely mono-tonic property of the Mittag-Leffler function E α . It is given in terms of theWright function W µ,ν , namely G t ( τ ) = t − α W − α, − α ( τ t − α ) , see [GLM99] formore details.For a general subordinator, the following lemma determines the t -Laplacetransform of G t ( τ ) , with k and K given in (2.4) and (2.5), respectively. Forthe proof see [Koc11] or Proposition 3.2 in [Toa15]. Lemma 2.4.
The t -Laplace transform of the density G t ( τ ) is given by Z ∞ e − λt G t ( τ ) d t = K ( λ ) e − τλ K ( λ ) . (2.11) The double ( τ, t )-Laplace transform of G t ( τ ) is Z ∞ Z ∞ e − pτ e − λt G t ( τ ) d t d τ = K ( λ ) λ K ( λ ) + p . (2.12)Here we would like to make the connection of the above abstract frame-work with general fractional derivatives. For any α ∈ (0 , the Caputo-Dzhrbashyan fractional derivative of order α of a function u is defined by(see e.g., [KST06] and references therein) (cid:0) D αt u (cid:1) ( t ) = ddt Z t k α ( t − τ ) u ( τ ) d τ − k α ( t ) u (0) , t > , (2.13)7here k α is given in Example 2.1, that is, k α ( t ) = g − α ( t ) = t − α Γ(1 − α ) , t > . Ingeneral, starting with a subordinator S and the kernel k ∈ L ( R + ) as givenin (2.4), we may define a differential-convolution operator by (cid:0) D ( k ) t u (cid:1) ( t ) = ddt Z t k ( t − τ ) u ( τ ) d τ − k ( t ) u (0) , t > . (2.14)The operator D ( k ) t is also known as general fractional derivative and its appli-cations to convolution-type differential equations was investigated in [Koc11]. Example 2.5 (Distributed order derivative) . Consider the kernel k definedby k ( t ) := Z g α ( t ) d α = Z t α − Γ( α ) d α, t > . (2.15)Then it is easy to see that K ( λ ) = Z ∞ e − λt k ( t ) d t = λ − λ log( λ ) , λ > . The corresponding differential-convolution operator D ( k ) t is called distributedorder derivative, see [APZ09, DGB08, Han07, Koc08, GU05, MS06] for moredetails and applications.We say that the functions f and g are asymptotically equivalent at infinity ,and denote f ( x ) ∼ g ( x ) as x → ∞ , meaning that lim x →∞ f ( x ) g ( x ) = 1 . We say that a function L is slowly varying at infinity (see [Fel71, Sen76]) if lim x →∞ L ( λx ) L ( x ) = 1 , for any λ > . Below C is constant whose value is unimportant and may change from lineto line.In the following we consider three classes of admissible kernels k ∈ L ( R + ) ,characterized in terms of their Laplace transforms K ( λ ) as λ → (i.e., aslocal conditions): K ( λ ) ∼ λ α − , < α < . (C1)8 ( λ ) ∼ λ − L (cid:18) λ (cid:19) , L ( y ) := C log( y ) − , C > . (C2) K ( λ ) ∼ λ − L (cid:18) λ (cid:19) , L ( y ) := C log( y ) − − s , s > , C > . (C3)We would like to emphasize that these three classes of kernels leads to dif-ferent type of differential-convolution operators. In particular, the Caputo-Djrbashian fractional derivative (C1) and distributed order derivatives (C2),(C3). Moreover, it is simple to check that the class of subordinators fromExample 2.2 falls into the class (C1) above. Remark . The asymptotic behavior of the function f ( t ) as t → ∞ may bedetermined, under certain conditions, by studying the behavior of its Laplacetransform ˜ f ( λ ) as λ → , and vice versa. An important situation wheresuch a correspondence holds is described by the Feller–Karamata Tauberian(FKT) theorem.We state below a version of the FKT theorem which suffices for ourpurposes, see the monographs [BGT87, Sec. 1.7] and [Fel71, XIII, Sec. 1.5]for a more general version and proofs. Theorem 2.7 (Feller–Karamata Tauberian) . Let U : [0 , ∞ ) −→ R be amonotone non-decreasing right-continuous function such that w ( λ ) := Z ∞ e − λt d U ( t ) < ∞ , ∀ λ > . If L is a slowly varying function and C, ρ ≥ , then the following are equiv-alent U ( t ) ∼ C Γ( ρ + 1) t ρ L ( t ) as t → ∞ , (2.16) w ( λ ) ∼ Cλ − ρ L (cid:18) λ (cid:19) as λ → + . (2.17) When C = 0 , (2.16) is to be interpreted as U ( t ) = o ( t ρ L ( t )) ; similarly for(2.17). In Section 3 and 4 we will focus our attention on deriving the asymptoticbehavior of the subordination u E ( t, x ) given in (1.1) for the inverse stable9ubordinator as well as for the classes (C1), (C2), and (C3) given above.On one hand, the results concerning the inverse stable subordinator as arandom time are well understood, due to the fact that the Laplace transform(in τ ) of the density G t ( τ ) is known (cf. Remark 2.3). On the other hand,for a general subordinator much less information about G t ( τ ) is known andexplicit results for the subordination u E ( t, x ) are not available. In order toget around this problem, and motivated by the results of Section 3, we studythe Cesaro limit of u E ( t, x ) for the general classes of random times.With the above considerations we are ready to state our main results. Theorem 2.8.
Let u E ( t, x ) be the subordination by the density G t ( τ ) associ-ated to the inverse stable subordinator. Denote by M t ( u E ( · , x )) := t R t u E ( s, x ) d s the Cesaro mean of u E ( t, x ) .1. If u ( t, x ) = t n , n ≥ , then the asymptotic behavior of u E ( t, x ) coincideswith the Cesaro limit and is equal to Ct nα as t → ∞ .
2. If u ( t, x ) = e − at , a > , then the asymptotic of u E ( t, x ) and its Cesarolimit are equal to Ct − α as t → ∞ . The proof of Theorem 2.8 is essentially the contents of Section 3 whilethe next theorem is shown in Section 4.
Theorem 2.9.
Let u E ( t, x ) be the subordination by the density G t ( τ ) associ-ated to the classes (C1), (C2), and (C3) and M t ( u E ( · , x )) := t R t u E ( s, x ) d s the Cesaro mean of u E ( t, x ) .1. Assume that u ( t, x ) = t n , n ≥ . Then the asymptotic of the Cesaromean for the three classes are: (C1). M t ( u E ( · , x )) ∼ Ct αn as t → ∞ , (C2). M t ( u E ( · , x )) ∼ C log( t ) − n as t → ∞ , (C3). M t ( u E ( · , x )) ∼ C log( t ) − (1+ s ) n as t → ∞ .2. If u ( t, x ) = e − at , a > , then the asymptotic of M t ( u E ( · , x )) for thedifferent classes are: C1). M t ( u E ( · , x )) ∼ Ct − α as t → ∞ , (C2). M t ( u E ( · , x )) ∼ C log( t ) − as t → ∞ , (C3). M t ( u E ( · , x )) ∼ C log( t ) − − s as t → ∞ . In this section we consider two elementary solutions of dynamical systems,namely u ( t ) = u ( t, x ) = t n , n ≥ and u ( t ) = u ( t, x ) = e − at , a > ,and investigate their subordination by the density G t ( τ ) of inverse stablesubordinator.Define the function u E ( t ) = u E ( t, x ) as the subordination of u ( t ) (of theabove type) by the kernel G t ( τ ) , that is, u E ( t ) := Z ∞ u ( τ ) G t ( τ ) d τ, t ≥ . (3.1)Our goal is to investigate the asymptotic behavior of u E ( t ) . At first we com-pute explicitly the function u E ( t ) by solving the integral (3.1) and obtain thetime asymptotic. Second we derive the Cesaro limit of u E ( t ) , more precisely,the asymptotic behavior of the Cesaro mean of u E ( t ) defined by M t ( u E ( · )) := 1 t Z t u E ( s ) d s. (3.2)It turns out that both asymptotic behaviors for the two functions u ( t ) givenabove coincide. Therefore, for the random time change associated to theinverse stable subordinator E ( t ) , t ≥ , the asymptotic behavior of u E ( t ) isthe same as the Cesaro limit. On the other hand, using the Cesaro limit wemay investigate a broad class of subordinators. In Section 4 we investigatethe Cesaro limit for the classes (C1), (C2), and (C3) while in this sectionconcentrate in the spacial case of inverse stable subordinators. Let us consider at first the subordination of the function u ( t ) = t n , n ≥ .Hence, u E ( t ) is given by u E ( t ) = Z ∞ τ n G t ( τ ) d τ. (3.3)11t follows from (2.10) that u E ( t ) is explicitly evaluated as u E ( t ) = ( − n d n d λ n E α ( − λt α ) (cid:12)(cid:12) λ =0 = n !Γ( αn + 1) t αn . The last equality follows easily from the power series of the Mittag-Lefflerfunction E α ( z ) = ∞ X n =1 z n Γ( αn + 1) . In addition, the asymptotic of the Mittag-Leffler function E α that gives u E ( t ) ∼ Ct nα as t → ∞ . (3.4)Now we turn to compute the asymptotic behavior of the Cesaro mean of u E ( t ) with the help of the FKT theorem. To this end we define the monotonefunction v ( t ) := Z t u E ( s ) d s. (3.5)The Laplace-Stieltjes transform of v ( t ) is given by ˜ v ( λ ) := Z ∞ e − λt d v ( t ) = Z ∞ e − λt u E ( t ) d t = Z ∞ e − λt Z ∞ τ n G t ( τ ) d τ d t. Using Fubini’s theorem and equation (2.11) we obtain ˜ v ( λ ) = Z ∞ τ n Z ∞ e − λt G t ( τ ) d t d τ = K ( λ ) Z ∞ τ n e − τλ K ( λ ) d τ. The r.h.s. integral can be evaluated as Z ∞ τ n e − τλ K ( λ ) d τ = ( λ K ( λ )) − (1+ n ) n ! which yields ˜ v ( λ ) = n ! λ − (1+ n ) K ( λ ) − n . (3.6)On the other hand, for the stable subordinator we have K ( λ ) = λ α − , cf.Example 2.1. Thus, we obtain ˜ v ( λ ) = n ! λ − (1+ αn ) = λ − ρ L (cid:18) λ (cid:19) , ρ = 1 + αn and L ( x ) = n ! is a trivial slowly varying function. ThenTheorem 2.7 yields v ( t ) ∼ Ct nα as t → ∞ and this implies the following asymptotic behavior for the Cesaro mean of u E ( t ) M t ( u E ( · )) = 1 t Z t u E ( s ) d s ∼ Ct αn as t → ∞ . (3.7) Remark . In conclusion, we find that the asymptotic behavior of the sub-ordination u E ( t ) of any monomial by the density G t ( τ ) (of the inverse stablesubordinator) as well as its Cesaro limit coincides. Note also the slower decayof the subordination u E ( t ) compared to u ( t ) due to < α < . Now we consider the solution u ( t ) = e − at , a > and proceed to study theasymptotic behavior of its subordination u E ( t ) by the kernel G t ( τ ) . Again adirect computation is possible in that case as well as the Cesaro mean.Hence, the subordination u E ( t ) is given by u E ( t, x ) = Z ∞ u ( τ ) G t ( τ ) d τ = Z ∞ e − aτ G t ( τ ) d τ. (3.8)It follows from equation (2.10) that u E ( t, x ) = E α ( − at α ) ∼ Ct − α as t → ∞ . (3.9)On the other hand, to derive the asymptotic behavior for the Cesaro meanof u E ( t ) (with the help of Theorem 2.7) we define the monotone function v ( t ) := Z t u E ( s ) d s. (3.10)The Laplace-Stieltjes transform of v ( t ) yields ˜ v ( λ ) := Z ∞ e − λt d v ( t ) = Z ∞ e − λt u E ( t, x ) d t = Z ∞ e − λt Z ∞ e − aτ G t ( τ ) d τ d t and using Fubini’s theorem and equation (2.10) we obtain ˜ v ( λ ) = K ( λ ) Z ∞ e − τ ( a + λ K ( λ )) d τ = K ( λ ) a + λ K ( λ ) . (3.11)13s K ( λ ) = λ α − for the class (C1) we may write ˜ v ( λ ) as ˜ v ( λ ) = λ − (1 − α ) a + λ α = λ − ρ L (cid:18) λ (cid:19) , ρ = 1 − α, L ( t ) := 1 a + t − α . It is simple to verify that L is a slowly varying function so that we may usethe FKT theorem to obtain v ( t ) ∼ Ct − α t − α a + t − α as t → ∞ . Dividing both sides by t leads to the asymptotic behavior of the Cesaro meanof u E ( t ) , that is, M t ( u E ( · )) = 1 t Z t u E ( s, x ) d s ∼ C t − α a + t − α ∼ Ct − α as t → ∞ . (3.12) Remark . We conclude that the asymptotic behavior u E ( t ) given in (3.9)coincides with the Cesaro limit of u E ( t, x ) . In addition, we notice that thestarting function u ( t ) = e − at has an exponential decay and its subordinationas a polynomial decay. In this section we study the asymptotic behavior of the subordination bythe density G t ( τ ) associated to the classes (C1), (C2), and (C3). Note thatExample 2.1 and 2.2 belong to the class (C1). As pointed out in Section 3here we only study the Cesaro limit of the subordination function u E ( t ) .As in Section 3, u E ( t ) is defined by u E ( t ) := Z ∞ τ n G t ( τ ) d τ (4.1)or u E ( t ) := Z ∞ e − aτ G t ( τ ) d τ (4.2)while v ( t ) is defined by v ( t ) := Z t u E ( s ) d s. The density G t ( τ ) in (4.1) and (4.2) is associated to each class (C1)–(C3)described above. We study the Cesaro limit of u E ( t ) for each class separately.14 .1 Subordination by the Class (C1) At first we study the asymptotic behavior of u E ( t ) given by (4.1). To thisend we use equality (3.6) to obtain the Laplace-Stieltjes transform of thefunction v ( t ) as ˜ v ( λ ) := Z ∞ e − λt d v ( t ) = λ − (1+ n ) ( K ( λ )) n . It follows from the behavior of K ( λ ) at λ = 0 of the class (C1) that ˜ v ( λ ) ∼ λ − (1+ αn ) n ! = λ − ρ L (cid:18) λ (cid:19) , where ρ = 1 + αn and L ( x ) = n ! is a slowly varying function. It follows fromthe FKT theorem that v ( t ) ∼ Ct ρ L ( t ) = Ct αn as t → ∞ . This implies the Cesaro limit of u E ( t ) as M t ( u E ( · )) ∼ Ct αn as t → ∞ . Note that this asymptotic is similar to the analogous for the inverse stablesubordinator, cf. (3.7).Let us now study the Cesaro limit of the function u E ( t ) given in (4.2).Using the equality (3.11) the Laplace-Stieltjes transform v ( t ) has the form ˜ v ( λ ) = K ( λ ) a + λ K ( λ ) . Replacing the local behavior of K ( λ ) at λ = 0 for the class (C1) gives ˜ v ( λ ) ∼ λ α − a + λ α = λ − ρ L (cid:18) λ (cid:19) , where ρ = 1 − α and L ( x ) = ax − α . An applications of the FKT theoremyields the asymptotic for v ( t ) , namely v ( t ) ∼ Ct ρ L ( t ) as t → ∞ . Finallydividing both sides by t gives the Cesaro limit of u E ( t ) , that is, M t ( u E ( · )) ∼ C t − α at − α ∼ Ct − α as t → ∞ . Again, we obtain the same asymptotic as for the inverse stable subordinator,see (3.12). In any case, since < α < , the time decaying is slower than theinitial function u ( t ) . 15 .2 Subordination by the Class (C2) Assume that u E ( t ) is the subordination given in (4.1). The Laplace-Stieltjestransform of v ( t ) (cf. equality (3.6)) has the form ˜ v ( λ ) := Z ∞ e − λt d v ( t ) = λ − (1+ n ) ( K ( λ )) − n . Using the behavior of K ( λ ) near λ = 0 for the class (C2) we obtain ˜ v ( λ ) ∼ λ − (cid:18) L (cid:18) λ (cid:19)(cid:19) n , where L ( x ) = C log( x ) − , C > , is a slowly varying function. Then it followsfrom the FKT theorem that v ( t ) ∼ Ct log( t ) − n and as a result the asymptotic behavior for the Cesaro mean of u E ( t ) follows M t ( u E ( · )) ∼ C log( t ) − n as t → ∞ . A similar analysis may be applied to study the asymptotic behavior forthe subordination u E ( t ) given in (4.2). The Laplace-Stieltjes transform ofthe monotone function v ( t ) may be evaluated using equality (3.11) to findthe following expression ˜ v ( λ ) = K ( λ ) a + λ K ( λ ) . Using the local behavior of K ( λ ) near λ = 0 from class (C2) yields ˜ v ( λ ) ∼ λ − L (cid:18) λ (cid:19) , where L ( x ) = C log( x ) − a + C log( x ) − which is a slowly varying function. Using theFKT theorem we obtain the longtime behavior for the Cesaro mean of u E ( t ) as M t ( u E ( · )) ∼ C log ( t ) − a + C log( t ) − ∼ C log( t ) − as t → ∞ . .3 Subordination by the Class (C3) At first we study the subordination u E ( t ) given in (4.1) for the class (C3).The Laplace-Stieltjes transform of the corresponding v ( t ) is computed usingequality (3.6) and we obtain ˜ v ( λ ) := Z ∞ e − λt d v ( t ) = λ − (1+ n ) ( K ( λ )) n . Using the behavior of K ( λ ) near λ = 0 for the class (C3) yields ˜ v ( λ ) ∼ λ − (cid:18) L (cid:18) λ (cid:19)(cid:19) n , where L ( x ) = C log( x ) − − s , C > , is a slowly varying function. Then itfollows from Theorem 2.7 that v ( t ) ∼ Ct log( t ) − (1+ s ) n and dividing both sides by t gives the asymptotic behavior for the Cesaromean of u E ( t ) , namely M t ( u E ( · )) ∼ C log( t ) − (1+ s ) n as t → ∞ . Let u E ( t ) be the subordination by u ( t ) = e − at , a > , that is, equality(4.2) with G t ( τ ) from the class (C3). It follows from equality (3.11) that theLaplace-Stieltjes transform of v ( t ) has the form ˜ v ( λ ) = K ( λ ) a + λ K ( λ ) . Using the local behavior of K ( λ ) near λ = 0 from class (C3) yields ˜ v ( λ ) ∼ λ − L (cid:18) λ (cid:19) , L ( x ) = C log ( x ) − − s a + C log( x ) − − s , where C, s > . As the function L is slowly varying at infinity, then by theFKT theorem we obtain the asymptotic behavior for the Cesaro mean of u E ( t ) as M t ( u E ( · )) ∼ C log ( t ) − − s a + C log( t ) − − s ∼ C log( t ) − − s as t → ∞ . cknowledgments This work has been partially supported by Center for Research in Mathemat-ics and Applications (CIMA) related with the Statistics, Stochastic Processesand Applications (SSPA) group, through the grant UIDB/MAT/04674/2020of FCT-Fundação para a Ciência e a Tecnologia, Portugal.The financial support by the Ministry for Science and Education of Ukrainethrough Project 0119U002583 is gratefully acknowledged.
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