RELAXATION PATTERNS AND SEMI-MARKOV DYNAMICS

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1 RELAXATION PATTERNS AND SEMI-MARKOV DYNAMICS MARK M. MEERSCHAERT AND BRUNO TOALDO Abstract. Exponential relaxation to equilibrium is a typical property of physical systems, but inhomogeneities are known to istort the exponential relaxation curve, leaing to a wie variety of relaxation patterns. Power law relaxation is relate to fractional erivatives in the time variable. More general relaxation patterns are consiere here, an the corresponing semi-markov processes are stuie. Our metho, base on Bernstein functions, unifies three ifferent approaches in the literature. Contents 1. Introuction 1 2. Relaxation patterns Some basic facts on Bernstein functions an suborinators Relaxation patterns Time-change processes Semi-Markov Dynamics Classification of states Examples 24 Acknowlegements 29 References Introuction Relaxation phenomena in complex systems can eviate from the traitional exponential moel. In a heterogeneous system, a linear combination of exponential curves with varying rates can lea to power law relaxation, or a variety of other forms. Power law (Cole-Cole) relaxation an Havriliak-Negami relaxation (transitioning between power laws frequency changes) are commonly seen in complex materials, incluing polymers, isorere crystals, supercoole liquis, an amorphous semiconuctors [21; 24; 54]. The connection between relaxation an continuous time ranom walk (CTRW) moels is reviewe in [6]. In the CTRW moel, particle motions X n are separate by ranom waiting times W n, an the long-time limiting particle ensity solves an evolution equation that incorporates Date: August 8, Mathematics Subject Classification. 6K15, 6J35, 34L1. Key wors an phrases. Relaxation, fractional calculus, Bernstein function, semi-markov process, continuous time ranom walk, semigroup. M. M. Meerschaert was partially supporte by ARO grant W911NF an NSF grants EAR an DMS

2 2 MARK M. MEERSCHAERT AND BRUNO TOALDO the relaxation curve. One famous example is the fractional Fokker-Planck equation for subiffusive particle motions in a potential well, where elays in particle motion cause by sticking or trapping with a power law istribute waiting time lea to a fractional time erivative in the evolution equation for the particle ensity [22; 33; 34; 45; 56]. More general waiting time istributions lea to a variety of pseuo-ifferential operators in time [39] that moel general relaxation patterns. In this paper we apply the theory of Bernstein functions to unify the three main approaches to relaxation moeling that are exemplifie by the work of Meerschaert an Scheffler [39], Toalo [57], an Magziarz an Schilling [35]. We show that all three approaches are equivalent, an we establish the corresponence between the moel evolution equations using conjugate Bernstein functions [55]. We establish some properties of solutions using regular variation theory [13; 19], an we apply these solutions to construct general semi-markov (CTRW) particle moels. The forwar an backwar Kolmogorov equations for Markov processes on a iscrete state space are generalize to the semi-markov case, an the classification of states into transient or recurrent is iscusse. More general evolution equations, solve by time-change (relaxe) semigroups on a Hilbert space, are also consiere. To illustrate the main ieas of this paper, we briefly consier a special case. For < β < 1, the time-fractional iffusion equation β t u(x, t) 2 xu(x, t) using the Caputo fractional erivative in time [4, Eq. (2.16)] is equivalent to t β D β t u(x, t) xu(x, 2 t) + u(x, ) Γ(1 β) using the Riemann-Liouville fractional erivative in time [4, Eq. (2.17)] since these two fractional erivatives are relate by β t f(t) D β t f(t) f()t β /Γ(1 β) for all < β < 1 [4, Eq. (2.33)]. Applying D 1 β t to both sies yiels a thir equivalent form t u(x, t) D 1 β t xu(x, 2 t) using a traitional first erivative on the left-han sie. The first form is compact, the secon highlights the initial conition, an the thir is most useful if one wishes to a a forcing term. Complete etails of the equivalence can be foun in Example 4.1. Our main goal in this paper is to establish an unerstan the corresponing equivalence for a general class of time-nonlocal iffusion equations. The main technical ifficulty is to fin an appropriate operator to apply to both sies, to convert the nonlocal time operator to a first erivative in time. It turns out that the key is to interpret the general case of these three equations in terms of conjugate Bernstein functions. 2. Relaxation patterns 2.1. Some basic facts on Bernstein functions an suborinators. In orer to go on with the results we recall basic facts from the theory of Bernstein functions an suborinators (see Bertoin [1, 11]; Schilling et al. [55] for more etails). A Bernstein function f : (, ) [, ) is efine to be of class C an such that ( 1) n 1 f (n) (φ) for all n N : {1, 2, 3,...}. A function f is a Bernstein

3 3 function if an only if [55, Thm 3.2] it can be written in the form ( f(φ) a + bφ + 1 e φs ) ν(s), (2.1) where a an b are non-negative constants an ν( ) is a measure on (, ) such that the integrability conition (s 1)ν(s) < (2.2) is fulfille. The triplet (a, b, ν) is sai to be a Lévy triplet. An integration by parts of (2.1) yiels φ 1 f(φ) b + e φs ν(s)s, (2.3) where ν(s) a + ν(s, ). It follows from [55, Corollary 3.7 (iv)] that (2.3) is a completely monotone function, i.e., it is C an such that ( 1) n n ( φ 1 φ n f(φ) ), for all n N {}. (2.4) A particular subset of the set of Bernstein functions is the set of special Bernstein functions. A Bernstein function f is sai to be special if f (φ) φ/f(φ) is again a Bernstein function. The function f, which is also special, is calle the conjugate of f an has the representation f (φ) φ f(φ) a + b φ + ( 1 e φs ) ν (s), (2.5) where [55, p. 93] { {, b >,, a >, b 1 a+ν(, ), b,, a 1 b+ tν(t), a. (2.6) In what follows we will also nee complete Bernstein functions. A Bernstein function f is sai to be complete [55, Def 6.1] if the ensity ν(s) of the Lévy measure ν(s) ν(s) s appearing in (2.1) exists an is a completely monotone function. Accoring to [55, Thm 6.2, (ii)] we have that a Bernstein function f is complete if an only if φ φ 1 f(φ) is a (non-negative) Stieltjes function, i.e., a function h : (, ) [, ) which can be written in the form h(φ) a φ + b + 1 s(s), (2.7) φ + s where s is a measure on (, ) such that (1 + s) 1 s(s) <. (2.8) It is also true that f is complete if an only if the conjugate f (φ) φ/f(φ) is complete, an hence every complete Bernstein function is special [55, Prop 7.1]. Bernstein functions are naturally associate with suborinators which are nonecreasing Lévy processes. The one-imensional istributions of a suborinator form a convolution semigroup of sub-probability measures on [, ), i.e., a family of measures {µ t ( )} t supporte on [, ) such that (1) µ t [, ) 1, (2) µ t µ s µ t+s for all s, t,

4 4 MARK M. MEERSCHAERT AND BRUNO TOALDO (3) µ t δ vaguely as t, an moreover the Laplace transform µ t (φ) L [µ t ( )] (φ) e φx µ t (x) e tf(φ), (2.9) where f(φ) is a Bernstein function. We will enote by σ f (t), t, the suborinator with Laplace exponent f. If f is a special Bernstein function then the corresponing suborinator is also calle special. The following facts will be use throughout the paper. A suborinator is special if an only if [55, Thm 1.3] its potential measure U σf (t) : E 1 {σ f (s) t}s cδ (t) + u σ f (t)t, (2.1) for some c an some non-increasing function u σ f : (, ) (, ) satisfying 1 u σ (t)(t)t <. In particular from [55, Corollary 1.8] we know that if b f an ν(, ) then c b, ν (, ), an u σ f (t) a + ν (t, ) ν (t). (2.11) 2.2. Relaxation patterns. Meerschaert an Scheffler [39] evelop limit theory for the continuous time ranom walk (CTRW) moel from statistical physics. Given an ii sequence of jumps J n an an ii sequence of waiting times W n, a particle jumps to location S n J J n at time T n W W n. Given a convergent triangular array of CTRW moels, they show [39, Theorem 2.1] that the limit process is of the form X(L f (t)) where X(t) is the limiting Lévy process for the ranom walk of jumps, time change by the inverse suborinator L f (t) inf { s : σ f (s) > t }. (2.12) Kolokoltsov [29] extene the moel to a Markov process limit X(t) by allowing the istribution of the jumps J n to vary in space. In both cases (uner some mil conitions, see [39, Theorem 4.1] an [29, Theorem 4.2]) the probability ensities p(x, t) of the CTRW limit solve a governing equation C f ( t )p(x, t) Ap(x, t), (2.13) where A is the generator of the Markov semigroup, an the Caputo-like operator C f ( t ) is efine so that L [C f ( t )u] (s) f(φ)ũ(φ) φ 1 f(φ)u() (2.14) where L [u] (φ) ũ(φ) e φt u(t) t is the Laplace transform, see [39, Remark 4.8]. If the waiting times W n belong to the omain of attraction of a stable suborinator with Laplace exponent f(φ) φ β, then (2.13) specializes to β t p(x, t) Ap(x, t), (2.15) where β t is the Caputo fractional erivative [41, Eq. (2.16)]. Other choices of f lea to istribute orer [38] an tempere [6; 14] fractional erivatives (see also Example 4.6). Let u be a real-value function on [, ). Toalo [57, Eq. (2.18)] introuce the operator D f u(t) b t u(t) + t u(s) ν(t s)s, (2.16)

5 5 where ν(s) a+ν(s, ) for a Lévy triplet (a, b, ν), an where s ν(s) is assume to be absolutely continuous on [s, ) for any s > (a generalize Riemann-Liouville erivative). The operator (2.16) can be regularize by subtracting an initial conition, as in [39, Remark 4.8], resulting in a generalization of the regularize Riemann-Liouville erivative D f t u(t) b t u(t) + t u(s) ν(t s)s ν(t)u(). (2.17) Use (2.3) to compute the Laplace symbol of (2.17) as [ ] ( ) L D f f(φ) t u(t) (φ) bφũ(φ) bu() + φl [u ν] (φ) φ b u() ( ) ( ) f(φ) f(φ) bφũ(φ) bu() + φũ(φ) φ b φ b u() f(φ)ũ(φ) φ 1 f(φ)u(). (2.18) This shows that (2.14) an (2.17) are the same operator at least for exponentially boune continuously ifferentiable functions u: inee the Laplace transforms agree an furthermore t D f t u(t) is a continuous function since by [57, Proposition 2.7] we can write D f t u(t) b t u(t) + u(s) ν(t s)s ν(t)u() (2.19) t b t u(t) + u (s) ν(t s) s (2.2) an therefore D f t u(t) is continuous, since u an ν are continuous, hence also u ν. Hence (2.17) provies an explicit efinition of the operator C f ( t ) in (2.13). Observe that (2.2) is a generalization of the classical Dzerbayshan-Caputo erivative (accoring to [57, Definition 2.4]). A thir approach was aopte in Magziarz an Schilling [35]: the authors pointe out that the istribution (one-imensional marginal) of B ( L f (t) ), a special case of the CTRW scaling limit where B is a Brownian motion, is the funamental solution to the generalize iffusion equation t q(x, t) 1 2 Φ t q(x, t), x2 x R, t >, (2.21) where Φ t is the integro-ifferential operator for a kernel M(t) such that Φ t u(t) t 2 u(s)m(t s)s, (2.22) L [M(t)] (φ) 1 f(φ). (2.23) A special case of (2.21) calle the fractional Fokker-Planck equation, with M(s) s α /Γ(1 α) for some < α < 1, was introuce by Metzler et al. [45] in the physics literature, see also Henry et al. [22]. The extension to a general waiting time istribution, an hence a general time-convolution operator Φ t, was pioneere by Sokolov an J. Klafter [56] in the context of statistical physics, see also Magziarz [34], an in the mathematical literature by Magziarz [33]. The form (2.21) of the

6 6 MARK M. MEERSCHAERT AND BRUNO TOALDO CTRW limit equation is neee when one wants to a a source/sink term with the natural units of x/t, see Baeumer et al. [5] for aitional iscussion. In this work we place these ifferent approaches in a unifying framework by appealing to the theory of special Bernstein functions. Let f be the special Bernstein function (2.1) with conjugate (2.5) where a an b are given by (2.6). Assume that ν(, ), so that b in view of (2.6). As in (2.3), an integration by parts in (2.5) yiels which implies that φ 1 f (φ) b + e φs ν (s)s, (2.24) L [ ν (t)] (φ) φ 1 f (φ) 1 f(φ), (2.25) since f (φ) φ/f(φ). We may write Φ t u(t) t Ψ tu(t), where Ψ t u(t) u(s)m(t s)s, (2.26) an L[Ψ t u(t)] f(φ) 1 ũ(φ) for continuously ifferentiable functions u. Hence by (2.23) an (2.25) the operator (2.22) is relate to the conjugate Bernstein function f, while (2.14) an (2.16) are relate to the Bernstein function f. In particular, if b, then L[D f t u(t)] f(φ)ũ(φ) an L[Ψ t u(t)] f(φ) 1 ũ(φ), so that D f Ψ t u(t) Ψ t D f u(t) for sufficiently smooth functions u. This shows that the operator (2.26) is the inverse of the operator (2.16) of Toalo [57], when b an M(t) ν (t). Then Φ t D f t Ψ td f t an heuristically (2.21) can be seen as the result of applying Φ t to both sies of (2.13) with A x. This will be mae precise in 2 Theorem 2.5. For sufficiently smooth functions u we can use (2.18) to say also that Ψ t D f u(t) u(t) + u(). Finally, note that we also have M(t) u σ f (t) in view of (2.11). Next we stuy the eigenstructure of the operator (2.17), an a corresponing property for (2.16), by consiering solutions t q(λ, t) C 1 ((, ), R), continuous at zero an exponentially boune, to the equations an { t q(λ, s) ν(t s)s ν(t)q(λ, ) λq(λ, t), t >, q(λ, ) 1, { t q(λ, t) λ t q(λ, ) 1, q(λ, s) ν (t s) s, t >, (2.27) (2.28) where ν(s) a + ν(s, ) an ν (s) a + ν (s, ). Note that in view of the iscussion above the operator on the right-han sie of (2.28) coincies with the operator (2.22) stuie in [35] if ν(, ) an M(t) ν (t). The prototype of our solutions is clearly the Mittag-Leffler function E α (λt α ) : k (λtα ) k /Γ(αk + 1), α (, 1), which is the eigenfunction of the Caputo fractional erivative (e.g. [41, p. 36]) an also solves (2.28) when the operator on the right-han sie is the Riemann-Liouville fractional erivative of orer 1 α (e.g. [31, p. 12]). Furthermore it is well known that this function is continuous on [, ) an completely monotone (an hence C 1 ((, ), R)) an it is not ifferentiable at zero (e.g. [32, Section 3.1]).

7 7 In the next theorem, we impose the aitional assumption that for some γ (, 2), C > an r > we have r s 2 ν(s) > Cr γ for all < r < r, (2.29) as in Orey [49]. For example, (2.29) hols with γ 2 α if ν(s) > Cs 1 α s, for all < s < r, which is true in all the examples iscusse in this paper. Note that Meerschaert an Scheffler [39], Toalo [57], an Magziarz an Schilling [35] all assume that ν(, ), an (2.29) is not much stronger. Theorem 2.1. Let f be a special Bernstein function having representation (2.1) with b an s ν(s) absolutely continuous on [s, ) for any s >. Assume that (2.29) hols an that if γ [ 3 2, 2) the Lévy measure ν has a boune ensity on [s, ) for any s >. Let f be the conjugate of f having representation (2.5). Let L f (t) be the inverse (2.12) of the suborinator σ f (t) with Laplace exponent f. Then for any λ the C 1 ((, ), R), continuous at zero an exponentially boune solution to (2.27) is unique an equal to the moment generating function an furthermore: q(λ, t) E[e λlf (t) ] (2.3) (1) The solution (2.3) to (2.27) is also the unique continuous an exponentially boune solution to (2.28); (2) [, ) θ q( θ, t) is completely monotone for each fixe t, an q(, t) 1 for all t ; (3) t q(λ, t) is completely monotone, for each fixe λ, if an only if s ν(s) is completely monotone; (4) if f(φ) is regularly varying at + with some inex ρ [, 1) then for all λ <, q(λ, t) ν(t) a λ as t, (2.31) both t q(λ, t) an ν(t) vary regularly at infinity with inex ρ, an q(λ, t)t for all λ <. (2.32) Proof. First we prove that (2.3) solves (2.27). Since ν(, ), [39, Thm 3.1] implies that L f (t) has a Lebesgue ensity x l(x, t). Now [39, Eq. (3.13)] shows that [ ] L [l(x, )] (φ) L x P {σ(x) } (φ) f(φ) φ e xf(φ) (2.33) an therefore, for θ > +, we have [39, Corollary 3.5] l (θ, φ) L [L [l(x, t)] (φ)] (θ) f(φ) φ 1 θ + f(φ). (2.34) By (2.18), with b, we have that [ t ] L q(λ, s) ν(t s)s ν(t) (φ) f(φ) q(λ, φ) f(φ) t φ. (2.35)

8 8 MARK M. MEERSCHAERT AND BRUNO TOALDO Taking Laplace transforms in (2.27) an solving for q(λ, φ) then yiels L [q(λ, )] (φ) q(λ, φ) f(φ) φ 1 f(φ) λ. (2.36) Comparing (2.34) to (2.36) shows that the moment generating function of L f is e λx l(x, t) x q(λ, t) Ee λlf (t), λ. (2.37) Now we prove that t E[e λlf (t) ] C 1 ((, ), R). Since (2.29) hols, it follows from Orey [49] that t P ( σ f (x) t ) has erivatives of all orers. Hence we have that E[e λlf (t) ] 1 + λ 1 + λ e λx ( x P ( σ f (x) t ) ) e λx P ( σ f (x) t ) x x e λx µ(s, x)xs, (2.38) where t µ(t, x) is the probability ensity of σ f (x) for any x >. Now it suffices to show that the function I(s) : e λx µ(s, x)x (2.39) is continuous. Since we assume (2.29) we have that the ensity µ(s, x) can be represente via the inversion formula µ(s, x) (2π) 1 e iξs e xϕ(ξ) ξ (2.4) where ϕ(ξ) f( iξ) is the characteristic exponent of the Lévy process σ f an further e xϕ(ξ) e Cx/4 ξ 2 γ for sufficiently large ξ (see Orey [49] at the beginning of page 937). Hence we have by the ominate convergence theorem that µ(s, x) is continuous on (s, x) (, ) (, ). What is more, it is boune on (s, x) (, ) (δ, ) for every δ >. Therefore if s n s >, as n, we have by the ominate convergence theorem that δ e λx µ(s n, x) x Also, by (2.11), we have for all s s >, Thus, given ɛ > there is δ >, such that δ R e λx µ(s, x) x as n. (2.41) µ(s, x)x ν (s) ν (s ) <. (2.42) δ We now show a similar inequality, for λ <, lim sup n δ µ(s n, x)x lim sup n µ(s, x)x < ɛ 2. (2.43) δ e λδ e λx µ(s n, x) x

9 lim sup n ( e λδ µ(s n, x)x δ ) e λx µ(s n, x)x ) ( e λδ µ(s, x)x e λx µ(s, x)x δ ( e λδ ( 1 e λx ) ) δ µ(s, x) x + e λx µ(s, x) x. 9 (2.44) The first equality above is justifie by (2.41) an since, with our assumptions on the Lévy measure, it follows from [25, Theorem 5.2] (see also the comments following that result) that the function u σ f (s) : µ(s, x)x, for s >, is continuous. Now we take λ (use (2.42) to justify the ominate convergence theorem) an we get that lim sup n δ µ(s n, x) x δ µ(s, x) x < ɛ 2. (2.45) Finally the continuity follows since using (2.41), (2.43) an (2.45), we obtain lim sup e λx µ(s n, x) x e λx µ(s, x)x ɛ. (2.46) n Hence (2.38) is continuous on [s, ) for all s >, an therefore t Ee λlf (t) is an element of C 1 ((, ), R). Now we can write f t EeλL (t) λ e λx µ(t, x)x. (2.47) Then it follows from the uniqueness of the Laplace transform (e.g., see Feller [19, Theorem 1, p. 43]) that (2.3) is the unique C 1 ((, ), R) an exponentially boune solution to the problem (2.27), which proves the first part of the theorem. Next we prove Item (1). For a C 1 an exponentially boune solution, by [2, Corollary 1.6.6], we can take Laplace transforms in (2.28) to get φ q(λ, φ) 1 λφl [q ν ] (φ) λφ ( φ 1 f (φ) b ) q(λ, φ). (2.48) Since f (φ) φ/f(φ), an b in view of (2.6), (2.48) can be rewritten ũ(λ, φ) f(φ)/φ f(φ) λ (2.49) which coincies with (2.36). This proves that (2.3) is also the unique C 1 an exponentially boune solution to (2.28), since they have the same Laplace transform. Next we prove Item (2). Since the assumptions imply that ν(, ), the suborinator σ f (t) is strictly increasing [52, Theorem 21.3], an hence L f () a.s. Then we also have q(λ, ) Ee λlf () 1. Since θ q( θ, t) is the Laplace transform of x l(x, t), it is completely monotone for each fixe t. Next we prove Item (3). If the function s ν(s) is completely monotone, we have that for some measure m( ) on (, ) an some non-negative constant a ν(s) a + e sw m(w) a + s w e yw m(w) y (2.5)

10 1 MARK M. MEERSCHAERT AND BRUNO TOALDO an therefore the function y v(y) e yw w m(w) (2.51) is the completely monotone ensity of the Lévy measure ν(y). This implies that f is a complete Bernstein function. Now [55, Thm 6.2 (vi)] implies that ϕ(z) z (2.52) z λ is a complete Bernstein function for λ, an therefore ϕ f is a complete Bernstein function in view of [55, Corollary 7.9]. Therefore we have for some measure k( ) on (, ) that ϕ f(φ) c + φ + an therefore, integrating by parts in (2.53), one has 1 (ϕ f) φ + e φt (c + φ + e φt (c + φ + e φt (c + ( 1 e φt ) e ts k(s) t (2.53) ) e ws k(s) w t ) s 1 e st k(s) t ) s 1 e st k(s) t. (2.54) t The constant in (2.54) is equal to zero. This can be ascertaine by observing that 1 φ (ϕ f) 1 f(φ) φ f(φ) λ as φ by [55, p. 23, Item (iv)] an that f, f, an λ. Then since 1 f(φ) φ f(φ) λ 1 φ (2.55) as φ, it follows that. Since by (2.36) we also have 1 (ϕ f) φ e φt q(λ, t)t, (2.56) an since t c + e ws k(s) w is obviously continuous, it follows from the t uniqueness theorem for Laplace transforms that q(λ, t) c + s 1 e st k(s). (2.57) This proves that t q(λ, t) is completely monotone, which establishes the irect half of Item (3). Now we prove the converse implication. By assumption we have q(λ, t) c + e tu µ(u), (2.58) for c an a measure µ( ) on (, ). In view of (2.36) we have 1 f(φ) φ f(φ) λ e φt q(λ, t) t (2.59)

11 11 an hence G(φ) : 1 f(φ) φ f(φ) λ c φ + e φt (c + ) e ts µ(s) t 1 µ(s), (2.6) φ + s which is a Stieltjes function provie that (1 + s) 1 µ(s) <. But such an integral converges since by Item (1), continuity of t q(λ, t) an the fact that q(λ, ) 1, it must be true that 1 q(λ, ) c + µ(s) (2.61) an therefore µ(s) is integrable. Now note that F (φ) 1/G(φ) is a complete Bernstein function by [55, Thm 7.3]. Then φ/f (φ) φg(φ) is also complete by [55, Proposition 7.1]. It follows that 1 φg(φ) f(φ) λ f(φ) 1 λ f(φ) is a Stieltjes function in view of [55, Thm 7.3]. Let g(φ) : 1 λ/f(φ) an use (2.7) to write then g(φ) a φ + b + λ/f(φ) g(φ) 1 a φ + b k(s), (2.62) φ + s 1 k(s) (2.63) φ + s an since we know that λ/f(φ) is non-negative (recall that f(φ) an λ ) also (2.63) must be non-negative for all φ (, ). In particular by letting φ we euce that b 1. We have thus prove that λ/f(φ) is a Stieltjes function. Therefore by applying again [55, Thm 7.3] to the Stieltjes function λ/f we euce that f(φ) is a complete Bernstein function an therefore f(φ) a + ( 1 e φs ) ν(s)s with ν(s) e st m(t), (2.64) for some measure m an some a (since we are assuming b ). From (2.64) we get that ν(s) a + s ν(w)w a + st m(t) e, (2.65) t an this proves that s ν(s) is completely monotone, which establishes the converse part of Item (3). Finally we prove Item (4). We say that a Borel measurable function f : (, ) [, ) varies regularly at infinity with inex ρ R if f(cx) lim x f(x) c ρ, (2.66) for any c >, see for example Bingham et al. [13, p. 1]. It follows that, for any ε >, for some x >, we have [19, Lemma VIII.8.2] x ρ ε < f(x) < x ρ+ε for all x x. (2.67)

12 12 MARK M. MEERSCHAERT AND BRUNO TOALDO If ρ, we say that f is slowly varying. If f(1/x) is regularly varying at infinity with inex ρ, then we say that f is regularly varying at zero with inex ρ. Suppose that U(x) is a nonecreasing right-continuous function on [, ) with Laplace transform Ũ(s) e sx U(x) for all s >. The Karamata Tauberian Theorem [19, Thm XIII.5.2] states that U(x) xρ L(x) Γ(1 + ρ) as x Ũ(s) s ρ L(1/s) as s, (2.68) where L(x) is slowly varying at infinity an ρ. Suppose that f(φ) varies regularly at φ with inex ρ 1 β for some β (, 1]. Note that if f varies regularly at zero, we must have ρ 1 β for some β [, 1] ue to the Lévy-Khintchine representation (2.1) [11, Proposition 1.5]. If ρ >, it follows from (2.67) that we must have f(+), an hence a in (2.1). If ρ, then a > is possible, in which case f(+) a. In either case, from (2.36) we have q(λ, φ) f(φ) φ 1 f(φ) λ f(φ) 1 φ a λ as φ +, where λ <, an then it is easy to check that φ q(λ, φ) varies regularly at φ + with inex β. Define so that Q(λ, t) q(λ, φ) q(λ, s) s e φt Q(λ, t). Apply the Karamata Tauberian Theorem to see that t Q(λ, t) varies regularly at infinity with inex β, an furthermore that Q(λ, t) tf(1/t) Γ(1 + β)(a λ) as t. Now apply the Monotone Density Theorem [13, Thm 1.7.2] to see that t q(λ, t) varies regulary at infinity with inex β 1, an furthermore tq(λ, t)/q(λ, t) β as t, so that q(λ, t) βf(1/t) Γ(1 + β)(a λ) as t. (2.69) Next observe that (2.3), along with the fact that b, implies that f(φ)/φ is the Laplace transform of ν(t). Then another application of the Karamata Tauberian Theorem shows that ν(t) βf(1/t) as t. (2.7) Γ(1 + β) Combining (2.69) an (2.7) shows that (a λ)q(λ, t) ν(t) as t, which proves the first statement of Item (4). Finally, since t Q(λ, t) varies regularly at infinity with inex β >, it follows from (2.67) that Q(λ, t) as t, which proves the secon statement of Item (4).

13 13 Remark 2.2. If s ν(s) is completely monotone, then we showe in the proof above that ν(s) a + e sw m(w) a + s w 1 e yw m(w) y an therefore the Lévy ensity of ν is also completely monotone. The corresponing Bernstein function f is thus a complete Bernstein function. Therefore the ajoint f is also complete (Proposition 7.1 in [55]) an has a Lévy ensity which is completely monotone with tail ν (s) a + a + s e tw m (w) t w 1 e sw m (w), for some measure m. Therefore s ν (s) is a completely monotone function an Item (3) of Theorem 2.1 may be restate as: the function t q(λ, t) is completely monotone if an only if s ν (s) is completely monotone. That is, ν is completely monotone if an only if ν is completely monotone. Remark 2.3. It follows from [39, Theorem 3.1] that the inverse suborinator (2.12) has a probability ensity l(x, t) ν(t s)µ(s, x) for any t >, where µ(s, x) is the probability istribution of the suborinator σ f (x) with Laplace symbol (2.1), an ν(s) a + ν(s, ). It follows that we can also write in view of (2.3). q(λ, t) e λx ν(t s)µ(s, x) x (2.71) Generalize relaxation equations an patterns have been also examine in [26; 27; 28; 59; 6]. Kochubei [28] consiere operators similar to that appearing in (2.27) but with ifferent kernels of convolution. By making assumptions on the Laplace transform of such kernels he etermine sufficient conitions for the complete monotonicity of the solution. In [26; 27] he also stuie istribute-orer relaxation patterns, i.e., the solution to 1 α u µ(α)α λu, λ <, (2.72) tα where µ is a non-negative continuous function on [, 1]. He pointe out that in this case the relaxation pattern is completely monotone. Observe that (2.72) is a particular case of (2.28) (see [58] for etails on this point). An important application of (2.72) is to ultraslow relaxation where f(φ) is slowly varying at φ, see [38] for more etails. Remark 2.4. The proof of [38, Theorem 3.9] provies a partial converse of Item (4) in Theorem 2.1 in the case of ultraslow iffusion. If the tail of the Lévy measure is of the form ν(t) 1 t η p(η)η,

14 14 MARK M. MEERSCHAERT AND BRUNO TOALDO where p varies regularly at zero with some inex α > 1, then ν(t) is slowly varying [38, Lemma 3.1], an then it follows that the Laplace symbol φ f(φ) is also slowly varying [38, Eq. (3.18)]. 1 Γ(1 η) φ η p(η)η 2.3. Time-change processes. Theorem 2.1 an the iscussion above suggests how the approaches of Meerschaert an Scheffler [39], Toalo [57], an Magziarz an Schilling [35] may be rearrange uner a unifying framework, by resorting to special Bernstein functions. Now we exten the equations (2.27) an (2.28) to a more general form. Suppose that A is a self-ajoint, issipative operator that generates a C -semigroup of operators T t on the (complex) Hilbert space (H,, ), an consier the generalize abstract Cauchy problem t g(s) ν(t s)s ν(t)g() Ag(t), t >, (2.73) or equivalently (as we will show in Theorem 2.5) t g(t) t Ag(s) ν (t s) s, t >. (2.74) Observe that if A λ (then g : [, ) R) the equations (2.73) an (2.74) reuce to that stuie in Theorem 2.1. In Theorem 2.5 below we will investigate solutions to (2.73) an (2.74), i.e., functions of the form g : [, ) H with g C 1 ((, ), H), g(t) continuous at zero, g(t) Dom(A) for any t an such that (2.73) an (2.74) are true. If A x then (2.74) with g(t) q(x, t) reuces 2 to the equation (2.21) in Magziarz an Schilling [35]. We follow Kolokoltsov [3, Section 1.9] an Schilling et al. [55, Chapter 12] for the basic theory of semigroups an generators. See Jacob [23, Chapter 2] or [55, Chapter 11] for a nice summary of the classical theory of linear self-ajoint operators on Hilbert spaces. By the efinition of a C -semigroup we have for all u H that (1) T u u (2) T t T s u T t+s u (3) lim t T t u u H. Note that since A is a self-ajoint generator an it is issipative we have that the spectrum is non-positive, i.e., for any u Dom(A) we have Au, u [55, Proposition 11.2 an formula (11.4)], an we can apply the spectral theorem [55, Thm 11.4]. Therefore we know that there exists an orthogonal projection-value measure E(B) : E(λ) (2.75) for Borel sets B R, supporte on the spectrum of A an therefore in this case on a subset of (, ], such that given a function we may write [55, Eq. (11.1)] Ξ(A)u B Ξ : (, ] R (2.76) (,] Ξ(λ)E(λ)u, (2.77)

15 15 for u Dom(Ξ(A)), where by [55, Eq. (11.11)] we have { } Dom(Ξ(A)) u H : Ξ(λ) 2 E(λ)u, u <. (2.78) (,] Therefore given any u H we can write T t u e At u (,] Then for all u H we have T t u H e λt E(λ)u H (,] e λt E(λ)u. (2.79) (,] E(λ)u u H, (2.8) H so that T t is a contraction semigroup (e.g., see [55, Example 11.5]). Since T t is a C -semigroup we also know that for u Dom(A) we have [3, Thm 1.9.1] t T tu AT t u T t Au, (2.81) an that the map t T t u is the unique classical solution to the abstract Cauchy problem [18, Proposition 6.2] { tg(t) Ag(t), (2.82) g() u. Theorem 2.5. Let f an q(λ, t) be as in Theorem 2.1 uner the same assumptions on ν. Let x l(x, t) be the ensity of inverse process (2.12) of the suborinator σ f (t) with Laplace symbol f. Let T t be a C -semigroup on the Hilbert space (H,, ) whose generator A is self-ajoint an issipative. The unique C 1 ((, ), H), continuous at zero an exponentially boune solution to (2.73), subject to g() u Dom(A) coincies with the C 1 ((, ), H), continuous at zero an exponentially boune solution of (2.74). This solution is the function q(a, t)u efine in the sense of (2.77) for all u H, an we also have a Bochner integral on H. q(a, t)u T s u l(s, t) s, (2.83) Proof. Using the functional calculus approach introuce above we efine q(a, t)u q(λ, t)e(λ)u. (2.84) (,] Now we recall from Theorem 2.1 that the function [, ) θ q( θ, t) is completely monotone an may be written as the Laplace transform of the ensity x l(x, t) of the inverse process (2.12) of the suborinator σ f (t) with Laplace symbol f. Therefore (2.84) becomes q(a, t)u q(λ, t)e(λ)u (,] (,] e λs l(s, t) s E(λ)u

16 16 MARK M. MEERSCHAERT AND BRUNO TOALDO (,] e λs E(λ)u l(s, t) s T s u l(s, t)s (2.85) by (2.79) an the Fubini-Tonelli Theorem use uner the scalar prouct, 2 H, by a simple polarization argument. Note that (2.85) hols for any function u H: inee [55, Example 11.5] q(a, t)u H T s u l(s, t)s T s u H l(s, t)s u H (2.86) H using (2.8) along with l(s, t)s 1. The fact that q(a, t) maps Dom(A) into itself may be ascertaine by using [51, p. 364, formula (15)] to say that q(a, t)a Aq(A, t) an then [51, p. 364, formula (1)], together with the fact that q(a, t) is a boune operator, to say that Dom (q(a, t)a) Dom(A) Dom (Aq(A, t)) {u : q(a, t)u Dom(A)}. The function t q(λ, t) is a monotone non-increasing function (see (2.38)) continuous on [, ) an continuously ifferentiable on (, ) with erivative q (λ, t) which is boune on t [t, ) for any t >. Hence continuity an ifferentiability properties of t q(a, t) (in the Hilbert space topology) are irect consequences of continuity an ifferentiability of q(λ, t) an of the representations q(a, t) q(a, s) 2 H (q(λ, t) q(λ, s)) 2 E(λ)u, u, (2.87) (,] 2 q(a, t)u q(a, s)u q (A, t) t s H ( 2 q(λ, t) q(λ, s) q (λ, t)) E(λ)u, u, (2.88) t s (,] taken as s t. In particular the secon equality, for strictly positive t, shows that (/t)q(a, t) q (A, t) since q (A, t)u exists in H for all u Dom(A) an t > : q (A, t)u H AT x u H µ(t, x) x u σ f (t) Au H <. (2.89) The fact that q(a, t) solves (2.73) an (2.74) can be ascertaine as follows. Since (/t)q(a, t) q (A, t), efine in the sense of (2.76), then for the Caputo type operator D f t hols that D f t q(a, t) D f t q(λ, t) E(λ)u. (2.9) (,] By using (2.17) an Theorem 2.1 we have that D f t t q(a, t) q(λ, s) ν(t s) s E(λ) ν(t)q(λ, )E(λ) (,] t ( <,] ( t ) q(λ, s) ν(t s) s ν(t)q(λ, ) E(λ) (,] t λq(λ, t)e(λ) (,] Aq(A, t), (2.91)

17 17 an this proves (2.73). The same arguments can be also applie to the function (, ] λ q (A, t) since (/t)q(a, t) q (A, t), which is of the form (2.76), an thus by using again Theorem 2.1 we get q (A, t) q (λ, t) E(λ)u t (,] (,] t λq(λ, s) ν (t s)s E(λ)u Aq(A, s) ν (t s) s. (2.92) Finally we prove uniqueness. From [57, Eq. (5.13)] it follows that, for any u Dom(A), the solution q(t) of the generalize Cauchy problem has Laplace transform (t λ) D f t q Aq, q() u, (2.93) 1 f(λ) q(λ) (f(λ) A) u. (2.94) λ In view of (2.17) the generalize Cauchy problem (2.93) is another way to write (2.73), an hence (2.94) also hols for any exponentially boune solution to (2.73), for any u Dom(A). The remainer of the argument is ue to Baeumer [7]. Since A generates a C -semigroup, the resolvent (f(λ) A) 1 is a boune operator for all f(λ) in the right half plane. In particular (f(λ) A) 1 an hence by the uniqueness of the Laplace transform, we have q for initial ata u. Then, given two exponentially boune solutions q 1, q 2 to (2.93), their ifference q q 1 q 2 solves (2.93) with u, an hence q 1 q 2. Therefore, the exponentially boune solution to (2.73) is unique. An argument similar to (2.49) shows that the exponentially boune solution to (2.74) for any u Dom(A) has the same Laplace transform (2.94), hence it is also unique. Remark 2.6. Fractional Cauchy problems of the form (2.15) with p(x, ) f(x) Dom(A) were consiere by Bazhlekova [8] an Baeumer an Meerschaert [4]. In this case, we have f(φ) φ β for some < β < 1. Distribute orer fractional Cauchy problems with f(φ) 1 φ β p(β)β, were consiere by Mijena an Nane [47] an Bazhlekova [9]. Solutions to the generalize Cauchy problem (2.13), which is equivalent to (2.73) or (2.74), were evelope by Toalo [57]. 3. Semi-Markov Dynamics In this section we construct a semi-markov process (3.6) on a countable state space whose ynamics are governe by the operator equations an t g(s) ν(t s) s ν(t)g() Ag(t) (3.1) t g(t) t Ag(s) ν (t s)s (3.2)

18 18 MARK M. MEERSCHAERT AND BRUNO TOALDO where A is an S S matrix (we allow a countably infinite state space S ) an g(t) q(a, t) is the operator of Thm 2.5 efine by (2.84) using functional calculus. We will work all throughout this section uner the following assumptions. A1) X(t) is a continuous-time Markov chain with countable state-space S, generate by A an associate to the semigroup of matrices {P t } t. We assume that A is symmetric, an that for its elements a i,j it is true that sup { a i,i } <. The assumption sup { a i,i } < implies that X(t) is non explosive [48, Thm 2.7.1]. Furthemore within such a framework it is true that P t solves the so-calle Kolmogorov backwar equation [48, Thm 2.8.3] an also the forwar one [48, Thm 2.8.6] t P t AP t, (3.3) t P t P t A, (3.4) both subject to P 1. If S is finite then A is a finite matrix (hence A is boune) an the representation P t e At is true. Since we o not assume that S is finite (but only countable) we can use the fact that A is symmetric an therefore the representation P t e At is true in the sense of (2.77) which becomes in this case P t e At j e λjt v j v j, (3.5) where λ j are the eigenvalues of A an the v j are a orthonormal basis of eigenvectors of A. A2) Y n is a (homogeneous) iscrete-time Markov chain on the countable state space S with symmetric transition matrix H an we enote by h i,j the elements of H. A3) The r.v. s J i are i.i.. with c..f. F J (t) 1 q(λ, t) for some λ <, where t q(λ, t) is completely monotone by Thm 2.1. Assume also that the conitions of Item (4) are fulfille. We efine T n n i1 J i an Y (t) Y n, for T n t < T n+1 (3.6) an assume that the i.i.. r.v. s J i are also inepenent from Y n an therefore T n an Y n are inepenent. By (2.31) an (2.5) we have that lim F J(t) 1 lim t t ν(t) a λ 1 a a λ, (3.7) an hence we will assume that a. A4) With σ f (t) we enote the suborinator with Laplace exponent (2.1) for b. Since in A3) we assume that t q(λ, t) is that of Thm 2.1 (an is completely monotone) we must have that ν(, ) an that s ν(s) is completely monotone. Given a suborinator σ f (t), t, with Laplace symbol f we enote the inverse process (2.12) by L f (t), t. Furthermore we assume that f(φ) is regularly varying at + for some inex ρ [, 1) an therefore by Item (4) of Theorem 2.1 the functions t ν(t) an t q(λ, t) are regularly varying at infinity with inex ρ, an EJ i for all i.

19 19 Note that equations (3.1) an (3.2) generalize the Kolmogorov backwar equation (3.3) to semi-markov processes. The corresponing generalizations of (3.4) are an t g(s) ν(t s) s ν(t)g() g(t)a (3.8) t g(t) t g(s)a ν (t s)s. (3.9) Corollary 3.1. Let A be as in A1). The matrix q(a, t), efine in the sense of (2.77), where q(λ, t) is the function of Theorem 2.1 (uner the same assumptions on ν) is the unique solution to (3.1) an (3.8) as well as (3.2) an (3.9) with initial atum g() 1 (ientity matrix). Furthermore we have, using the Bochner integral, that q(a, t) P s l(s, t)s. (3.1) Proof. This Corollary is a irect consequence of Thm 2.5. To clarify the arguments, here we provie some etails where the proof becomes simpler in the present case. Observe that now the generator A is a matrix with non-positive eigenvalues an, for an orthonormal basis v j of eigenvectors of A, the spectral representation (2.77) here is the matrix Ξ(A) j The function q(a, t) is therefore the matrix Ξ(λ j )v j v j. (3.11) q(a, t) j q(λ j, t)v j v j j e λjs l(s, t)s v j v j P s l(s, t)s. (3.12) It is clear that equation (3.8) is also satisfie since P t A AP t. In our case this may be easily checke Aq(A, t) λ j v j v j e λis l(s, t)s v i v i j i λ j e λis v j v jv i v i l(s, t)s j j i λ j e λjs v j v jl(s, t)s which finishes the proof. e λjs v j v jl(s, t)s λ i v i v i j i q(a, t)a (3.13)

20 2 MARK M. MEERSCHAERT AND BRUNO TOALDO Theorem 3.2. The process Y (t) introuce in A3) is semi-markov an such that the S S matrix with elements q i,j (t) P {Y (t) j Y () i} (3.14) satisfies (3.1) an (3.8) as well as (3.2) an (3.9) with initial atum g() 1 for A λ(h 1), where H is the transition matrix of the iscrete-time Markov chain Y n on S introuce in A2). Furthermore (q i,j (t)) i,j q(a, t) e λ(h 1)s l(s, t) s, (3.15) where q(a, t) is the function of Thm 2.5 an s l(s, t) is the ensity of the process L f (t) in A4). Proof. First note that since Y (t) Y n, for T n t < T n+1, we have that Y (t) Y N ν (t) where an therefore Y (t) is a semi-markov process since N ν (t) max {n N : T n t}, (3.16) P (Y n j, J n t (Y, T ) (Y n 1, T n 1 )) P (Y n j Y n 1 i) (1 q(λ, t)), (3.17) ue to the inepenence between the r.v. s J i an the chain Y n. Therefore the q i,j satisfy the (backwar) renewal equation [16, Chapter 1, formula (5.5)] q i,j (t) q(λ, t)δ i,j + l S h i,l q l,j (s) f J (t s)s (3.18) where h i,j are the elements of the symmetric transition matrix H of the iscretetime chain an f J (t) (1 q(λ, t)). (3.19) t Next we prove that t q i,j (t) is continuous. Since t q(λ, t) is completely monotone uner A3), we can write q(λ, t) tw m(w) e w for some measure m(w), an hence we also have f J (t) (3.2) e tw m(w) (3.21) an this means that also t f J (t) is completely monotone. Then (3.18) yiels for any i, j S an t, h > q i,j (t) q i,j (t + h) q(λ, t) q(λ, t + h) + h i,l q l,j (s) (f J (t s) f J (t + h s)) s l S +h + h i,l q l,j (s)f J (t + h s)s t l S

21 q(λ, t) q(λ, t + h) + (f J (t s) f J (t + h s)) s + h 21 f J (s)s (3.22) where we use that l S h i,lq l,j (s) l S h i,l 1. Then the first an the last terms in (3.22) go to zero as h + since t q(λ, t) an t f J (t) are completely monotone functions. For the integral we can use (3.21) to say that (f J (t s) f J (t + h s)) f J (t s) (3.23) an f J(t s)s < for all t >. Therefore the integral in (3.22) goes to zero as h + by ominate convergence theorem. For h < the argument is similar. Hence t q i,j (t) is continuous. Therefore the q i,j (t), i, j S, are the unique continuous functions whose Laplace transforms satisfy (we here use (2.35) an (3.18)) q i,j (φ) f(φ)/φ f(φ) λ δ i,j l S λ h i,l q l,j (φ) f(φ) λ. (3.24) Now multiply by f(φ) λ on both sies of (3.24) an substract q i,j () δ i,j to get ( ) f(φ) q i,j (φ) f(φ) φ q i,j() λ q l,j (φ)h i,l q i,j (φ). (3.25) Now multiply by f (φ) φ/f(φ) to get ( ) φ q i,j (φ) q i,j () λf (φ) q l,j (φ)h i,l q i,j (φ). (3.26) By Laplace inversion of (3.25) an (3.26) we have using (2.3) that q i,j (t) satisfies for all i, j S { t q i,j(s) ν(t s)s ν(t)q i,j () λ ( l S q l,j(t)h i,l q i,j (t) ), (3.27) q i,j () δ i,j, an { t q i,j(t) λ t q i,j () δ i,j. l S l S ( l S q l,j(s)h i,l q i,j (s) ) ν (t s)s, Note that the solution to the matrix problem, for A λ(h 1), { t q(a, s) ν(t s)s ν(t)q(a, ) Aq(A, t), q(a, ) 1, (3.28) (3.29) is a matrix such that each entry satisfy the backwar equation (3.27). The backwar equation therefore is prove. The forwar equation follows by (3.13). To verify in this special case, since H is symmetric then so is H 1 an the eigenvalues of H 1 are non-positive since H is a transition matrix. Therefore we can write as in (3.12) an we know that in view of (3.13). q λ(h 1) (t) e λ(h 1)s l(s, t)s, (3.3) λ(h 1)q( λ(h 1), t) q( λ(h 1), t)( λ)(h 1), (3.31)

22 22 MARK M. MEERSCHAERT AND BRUNO TOALDO If we interpret the i.i.. r.v. s J i as waiting times between events in some point process then the sequence J i is a renewal process an the r.v. T n is the instant of the n-th event. The process counting the number of events occurre up to a certain time t is the counting process N ν (t) max {n N : T n t}. Clearly if one consiers exponentially istribute waiting times then the corresponing counting process is the Poisson process. When the waiting times are Mittag-Leffler istribute, i.e., P {J > t} E α (λt α ), α (, 1), λ <, a particular semi-markov moel on a graph have been consiere in [2; 5]. The authors showe that the governing equation is time-fractional. In general if the i.i. waiting times J i have finite mean µ J then one has by a simple argument using the strong law of large numbers that a.s. N ν (t) lim t t 1 µ J, (3.32) an the elementary renewal theorem [3, Proposition 1.4] states that EN ν (t) lim t t 1 µ J. (3.33) These facts may be interprete as an equivalence (in the long time behaviour) between the Poisson process an a general renewal process with finite-mean waiting times. Note that (3.32) an (3.33) means that if EJ i < then as t we have, a.s., N(t) N ν (t). This heuristically means that a renewal process with finite mean waiting times is inistinguishable after a long time from the Poisson process. Therefore when you observe the process Y (t) efine in A3) with sojourn times J n T n+1 T n having finite mean then, after a transient perio it behaves like the case in which T n+1 T n are exponential r.v. s. We have here introuce a class of renewal processes associate with waiting times J such that P {J > t} q(λ, t) an uner A3) we know that EJ. Therefore they never behave as a Poisson process. Corollary 3.1 an Theorem 3.2 implies that the time-change Markov chain X ( L f (t) ), t, may be equivalently constructe starting from an embee Markov chain Y n an by inserting between jumps the heavy-taile waiting times J i. In equation (3.6) we efine the renewal process Y (t) that jumps to the state Y n for the unerlying iscrete time Markov process at the arrival time T n of the renewal process with waiting time istribution P[J > t] q(λ, t). Then we have Y (t) Y N ν (t), a time change using the renewal process (3.16). Next we show that the same process can also be constructe by a time change using the inverse suborinator (2.12). Proposition 3.3. Let N(t) be a homogeneous Poisson process with rate θ λ. The time-change process N ( L f (t) ) an the process N ν (t) are the same process. Hence the semi-markov process (3.6) is the same process as the time-change Markov chain Y N(Lf (t)). Proof. This is a consequence of [4, Thm 4.1] since P {J i > t} q(λ, t) for any i an in view of Thm 2.1 it is true that q(λ, t) Ee λlf (t) Classification of states. We here investigate whenever a state i is transient or recurrent for the semi-markov process Y (t), by making assumptions on the embee chain Y n. We recall that a state i is recurrent if P (the set {t : Y (t) i} is unboune Y () i) 1, (3.34)

23 23 an is transient if the probability in (3.34) is zero. Since we take the probability of a tail event, an 1 are the only possibilities. We have the following result. Theorem 3.4. Uner A2), A3), an A4) it is true that (1) If the state i is recurrent for Y n then it is recurrent for Y (t) (2) If the state i is transient for Y n then it is transient for Y (t) (3) q i,i (t)t inepenently from the fact that the state i is transient or recurrent. We recall that q i,i (t) P {Y (t) i Y () i} P i {Y (t) i}. Proof. Note that in view of Proposition 3.3 we can write Y (t) Y N ν (t) Y N(L f (t)). (3.35) Since t σ f (t) is a.s. right-continuous, unboune, an strictly increasing, it follows from the change of variable formula in Meerschaert an Straka [43, p. 177] that 1 {t :Y (t)i} t n 1 { t :Y N(L f (t)) i} t 1 {t :YN(t) i} σf (t) 1 {Yni} τ n t<τ n+1 e(t), (3.36) where e(t) is the Poisson point process unerlying the suborinator σ f with characteristic measure ν(s)s. Note that if the state i is recurrent for Y n then P (Y n i for infinitely many n) 1 (3.37) an the number of summans in (3.36) is a countable infinity. Furthermore the sequence τ n t<τ n+1 e(t) (3.38) is a sequence of i.i.. r.v. s since e(s) is a Poisson point process. Therefore (3.36) is the sum of a countable infinity of i.i.. positive r.v. s. which iverges with probability one. This proves Item 1. If instea the state i is transient, then the number of summans in (3.36) is finite, an since our suborinators are here assume to be not subject to killing, it is true that for all t 1 < t 2 < < t 1 s t 2 e(s) < a.s. (3.39) an the sum (3.36) is finite since it is the sum of a finite number of finite summans. This proves 2. Observe now that an this proves Item 3. q i,i (t) t E i 1 {Y (t)i} t E i J 1 (3.4)

24 24 MARK M. MEERSCHAERT AND BRUNO TOALDO It is instructive to compare Part (3) of Theorem 3.4 with the well-known characterization of transient an recurrent states in a semi-markov process with finite mean waiting time between jumps, using the occupation measure (or -potential) [16]. The semi-markov process is forme by inserting a ranom waiting time J i before jumping from state Y i 1 to state Y i in the unerlying Markov chain. Hence the number of times the process returns to its starting point is not affecte. Hence the state is recurrent for the semi-markov process if an only if it is recurrent for the unerlying Markov chain. If the occupation times (waiting times) in each state have a finite mean, then the total expecte occupation time in the starting state is proportional to the number of visits. This happens if an only if q i,i (t)t. However, when the waiting times between state transitions are heavy-taile with infinite mean, the mean time spent in the starting point by the process is always infinite. 4. Examples In this section, we provie some practical examples, to illustrate the application of the results in this paper. Example 4.1. Consier a CTRW with ii particle jumps X n inepenent of the ii waiting times W n. Then a particle arrives at location S(n) X X n at time T n W 1 + +W n. If E[X n ] an E[Xn] 2 <, then n 1/2 S([nt]) B(t), a Brownian motion, by Donsker s Theorem [17, Theorem 8.7.5]. If P[W n > t] t β /Γ(1 β) for some < β < 1, then n 1/β T [nt] σ f t, a β-stable suborinator with Laplace symbol f(φ) φ β [41, Theorem 3.41]. The number of jumps by time t > is N t max{n : T n t} an a simple argument [37, Theorem 3.2] shows that this inverse process has an inverse limit n β N nt L f t where the inverse stable suborinator L f t is efine by (2.12). Then we have n β/2 S(N nt ) (n β ) 1/2 S(n β n β N nt ) (n β ) 1/2 S(n β L f t ) B(L f t ), as n, see [37, Theorem 4.2] for a rigorous argument. The probability ensity p(x, t) of the limit process B(L f t ) solves the time-fractional iffusion equation (2.13) with C f ( t ) β t, a Caputo fractional erivative of orer < β < 1, an A D x 2 where D E[Xn]/2, 2 see [37, Theorem 5.1] an [41, Eq. (1.8)]. Since the Caputo fractional erivative [41, Eq. (2.16)] β t p(x, t) 1 Γ(1 β) t p(x, t u)u β u is relate to the Riemann-Liouville fractional erivative [41, Eq. (2.17)] by [41, Eq. (2.33)] D β 1 t p(x, t) Γ(1 β) t t β p(x, t u)u β u D β t p(x, t) p(x, ) Γ(1 β) β t p(x, t), (4.1) we can also write the governing equation (2.13) in the form t β D β t p(x, t) p(x, ) A p(x, t). (4.2) Γ(1 β)

25 25 The Laplace symbol f(φ) φ β can be compute irectly from (2.1) with a b an ν(t) βt β 1 t/γ(1 β) using integration by parts an the efinition of the Gamma function [41, Proposition 3.1]. Then ν(t) t β /Γ(1 β) an the operator D f of Toalo [57] efine by (2.16) reuces to the Riemann-Liouville fractional erivative. Similarly, the operator D f t of [57] efine by (2.17) reuces to the Caputo fractional erivative. Then the governing equation (2.73) of Toalo [57] with g(t) p(x, t) reuces to (4.2). Since the conjugate Bernstein function f (φ) φ/f(φ) φ 1 β, the same calculation as for f shows that ν (t) t β 1 /Γ(β), an then the operator Φ t of Magziarz an Schilling [35] efine by (2.22) with M(t) ν (t) is the Riemann-Liouville fractional integral I β t p(x, t) 1 Γ(β) p(x, t u)u β 1 u 1 Γ(β) p(x, u)(t u) β 1 u of orer β [41, p. 25]. Then the governing equation (2.74) of Magziarz [35] with g(t) p(x, t) reuces to Since t Iβ t D 1 β t 1 p(x, t) t Γ(β) t Ap(x, s) (t s) β 1 s t A Iβ t p(x, t). (4.3) we can also write (4.3) in the form t p(x, t) D1 β t A p(x, t) A D 1 β t p(x, t), which is commonly seen in applications [22; 45]. The heuristic erivation of (4.3) is to simply apply the operator D 1 β t to both sies of (4.2), or the equivalent form β t p Ap, but the initial conition requires some care. Example 4.2. Replacing the ii jumps X n in Example 4.1 with a convergent triangular array, we obtain a limit B(L f t ) where B(t) is an arbitrary Lévy process [39, Theorem 3.6] with generator A given by the Lévy-Khintchine formula [36, Theorem ]. Then all the results of Example 4.1 hol with A D x 2 replace by this Lévy generator, in R 1 or in R for any finite imension [39, Theorem 4.1]. The same is true more generally for Markov generators A, where the jump istribution epens on the current state [29, Theorem 4.2]. Example 4.3. For a CTRW with eterministic particle jumps X n 1, an the same waiting times W n as in Example 4.1, we have S(n) n an the CTRW S(N t ) N t converges to the inverse stable suborinator: n β N nt L f t. The probability ensity l(x, t) of L f t solves the time-fractional equation (2.13) with C f ( t ) β t an A x [42, Eq. (5.7)]. Several equivalent governing equations for the inverse stable suborinator L f t are also iscusse in [42], incluing equation (2.73) of Toalo [57] (see [42, Eq. (5.9)]) an equation (2.74) of Magziarz [35] (see [42, Eq. (5.18)]) with g(t) l(x, t) an A x. In this case, the moment generating function (2.3) can be written explicitly as q(λ, t) E β (λt β ) in terms of the Mittag-Leffler function E β (z) j z j Γ(1 + βj), (4.4)