Non-Markovian diffusion equations and processes: analysis and simulations

Non-Mark o vian diusion equations and pro esses: analysis and sim ulations An tonio MURA 1 , Murad S. T A QQU 2 and F raneso MAINARDI 1 1 . Dep artment of Physis, University of Bolo gna, and INFN, Via Irnerio 46, I-40126 Bolo gna, Italy URL: h ttp:// www.fraalmo.org 2 . Dep artment of Mathematis, Boston University, Boston, MA 02215, USA URL: h ttp://math.bu.edu/p eople/m urad/ Revised V ersion: Ma y 2008 in press on Ph ysia A (2008), doi:10.1016/j.ph ysa.2008.04. 035 Keyw ords : Non-Mark o vian pro esses, frational deriv ativ es, anomalous diusion, sub ordination, frational Bro wnian motion. Abstrat : In this pap er w e in tro due and analyze a lass of diusion t yp e equations related to ertain non- Mark o vian sto  hasti pro esses. W e start from the forw ard drift equation whi h is made non-lo al in time b y the in tro dution of a suitable  hosen memory k ernel K ( t ) . The resulting non-Mark o vian equation an b e in terpreted in a natural w a y as the ev olution equation of the marginal densit y funtion of a random time pro ess l ( t ) . W e then onsider the sub ordinated pro ess Y ( t ) = X ( l ( t )) where X ( t ) is a Mark o vian diusion. The orresp onding time ev olution of the marginal densit y funtion of Y ( t ) is go v erned b y a non-Mark o vian F okk er-Plan k equation whi h in v olv es the memory k ernel K ( t ) . W e dev elop sev eral appliations and deriv e the exat solutions. W e onsider dieren t sto  hasti mo dels for the giv en equations pro viding path sim ulations. 1 In tro dution In this in tro dution, w e desrib e and motiv ate the themes dev elop ed in the pap er. Historial notes will b e presen ted in Setion 2 . Bro wnian motion B ( t ) , t ≥ 0 , is a sto  hasti pro ess with man y prop erties. It is at the same time Gaussian and Mark o vian, has stationary inremen ts and is self-similar. A pro ess X ( t ) , t ≥ 0 , is said to b e self-similar with self-similarit y exp onen t H if, for all a ≥ 0 , the pro esses X ( at ) , t ≥ 0 , and a H X ( t ) , t ≥ 0 , ha v e the same nite-dimensional distributions. Bro wnian motion is self-similar with exp onen t H = 1 / 2 . In on trast, fra- tional Bro wnian motion B H ( t ) , t ≥ 0 , is Gaussian, has stationary inremen ts, is self-similar with self-similarit y exp onen t 0 < H < 1 , but is not Mark o vian, unless H = 1 / 2 , in whi h ase the frational Bro wnian motion b eomes Bro wnian motion. When 1 / 2 < H < 1 , the inremen ts of frational Bro wnian motion ha v e long-range dep endene [49℄. Beause Bro wnian motion is Mark o vian with stationary inremen ts, its nite-dimensional distributions an b e obtained from the marginal densit y funtion f B ( x, t ) = 1 √ 4 π t e − x 2 / 4 t , x ∈ R (1) at time t ≥ 0 . This densit y funtion is the fundamen tal solution of the standard diusion equation: ∂ t u ( x, t ) = ∂ xx u ( x, t ) , (2) 1 whi h in in tegral form reads: u ( x, t ) = u 0 ( x ) + Z t 0 ∂ xx u ( x, s ) ds, u 0 ( x ) = u ( x, 0) . (3) Th us, f B ( x, t ) is a solution of Eq. (3) with u 0 ( x ) = δ ( x ) , where δ ( x ) is the Dira delta distribution. W e allo w, throughout the pap er, funtions to b e distributions. Remark 1.1. W e follo w the ph ysis on v en tion of not inluding the fator 1 / 2 in Eq. ( 2 ). Therefore, in this pap er, standard Bro wnian motion B ( t ) , t ≥ 0 , is su h that, for ea h time t ≥ 0 , B ( t ) ∼ N (0 , 2 t ) . The tilde notation X ∼ f X ( x ) indiates that the random v ariable X has the probabilit y densit y funtion f X ( x ) . Our goal is to extend Eq. (3) to non-Mark o vian settings. W e will onsider non-lo al, frational and stret hed mo diations of the diusion equation. These mo died equations will b e alled Non-Markovian diusion e qua- tions , b eause, while they originate from a diusion equation, the orresp onding pro ess, whose probabilit y densit y funtion is a solution of these mo died equations, will b e t ypially non-Mark o vian. T o motiv ate the mo diations, onsider rst the non-random pro ess l ( t ) = t , t ≥ 0 , whi h depits a non-random linear time ev olution and let f l ( τ , t ) denote its densit y funtion at time t . Therefore one has f l ( τ , t ) = δ ( τ − t ) where δ ( x ) is the Dira distribution. It is natural to in terpret f l ( τ , t ) as the fundamen tal solution of the standard forw ard drift equation: ∂ t u ( τ , t ) = − ∂ τ u ( τ , t ) , τ , t ≥ 0 , (4) whi h in in tegral form reads: u ( τ , t ) = u 0 ( τ ) − Z t 0 ∂ τ u ( τ , s ) ds, u 0 ( τ ) = u ( τ , 0) . (5) The general solutions are of the form u ( τ , t ) = u 0 ( τ − t ) and th us, when u 0 ( τ ) = δ ( τ ) , the solution of Eq. ( 4) is indeed u ( τ , t ) = δ ( τ − t ) . Observ e that the v ariable τ ≥ 0 pla ys the role of a spae v ariable. W e will onsider the follo wing generalization of the forw ard drift equation ( 5) u ( τ , t ) = u 0 ( τ ) − Z t 0 K ( t − s ) ∂ τ u ( τ , s ) ds, τ , t ≥ 0 , (6) where K ( t ) , with t ≥ 0 , is a suitable k ernel  hosen su h that the fundamen tal solution of Eq. ( 6) is a probabilit y densit y funtion at ea h t ≥ 0 . W e refer to Eq. (6 ) as the non-Markovian forwar d drift e quation . The presene of the memory k ernel K in Eq. (6) suggests a orresp onding mo diation of the diusion equation (3). Namely , w e will onsider the equation: u ( x, t ) = u 0 ( x ) + Z t 0 K ( t − s ) ∂ xx u ( x, s ) ds, x ∈ R , t ≥ 0 . (7) Its fundamen tal solution turns out to b e: f ( x, t ) = Z ∞ 0 G ( x, τ ) h ( τ , t ) dτ , (8) where G ( x, t ) = 1 √ 4 π t exp( − x 2 / 4 t ) , (9) and h ( τ , t ) is the fundamen tal solution of Eq. (6 ). 2 The solution (8) is a marginal (one-p oin t) probabilit y densit y funtion. W e will onsider dieren t random pro esses whose marginal probabilit y densit y funtion oinides with it. As illustration, onsider the follo wing examples 1 . Example 1.1. If w e  ho ose: K ( t ) = t − 1 / 2 √ π , t ≥ 0 , (10) then w e ha v e, see Eq. (65 and Eq. (70 ): h ( τ , t ) = 1 √ π t exp  − τ 2 4 t  , τ ≥ 0 , t ≥ 0 , (11) as the fundamen tal solution of Eq. (6 ). No w onsider the pro ess D ( t ) = B ( l ( t )) , t ≥ 0 , (12) where B is a standard Bro wnian motion and l ( t ) ≥ 0 is a random time- hange (not neessarily inreasing), indep enden t of B , whose marginal densit y funtion is giv en b y h ( τ , t ) . One p ossible  hoie for the random time pro ess is simply: l ( t ) = | b ( t ) | , t ≥ 0 , where b ( t ) , t ≥ 0 , is a standard Bro wnian motion [9, 18℄. Su h a random time pro ess l ( t ) , t ≥ 0 , is self-similar of order H = 1 / 2 . 2 Let no w B ( t ) , t ≥ 0 , b e another standard Bro wnian motion indep enden t of b ( t ) . Th us, the pro ess (see also [3℄) D ( t ) = B ( | b ( t ) | ) , t ≥ 0 , (13) has marginal densit y dened b y Eq. (8) with h ( τ , t ) giv en b y Eq. (11). But, D ( t ) is not the only pro ess with densit y funtion f ( x, t ) , giv en b y Eq. ( 8 ). F or example, the pro ess Y ( t ) = p | b (1) | B 1 / 4 ( t ) , t ≥ 0 , (14) where B 1 / 4 is an indep enden t frational Bro wnian motion with self-similarit y exp onen t H = 1 / 4 , has the same one-dimensional probabilit y densit y funtions as the previous pro ess D ( t ) , t ≥ 0 , see Eq. (40 ) with β = 1 / 2 . Example 1.2. The frational Bro wnian motion in Eq. (14 ) has a self-similarit y exp onen t H < 1 / 2 . The inremen ts of su h a pro ess are kno wn to b e negativ ely orrelated [31, 32, 49℄. T o allo w for the presene of frational Bro wnian motion B H ( t ) with 0 < H < 1 , w e in tro due a seond (non-random) time- hange t → g ( t ) , where g (0) = 0 and g ( t ) is smo oth and inreasing, that is w e onsider the non-Mark o vian diusion equation u ( x, t ) = u 0 ( x ) + Z t 0 g ′ ( s ) K ( g ( t ) − g ( s )) ∂ xx u ( x, s ) ds. (15) whose fundamen tal solution is no w: f ( x, t ) = Z ∞ 0 G ( x, τ ) h ( τ , g ( t )) dτ , (16) where h is the fundamen tal solution of Eq. (6). If K ( t ) is as in Eq. (10 ) and g ( t ) = t 2 α , with 0 < α < 2 , then the pro esses: D ( t ) = B ( | b ( t 2 α ) | ) , t ≥ 0 , Y ( t ) = p | b (1) | B α/ 2 ( t ) , t ≥ 0 , ha v e a marginal densit y funtion dened b y Eq. (16 ) with h ( τ , t ) as in Eq. ( 11), whi h is the fundamen tal solution of Eq. (15). In this ase Y ( t ) is dened through an indep enden t frational Bro wnian motion B α/ 2 with Hurst's parameter H = α/ 2 and th us 0 < H < 1 . This is a sp eial ase of Eq. (78). 1 In these examples w e refer to fats whi h are justied later in the pap er through forw ard referenes. The reader ma y w an t to fo us at this p oin t only on the examples and ignore the referenes. 2 Another p ossible  hoie for a random time pro ess with marginal densit y giv en b y Eq. (11 ) is the lo  al time in zero of a standard Bro wnian motion [4℄. In this ase the time- hange pro ess l ( t ) is inreasing. 3 The preeding examples illustrate the themes pursued in the pap er. W e will fo us, ho w ev er, not only on p o w er-lik e k ernels su h as those dened in Eq. (10 ), but also on exp onen tial-lik e k ernels su h as: K ( t ) = e − at , a ≥ 0 . (17) W e also onsider what happ ens when the Bro wnian motion B ( t ) , t ≥ 0 , is replaed b y a more general linear (time-homogeneous) diusion Q ( t ) , t ≥ 0 , go v erned b y the F okk er-Plan k equation 3 , ∂ t u ( x, t ) = P x u ( x, t ) , (18) where P x is a linear op erator indep enden t of t ating on the v ariable x ∈ R . In other w ords w e onsider the non-Mark o vian diusion equation: u ( x, t ) = u 0 ( x ) + Z t 0 g ′ ( s ) K ( g ( t ) − g ( s )) P x u ( x, s ) ds. (19) W e sho w that its fundamen tal solution is: f ( x, t ) = Z ∞ 0 G ( x, τ ) h ( τ , g ( t )) dτ , (20) where G ( x, t ) is the fundamen tal solution of Eq. (19 ), while h ( τ , t ) is the fundamen tal solution of Eq. (6). W e also pro vide expliit solutions when P x is the dieren tial op erator asso iated with Bro wnian motion with drift, when it is asso iated with Geometri Bro wnian motion and when the k ernel K ( t ) is the p o w er k ernel and the exp onen tial k ernel. In order not to dw ell on te hnialities, w e supp ose impliitly , throughout the pap er, that w e ha v e suien t regularit y onditions, to justify the algebrai manipulations that are p erformed. The pap er is organized as follo ws:  Historial notes are presen ted in Setion 2 .  In Setion 3 w e study the non-Mark o vian forw ard drift equation (6) and its orresp onding random time pro ess l ( t ) . W e deriv e suitabilit y onditions on the k ernel K ( t ) . W e end the setion b y noting that a self-similar time- hange pro ess, for instane with self-similarit y parameter H = β , requires the  hoie K ( t ) = C t β − 1 / Γ( β ) with 0 < β ≤ 1 .  In Setion 4 w e study the non-Mark o vian diusion equation (15 ) and its solutions, and w e disuss its v arious sto  hasti in terpretations.  In Setion 5 w e illustrate the fat that the sto  hasti represen tation is not unique.  In Setion 6 w e study the more general non-Mark o vian F okk er-Plan k equation and deriv e its solution Eq. (20 ).  In Setion 7 w e go thorough sev eral examples with P x u ( x, t ) = ∂ xx u ( x, t ) , that is, when the underlying diusion pro es is Bro wnian motion. W e onsider non-Mark o vian diusion equations, asso iated with the β -p ower k ernel K ( t ) = t β − 1 / Γ( β ) , 0 < β ≤ 1 , and with the exp onential-de  ay k ernel K ( t ) = e − at , a ≥ 0 . W e also onsider dieren t  hoies of the deterministi saling funtion g ( t ) , for example a logarithmi time sale g ( t ) = lo g( t + 1) is onsidered.  In Setion 8 w e fo us on appliations when the underlying diusion pro ess is not standard Bro wnian motion. W e onsider the ase of Bro wnian motion with drift and Geometri Bro wnian motion and w e study the orresp onding equations with the β -p ower k ernel and the exp onential-de  ay k ernel.  Setion 9 on tains a summary and onluding remarks. 3 Also kno wn as the forw ard K olmogoro v equation. 4 2 Historial notes Non-Mark o vian equations lik e Eq. ( 7), or more generally Eq. (19 ), are often enoun tered when studying ph ysial phenomena related to relaxation and diusion problems in omplex systems (see Srok o wsky [47℄ for examples). Equations of the t yp e (7) ha v e b een studied for example b y K olsrud [22℄. He obtained Eq. 8), but without pro viding sp ei examples. A similar study w as done b y W yss [51℄ who, ho w ev er, fo used only on p o w er-lik e k ernels K ( t ) = C t β − 1 . Sok olo v [45℄ (see also Srok o wsky [47℄), studied the non-Mark o vian equation ∂ t P ( x, t ) = Z t 0 k ( t − s ) L x P ( x, s ) ds, (21) where L x is a linear op erator ating on the v ariable x . He pro vided a formal solution in the form of Eq. 20 ). Observ e, ho w ev er, that our equation (19 ) diers from Eq. (21 ), not only b y the presene of the saling funtion g ( t ) , but also b y the  hoie of the memory k ernel. Our k ernel K ( t ) and Sok olo v's k ernel k ( t ) are related b y the equation: K ( t ) = Z t 0 k ( s ) ds ⇒ e K ( s ) = e k ( s ) /s, s > 0 , (22) where the tilde indiates the Laplae transform, see Eq. (25 ). The suitabilit y onditions for these memory k ernels are th us not the same (these onditions are dev elop ed in Setion 3 ). F or example, onsider the simple exp onential-de  ay k ernel e − at , a ≥ 0 . This  hoie of the k ernel is safe in the on text of Eq. (19), i.e. for the  hoie K ( t ) = e − at , but is dangerous if one onsiders Eq. ( 21 ) with the k ernel k ( t ) = e − at . In the ase of Eq. (19 ), the exp onential-de  ay k ernel orresp onds to a system for whi h non-lo al memory eets are initially negligible. In fat, K ( t ) = e − at → 1 as t → 0 and th us the system app ears Mark o vian at small times. On the other hand, the  hoie k ( t ) = e − at orresp onds to the k ernel K ( t ) = a − 1 (1 − e − at ) whi h for small times b eha v es lik e t . In this ase Sok olo v [45℄ notied that the orresp onding equations are only reasonable in a restrited domain of the mo del parameters and for ertain initial and b oundary onditions. Our starting p oin t is dieren t from that of the previous authors. Instead of starting diretly from the F okk er- Plan k equations (18), w e start from the forw ard drift equation ( 5) whi h is then generalized b y in tro duing a memory k ernel K ( t ) , Eq. ( 6). One is then naturally led to the non-Mark o vian diusion equations (15 and (19 ) after the in tro dution of the saling funtion g ( t ) . In fat, in sp ei ases, it is sometimes simpler to solv e rst the non-Mark o vian forw ard drift equation (6) and then use the solution to solv e the non-Mark o vian diusion equation (15 ) or (19) b y using (16) or (20 ). The form of the solution (16 ) or (20) has no w a ready-made in terpre- tation. F or example, in Eq. (20) the funtion G ( x, t ) is the fundamen tal solution of the Mark o vian equation ( 18 ) and the funtion h ( τ , t ) is the fundamen tal solution of the non-Mark o vian equation (6 ) and it is these t w o solu- tions that on tribute to Eq. (20) whi h is the fundamen tal solution of the non-Mark o vian diusion equation ( 19 ). F urthermore, the form (16 ) or (20 ) has a natural in terpretation in terms of sub or dinate d pro esses, see Eq. (12). A ording to Whitmore and Lee [23℄, the term sub ordination w as in tro dued b y Bo  hner [5, 6℄. It refers to pro esses of the form Y ( t ) = X ( l ( t )) , t ≥ 0 , where X ( t ) , t ≥ 0 , is a Mark o v pro ess and l ( t ) , t ≥ 0 , is a (non-negativ e) random time pro ess indep enden t of X . The marginal distribution of the sub ordinated pro ess is learly: f Y ( x, t ) = Z ∞ 0 f X ( x, τ ) f l ( τ , t ) dτ , t ≥ 0 , x ∈ R , (23) where f X ( x, t ) and f l ( τ , t ) represen t the marginal densit y funtions of the pro esses X and l . Therefore, Eq. (16) or Eq. (20 ) an b e in terpreted in terms of sub ordinated pro esses, with Eq. (6)  haraterizing the random time pro ess l ( t ) and Eq. (18 )  haraterizing the Mark o v paren t pro ess X ( t ) . 5 The sto  hasti in terpretation through sub ordinated pro esses, rst suggested b y K olsrud, is v ery natural b eause Y ( t ) = X ( l ( t )) has a diret ph ysial in terpretation. F or example, in equipmen t usage, X ( t ) an b e the state of a ma hine at time t and l ( t ) the eetiv e usage up to time t . In an eonometri study , X ( t ) ma y b e a mo del for the prie of a sto  k at time t . If l ( t ) measures the total eonomi ativit y up to time t , the prie of the sto  k at time t should not b e desrib ed b y X ( t ) but b y the sub ordinated pro ess Y ( t ) = X ( l ( t )) . The resulting sub ordinated pro ess Y ( t ) is in general non-Mark o vian. In this w a y , the non-lo al memory eets are attributable to the random time pro ess l ( t ) and to its dynamis whi h is in general non-lo al in time, see Eq. ( 6). Note, ho w ev er, that the solution of Eq. (19 ) represen ts only the marginal (one-p oin t) densit y funtion of the pro ess and therefore annot  haraterize the full sto  hasti struture of the pro ess. As w e note in the pap er, there are also pro esses that are not sub ordinated pro esses that serv e as sto  hasti mo dels for non-Mark o vian diusion equations lik e Eq. (19) or Eq. (21). F or example, onsider in Eq. ( 7) the β -p ower k ernel K ( t ) = t β − 1 / Γ( β ) , with 0 < β ≤ 1 . F rom a sto  hasti p oin t of view, the fundamen tal solution of this equation, also alled the time-frational diusion equation of order β , an b e in terpreted as the marginal densit y funtion of a self-similar sto  hasti pro esses with parameter H = β / 2 . This pro ess, for example, an b e tak en to b e a sub ordinated pro ess Y ( t ) = B ( l ( t )) , with a suitable  hoie of the random time l . In K olsrud [22℄, the random time l is tak en to b e related to the lo al time of a d = 2(1 − β ) -dimensional frational Bessel pro ess, while in Meers haert et al. [34℄ (see also [1, 15 17, 21, 40, 48℄), in the on text of a Con tin uous Time Random W alk (CTR W), it is  hosen to b e the in v erse of the totally sk ew ed stritly β -stable pro ess. The in terested reader is referred to the wide literature onerning the relationship b et w een CTR W and non-Mark o vian diusion equations and its appliations. See for instane, [2, 13, 14, 19, 28, 35, 36 , 41 , 42 , 5 0, 5 2℄ and referenes therein. S hneider [43℄, moreo v er, in a v ery general mathematial onstrution, in tro dued the so-alled Gr ey Br ow- nian motion . This pro ess is a self-similar pro ess with stationary inremen ts whi h, as turns out, an b e represen ted b y Y ( t ) = Λ β B H ( t ) , t ≥ 0 , where B H is a frational Bro wnian motion with H = β / 2 and Λ β is a suitable  hosen random v ariable indep enden t of B H (see Mura et al. for details [38, 39℄). This pro ess has a marginal densit y funtion that ev olv es in time aording to the time-frational diusion equation of order β . In this ase the non-Mark o vian prop ert y is due to the presene of the frational Bro wnian motion. As w e sho w in the pap er, long-range dep endene an b e made to app ear through the time-saling funtion g ( t ) , see Eq. (15 ) and Example 1.2 . Figures 4, 5 and 6 displa y tra jetories of the pro esses D ( t ) and Y ( t ) and orresp onding densit y funtions. 3 The non-Mark o vian forw ard drift equation W e start with the follo wing generalization of Eq. ( 5), namely: u ( τ , t ) = u 0 ( τ ) − Z t 0 K ( t − s ) ∂ τ u ( τ , s ) ds, τ , t ≥ 0 , (24) where K ( t ) , with t ≥ 0 , is a suitable  hosen k ernel. W e then  ho ose a random time pro ess l ( t ) su h that, for ea h t ≥ 0 , its marginal densit y f l ( τ , t ) is the fundamen tal solution of Eq. ( 24 ). Observ e that Eq. (24 ) is non-lo al b eause u ( τ , t ) in v olv es u ( τ , s ) at all 0 ≤ s ≤ t . Equation ( 24 ) will b e alled non-Markovian forwar d drift e quation , see Setion 1 , Eq. (6). It is on v enien t to w ork with Laplae transforms. W e indiate b y L { ϕ ( x, t ); t, s } the Laplae transform of the funtion ϕ with resp et to t ev aluated in s ≥ 0 , namely: L { ϕ ( x, t ); t, s } = Z ∞ 0 e − ts ϕ ( x, t ) dt, s ≥ 0 . (25) If the funtion ϕ dep ends only on the v ariable t w e write simply e ϕ ( s ) , b eause in this ase there is no am biguit y onerning the in tegration v ariable. In partiular w e let e K ( s ) denote the Laplae transform of the k ernel K . 6 Prop osition 3.1. L et f l ( τ , t ) denote the fundamental solution of Eq. (24 ). Then, L { f l ( τ , t ); t, s } = 1 s e K ( s ) exp − τ e K ( s ) ! , τ , s ≥ 0 , (26) and zer o for τ < 0 . Pro of : w e tak e the Laplae transform with resp et to the v ariable t in Eq. ( 24 ): ∂ τ e u ( τ , s ) = u 0 ( τ ) s e K ( s ) − e u ( τ , s ) e K ( s ) , (27) th us Eq. (26 ) is a solution, in the distributional sense, when u 0 ( τ ) = δ ( τ ) . Indeed the general solution ofEq. 27 ) with u 0 ( τ ) = δ ( τ ) is: ϕ ( τ , s ) = θ ( τ ) s e K ( s ) exp − τ e K ( s ) ! + C exp − τ e K ( s ) ! , τ ∈ R , where C is a real onstan t and where θ ( x ) =  1 , x ≥ 0 , 0 , x < 0 (28) is the Hea viside's step funtion. Sine w e require ϕ ( τ , t ) = 0 for τ < 0 , w e get C = 0 i.e. Eq. ( 26 ).  3.1 Suitabilit y onditions on the k ernel K W e m ust  ho ose the k ernel K su h that the fundamen tal solution of Eq. ( 24 ) is a probabilit y densit y in τ ≥ 0 . W e observ e that if f l ( τ , t ) satises Eq. (24) and Eq. (26), then it is automatially normalized for ea h t ≥ 0 . In fat, for a funtion ϕ ( x, t ) for whi h it is alw a ys p ossible to  hange the order of in tegration, one has: Z R ϕ ( x, t ) dx = 1 ⇐ ⇒ Z R e ϕ ( x, s ) dx = s − 1 . (29) Sine Eq. (26 ) satises the righ t-hand side of Eq. (29 ), w e get R R + f l ( τ , t ) dτ = 1 . One still needs, ho w ev er, to  ho ose the k ernel K su h that f l ( τ , t ) ≥ 0 for all τ , t ≥ 0 . In order to get a suitable ondition on the k ernel K , w e mak e use of the notion of ompletely monotone funtion. Reall that a funtion ϕ ( t ) is  ompletely monotone if it is non-negativ e and p ossesses deriv ativ es of an y order and: ( − 1) k d k dt k ϕ ( t ) ≥ 0 , t > 0 , k ∈ Z + = { 0 , 1 , 2 , . . . } . (30) W e observ e that as t → 0 , the limit of d k ϕ ( t ) /dt k ma y b e nite or innite. T ypial non-trivial examples are ϕ ( t ) = exp( − at ) , with a > 0 , ψ ( t ) = 1 / t and φ ( t ) = 1 / (1 + t ) . It is easy to sho w that if ϕ and ψ are ompletely monotone then their pro dut ϕψ is as w ell. Moreo v er, if ϕ is ompletely monotone and ψ is p ositiv e with rst deriv ativ e ompletely monotone then the funtion ϕ ( ψ ) is ompletely monotone. W e ha v e the follo wing  haraterization of ompletely monotone funtions [10℄: Lemma 3.1. A funtion ϕ ( s ) , dene d on the p ositive r e al line, is  ompletely monotone if and only if is of the form: ϕ ( s ) = Z ∞ 0 e − ts F ( dt ) , s ≥ 0 , wher e F is a nite or innite non-ne gative me asur e on the p ositive r e al semi-axis. Hene, to ensure that f l ( τ , t ) ≥ 0 for all τ , t ≥ 0 , it is enough to require that the funtion dened in Eq. (26) m ust b e ompletely monotone, as a funtion of s , for an y τ ≥ 0 , and th us that the k ernel K satises the follo wing: 7 Suitabilit y onditions 1. s e K ( s ) is p ositiv e with rst deriv ativ e ompletely monotone, 2. 1 / e K ( s ) is p ositiv e with rst deriv ativ e ompletely monotone. Indeed, w e an view Eq. 26 ) as the pro dut of the t w o ompletely monotone funtions 1 /u and exp( − τ u ) , the rst ev aluated at u = s e K ( s ) and the seond ev aluated at u = 1 / e K ( s ) . 3.2 Examples Example 3.1 ( β -p ower kernel) . If w e  ho ose: K ( t ) = t β − 1 Γ( β ) , w e get e K ( s ) = s − β . In this ase s e K ( s ) = s 1 − β is p ositiv e and has rst deriv ativ e (1 − β ) s − β ompletely monotone if and only if 0 < β ≤ 1 . Moreo v er, 1 / e K ( s ) = s β is p ositiv e with rst deriv ativ e β s β − 1 ompletely monotone if and only if 0 < β ≤ 1 . Therefore, a go o d  hoie for the k ernel K is: K ( t ) = t β − 1 Γ( β ) , 0 < β ≤ 1 . (31) Example 3.2 ( Exp onential-de  ay kernel) . Cho osing: K ( t ) = exp( − at ) , a ≥ 0 , (32) w e get s e K ( s ) = s/ ( s + a ) whi h is p ositiv e with rst deriv ativ e a ( s + a ) − 2 ompletely monotone for an y a ≥ 0 . Moreo v er, 1 / e K ( s ) = ( s + a ) is p ositiv e if a ≥ 0 with rst deriv ativ e ompletely monotone. Example 3.3 ( β -p ower with exp onential-de  ay kernel) . Cho osing: K ( t ) = t β − 1 Γ( β ) exp( − at ) , 0 < β ≤ 1 , a ≥ 0 , (33) w e ha v e e K ( s ) = ( s + a ) − β . Therefore, s e K ( s ) = s ( s + a ) − β whi h is p ositiv e if a ≥ 0 with rst deriv ativ e ( s + a ) − β (1 − β s ( s + a ) − 1 ) ompletely monotone if 0 < β ≤ 1 . Moreo v er, 1 / e K ( s ) = ( s + a ) β is p ositiv e if a ≥ 0 with rst deriv ativ e β ( s + a ) β − 1 ompletely monotone if 0 < β ≤ 1 . The follo wing theorem states that a self-similar random time pro ess l ( t ) , t ≥ 0 , is asso iated with the k ernel K ( t ) in Example 3.1 : Theorem 3.1. If the time-hange pr o  ess l ( t ) , t ≥ 0 , is self-similar (for instan e of or der H = β ), with mar ginal pr ob ability density f l ( τ , t ) satisfying Eq. (26 ), then we must have: K ( t ) = C t β − 1 Γ( β ) , 0 < β ≤ 1 , (34) for some p ositive  onstant C . Pro of : The self-similarit y ondition en tails that for an y τ , t ≥ 0 and for an y a > 0 : a − β f l ( a − β τ , t ) = f l ( τ , at ) . If w e tak e the Laplae transform and set e f ( τ , s ) = L{ f l ( τ , t ); t, s } , w e ha v e: a − β e f l ( a − β τ , s ) = 1 a e f l  τ , s a  . 8 Using Eq. 26 ) w e get that for an y τ , s ≥ 0 and a > 0 : a − β e K ( s ) exp − a − β τ e K ( s ) ! = 1 e K ( s a ) exp − τ e K ( s a ) ! . Sine this relation is v alid for an y  hoie of τ ≥ 0 and s ≥ 0 , putting τ = 0 and s = a , w e get: a − β e K ( a ) = 1 e K (1) . Th us, for an y a > 0 : e K ( a ) = e K (1) a − β , whi h is the Laplae transform ofEq. 34 ). If w e add moreo v er the ondition of omplete monotoniit y w e nd: 0 < β ≤ 1 as indiated in Example 3.1 .  4 Non-Mark o vian diusion equation W e fo us here on the non-Mark o vian diusion equation ( 15) in tro dued in the rst setion. There are t w o ingredien ts: 1. The fundamen tal solution ofEq. 24 ), denoted here b y h ( τ , t ) and dened b y Eq. (26 ). 2. The fundamen tal solution G ( x, t ) , dened b y Eq. (9), of the standard diusion equation whi h is the one-dimensional densit y of the standard Bro wnian motion. The follo wing theorem om bines these t w o ingredien ts and pro vides the fundamen tal solution of a orresp onding non-Mark o vian diusion equation. Theorem 4.1. L et h ( τ , t ) denote the fundamental solution of Eq. 24 ), so that by Pr op osition 3.1 , one has: L { h ( τ , t ); t, s } = 1 s e K ( s ) exp − τ e K ( s ) ! , τ , s ≥ 0 , (35) for a suitable hoi e of K . L et g b e a stritly inr e asing funtion with g (0) = 0 and let G ( x, t ) b e dene d by Eq. (9 ). Then, f ( x, t ) = Z ∞ 0 G ( x, τ ) h ( τ , g ( t )) dτ , (36) is the fundamental solution of the non-Markovian diusion e quation: u ( x, t ) = u 0 ( t ) + Z t 0 g ′ ( s ) K ( g ( t ) − g ( s )) ∂ xx u ( x, s ) ds. (37) Pro of : see Setion 6 .  W e ha v e immediately the follo wing: Corollary 4.1. If H ( x, t ) is a solution of the standar d diusion e quation with initial  ondition H ( x, 0 ) = u 0 ( x ) , then the funtion: u ( x, t ) = Z ∞ 0 H ( x, τ ) h ( τ , g ( t )) dτ (38) is a solution of Eq. (37 ). 9 Pro of : If, for an y t ≥ 0 , the funtion f ( x, t ) dened in Eq. (36 ) is the fundamen tal solution of Eq. (37 ) then a general solution is giv en b y: u ( x, t ) = Z R f ( x − y , t ) u 0 ( y ) dy = Z R Z ∞ 0 G ( x − y , τ ) u 0 ( y ) h ( τ , g ( t )) dτ dy = Z ∞ 0  Z R G ( x − y , τ ) u 0 ( y ) dy  h ( τ , g ( t )) dτ = Z ∞ 0 H ( x, τ ) h ( τ , g ( t )) dτ .  W e observ e that: 1. The equation (35 ) states that h ( τ , t ) is the fundamen tal solution of Eq. (24 ). 2. While G ( x, t ) is the fundamen tal solution of the standard diusion equation obtained when u 0 ( x ) = δ ( x ) , the general solution, denoted H ( x, t ) in the ab o v e theorem, results from a general initial ondition u 0 ( x ) . Man y ph ysial phenomena, esp eially related to relaxation pro esses in omplex systems, are desrib ed b y non- Mark o vian master equations lik e Eq. ( 37 ). K ( t ) is a memory k ernel and g ( t ) is just a time-saling funtion. Su h equations are often argued b y phenomenologial onsiderations and an b e more or less rigorously deriv ed starting from a mirosopi desription [7, 20, 47, 53℄. 5 The sto  hasti represen tation is not unique The solution of the non-Mark o vian diusion equation an b e view ed as the marginal densit y funtion of the sub ordinated pro ess, see Eq. (12) D ( t ) = B ( l ( g ( t ))) , t ≥ 0 , sine its marginal densit y is: f D ( x, t ) = Z ∞ 0 G ( x, τ ) f l ( τ , g ( t )) dτ . Here, for ea h t ≥ 0 , D ( t ) ∼ f D ( x, t ) , B ( t ) ∼ G ( x, t ) and l ( t ) ∼ f l ( τ , t ) . In the notation of Theorem 4.1, w e ha v e f D ( x, t ) = f ( x, t ) and f l ( τ , t ) = h ( τ , t ) . The Laplae transform of f l ( τ , t ) with resp et to t is giv en b y Eq. (35). This sto  hasti represen tation is not unique (see Example 1.1 , Example 1.2 and examples b elo w). Indeed, the non-Mark o vian diusion equation  haraterizes only the marginal, that is one-p oin t, probabilit y densit y funtion. Ho w ev er, pro esses with a dieren t dep endene struture an ha v e the same marginal densit y f ( x, t ) . A dditional requiremen ts ould b e imp osed so as to sp eify the sto  hasti mo del more preisely . Example 5.1. If w e require the random time pro ess l β ( t ) , t ≥ 0 , to b e self-similar of order β , then in view of Theorem 3.1 , the k ernel m ust b e  hosen as in Eq. (34 ) and w e m ust ha v e 0 < β ≤ 1 . W e will study this ase more in details in Setion 7 . Here w e just observ e that if w e onsider a standard frational Bro wnian motion B β / 2 of order β / 2 , then f ( x, t ) is also the marginal distribution of Y ( t ) = q l β (1) B β / 2 ( t ) , (39) where B β / 2 ( t ) is assumed to b e indep enden t of l β (1) . In fat, b eause l β ( t ) , t ≥ 0 , is self-similar of order H = β , one has: D ( t ) = B ( l β ( t )) = d q l β ( t ) B (1) = d q l β (1) t β / 2 B (1 ) = d q l β (1) t β / 2 B β / 2 (1) = d q l β (1) B β / 2 ( t ) = Y ( t ) , (40) where = d denotes here the equalit y of the marginal distributions. Both D ( t ) , t ≥ 0 , and Y ( t ) , t ≥ 0 , are self-similar pro esses with Hurst's exp onen t H = β / 2 . Ho w ev er, while Y ( t ) , t ≥ 0 , has alw a ys stationary inremen ts, this is not in general true in the ase of the pro ess D ( t ) , t ≥ 0 . 10 6 Non-Mark o vian F okk er-Plan k equation W e onsidered up un til no w pro esses of the t yp e B ( l ( g ( t ))) , where B is a standard Bro wnian motion. What happ ens if w e replae B b y a more general diusion? Namely , what happ ens if instead of starting with the standard diusion equation (2 ) w e start with a more general Mark o vian F okk er-Plan k equation: ∂ t u ( x, t ) = P x u ( x, t ) , x ∈ R , t ≥ 0 , (41) where P x is a linear op erator, indep enden t of t , ating on the v ariable x ? W e ha v e the follo wing generalization of Theorem 4.1 : Theorem 6.1. Supp ose that h ( τ , t ) is a pr ob ability density funtion satisfyingEq. 26 ) L { h ( τ , t ); t, s } = 1 s e K ( s ) exp − τ e K ( s ) ! , τ , s ≥ 0 , (42) for a suitable hoi e of K . L et g b e a stritly inr e asing funtion with g (0) = 0 and G ( x, t ) b e the fundamental solution of Eq. (41 ). Then the fundamental solution of the inte gr al e quation: u ( x, t ) = u 0 ( t ) + Z t 0 g ′ ( s ) K ( g ( t ) − g ( s )) P x u ( x, s ) ds (43) is f ( x, t ) = Z ∞ 0 G ( x, τ ) h ( τ , g ( t )) dτ . (44) W e pro vide t w o v ersions of the pro of. The rst starts with the solution f ( x, t ) in Eq. ( 44 ) and v eries that it satises Eq. (43). The seond starts from the partial in tegro-dieren tial equation ( 43) and deriv es the solution f ( x, t ) under ertain assumptions stated b elo w Eq. (49 ). Pro of 1 : F or rst w e observ e that L { f ( x, t ); g ( t ) , s } = 1 s e K ( s ) L {G ( x, t ); t, e K ( s ) − 1 } . (45) With the  hange of v ariables g ( s ) = z , w e write: u ( x, g − 1 ( w )) = u 0 ( x ) + Z w 0 K ( w − z ) P x u ( x, g − 1 ( z )) dz , w = g ( t ) . (46) W e w an t to sho w that Eq. ( 44) with the  hoie (42 ) solv es Eq. (43 ). If w e tak e the Laplae transform of Eq. (43) using Eq. (46 ), w e get: L { u ( x, t ); g ( t ) , s } = u 0 ( x ) s + e K ( s ) P x L { u ( x, t ); g ( t ) , s } that is: s L { u ( x, t ); g ( t ) , s } − u 0 ( x ) = s e K ( s ) P x L { u ( x, t ); g ( t ) , s } . (47) No w, if w e substitute on Eq. (47 ) a solution of the form (44 ), u ( x, t ) = Z ∞ 0 H ( x, τ ) h ( τ , g ( t )) dτ , (48) w e ha v e: e K ( s ) − 1 L {H ( x, t ); t, e K ( s ) − 1 } = u 0 ( x ) + P x L {H ( x, t ); t, e K ( s ) − 1 } 11 i.e. w e ha v e, with ob vious notations: τ e H ( x, τ ) = u 0 ( x ) + P x e H ( x, τ ) , in whi h one readily reognizes the Laplae transform of the Mark o vian F okk er-Plan k equation with the same initial ondition u 0 ( x ) . Therefore: ∂ t H ( x, t ) = P x H ( x, t ) , H ( x, 0) = u 0 ( x ) . This argumen t sho ws not only that Eq. (44 ) is the fundamen tal solution of Eq. (43 ), but also that a general solution is giv en b y Eq. (48 ) (see Corollary 4.1 ). This result is summarized in Corollary 6.1 (see b elo w). Pro of 2 : W e no w start from Eq.( 43 ) and w e use in tegral transforms in order to get the fundamen tal solu- tion. Let F denote the F ourier transform op erator and let: ( F ϕ )( k , t ) = b ϕ ( k, t ) = Z R e ikx ϕ ( x, t ) dx . Sine b u 0 ( k ) = 1 , and sine ( F P x u )( k , t ) = ( F P x F − 1 F u )( k , t ) = b P k b u ( k , t ) , where b P k = ( F P x F − 1 ) k denotes the F ourier transform of the op erator P x , w e ha v e: b u ( k , g − 1 ( w )) = 1 + Z w 0 K ( w − z ) b P k b u ( k , g − 1 ( z )) dz . T aking the Laplae transform w e ha v e: L { b u ( k , g − 1 ( w )); w , s } = s − 1 + b P k e K ( s ) L { b u ( k , g − 1 ( w )); w , s } , whi h is the same as: L { b u ( k , t ); g ( t ) , s } = s − 1 + b P k e K ( s ) L { b u ( k , t ); g ( t ) , s } . Therefore:  e K ( s ) − 1 − b P k  L { b u ( k , t ); g ( t ) , s } = s − 1 e K ( s ) − 1 . Denoting 1( k ) = 1 , w e ha v e: L { b u ( k , t ); g ( t ) , s } = 1 s e K ( s )  e K ( s ) − 1 − b P k  − 1 1( k ) , (49) where w e supp ose that the op erator  e K ( s ) − 1 − b P k  − 1 is w ell dened and ats on the onstan t funtion 1( k ) = 1 . Observ e that the F okk er-Plan k equation (41 ) is obtained from Eq. (43 ) b y setting K ( t ) = 1 , for ea h t ≥ 0 , that is e K ( s ) = s − 1 , and g ( t ) = t , for ea h t ≥ 0 . In this ase Eq. (49 ) b eomes: L { b G ( k , t ); t, s } = ( s − b P k ) − 1 1( k ) . (50) where G ( x, t ) is the fundamen tal solution. T aking the in v erse F ourier transform, w e get: L {G ( x, t ); t, s } = F − 1 n ( s − b P k ) − 1 1( k ) ; k , x o , (51) where: F − 1 { ϕ ( k, s ) ; k , x } = 1 2 π Z R e − ikx ϕ ( k , s ) dk . (52) Replaing s b y e K ( s ) − 1 in Eq. (51 ), one has: L {G ( x, t ); t, e K ( s ) − 1 } = F − 1 n ( e K ( s ) − 1 − b P k ) − 1 1( k ) ; k , x o . (53) 12 Going ba k to Eq. (49 ) and in v erting the F ourier transform w e obtain in view of Eq. (53 ): L { u ( x, t ); g ( t ) , s } = 1 s e K ( s ) F − 1   e K ( s ) − 1 − b P k  − 1 1( k ) ; k, x  = 1 s e K ( s ) L {G ( x, t ); t, e K ( s ) − 1 } . that is Eq. (45 ).  Remark 6.1. If the Mark o vian pro ess is a Bro wnian motion one has P x = ∂ 2 ∂ x 2 . The F ourier transform of P x is b P k = − k 2 and Eq. (49 ) b eomes: L { b u ( k , t ); g ( t ) , s } = 1 s e K ( s )  e K ( s ) − 1 + k 2  − 1 1( k ) , where  e K ( s ) − 1 + k 2  − 1 1( k ) = 1  e K ( s ) − 1 + k 2  , whi h is w ell dened b eause e K ( s ) − 1 is p ositiv e. Corollary 6.1. If H ( x, t ) is a gener al solution of the Markovian F okker-Plank e quation (41 ) with initial  ondition H ( x, 0) = u 0 ( x ) , then the funtion: u ( x, t ) = Z ∞ 0 H ( x, τ ) h ( τ , g ( t )) dτ (54) is a gener al solution of Eq. (43). >F rom a sto  hasti p oin t of view, f ( x, t ) ould b e seen as the marginal distribution at time t of the sub or- dinated pro ess: D ( t ) = Q ( l ( g ( t ))) (55) where Q is the diusion go v erned b y the F okk er-Plan k equation ( 41 ) and l ( t ) is the random time pro ess, indep enden t of Q ( t ) , with marginal distributions dened b y h ( τ , t ) . 7 Examples in v olving standard Bro wnian motion In the follo wing examples, w e onsider sto  hasti mo dels where the op erator P x in Eq. (41 ) is ∂ xx , namely the op erator orresp onding to standard Bro wnian motion. W e will study more general op erators in the next setion. W e shall  ho ose v arious k ernels K ( t ) and v arious stret hing funtions g ( t ) . W e let h ( τ , t ) denote the fundamen tal solution of the non-Mark o vian forw ard drift equation ( 24 ). Sine the orresp onding sto  hasti mo dels are not unique, w e will mainly fo us on the sub ordinated pro ess B ( l ( t )) , t ≥ 0 . Ho w ev er, w e also giv e examples of other appropriate sto  hasti mo dels. 7.1 Time-frational diusion equation Let g ( t ) = t . Consider the β - p ower k ernel: K ( t ) = t β − 1 Γ( β ) , 0 < β ≤ 1 , (56) and let h ( τ , t ) denote the fundamen tal solution of the non-Mark o vian forw ard drift equation (24 ) with k ernel (56). Remark 7.1. In view of Theorem 3.1 , su h a k ernel arises if one requires h ( τ , t ) to b e the marginal densit y funtion of a self-similar random time pro ess l ( t ) of order β . 13 InsertingEq. 56 ) in Eq. (37 ) w e obtain the follo wing equation: u ( x, t ) = u 0 ( t ) + 1 Γ( β ) Z t 0 ( t − s ) β − 1 ∂ xx u ( x, s ) ds , (57) whi h is sometimes alled the time-fr ational diusion e quation [27, 44℄. In view of Theorem 4.1, the fundamen tal solution is: f ( x, t ) = Z ∞ 0 G ( x, τ ) h ( τ , t ) dτ , where h ( τ , t ) satises: L { h ( τ , t ); t, s } = s β − 1 e − τ s β , τ , s ≥ 0 . (58) Su h a funtion h ( τ , t ) an b e expressed as: h ( τ , t ) = t − β M β ( τ t − β ) , (59) where M β ( r ) , is dened for 0 < β < 1 b y the p o w er series [24, 25℄: M β ( r ) = ∞ X k =0 ( − r ) k k !Γ [ − β k + (1 − β )] = 1 π ∞ X k =0 ( − r ) k k ! Γ [( β ( k + 1))] sin [ π β ( k + 1)] , r ≥ 0 . (60) The ab o v e series denes a transenden tal funtion (en tire of order 1 / (1 − β ) ) [12℄. 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 τ t=1 M β ( τ ) β =1/4 β =1/2 β =3/4 Figure 1: Plot of the densit y funtion h ( τ , t ) = t − β M ( τ t − β ) at time t = 1 , for dieren t v alues of the parameter β = [1 / 4 , 1 / 2 , 3 / 4] . 14 Remark 7.2. The funtion h ( τ , t ) in Eq. ( 59 ) represen ts the fundamen tal solution of the time-frational forw ard drift equation (see also [15℄): u ( τ , t ) = u 0 ( τ ) − 1 Γ( β ) Z t 0 ( t − s ) β − 1 ∂ τ u ( τ , s ) ds. (61) This equation redues to the standard drift equation when β → 1 . 7.1.1 Prop erties of the M -funtion It is useful to reall some imp ortan t prop erties of the M -funtion [12, 29℄. These are b est expressed in terms of the funtion M β ( τ , t ) = t − β M β ( τ t − β ) , (62) dened for an y τ , t ≥ 0 and 0 < β < 1 . 1. The Laplae transform of M β ( τ , t ) with resp et to t is: L {M β ( τ , t ); t, s } = s β − 1 e − τ s β , τ , s ≥ 0 . (63) 2. The ab o v e equation suggests that in the singular limit β → 1 one has: M 1 ( τ , t ) = δ ( τ − t ) , τ , t ≥ 0 . (64) 3. If β = 1 / 2 : M 1 / 2 ( τ , t ) = 1 √ π t exp( − τ 2 / 4 t ) , τ , t ≥ 0 . (65) 4. The M -funtion is a partiular ase of a F o x H -funtion [30, 44℄. W e indiate with M { ϕ ( x ); x, u } = Z ∞ 0 ϕ ( x ) x u − 1 dx, (66) the Mellin transform of a funtion ϕ ( x ) , x ≥ 0 , with resp et to x ev aluated in u ≥ 0 . The F o x H -funtion H m,n p,q ( z ) = H m,n p,q  z    ( a i , α i ) i =1 ,...,p ( b j , β j ) j =1 ,...,q  , is  haraterized b y its Mellin transform as follo ws: M { H m,n p,q ( z ); z , u } = A ( u ) B ( u ) C ( u ) D ( u ) , (67) with A ( u ) = m Y i =1 Γ( b j + β j u ) , B ( u ) = n Y j =1 Γ(1 − a j − α j u ) , C ( u ) = q Y i = m +1 Γ(1 − b j − β j u ) , D ( u ) = p Y j = n +1 Γ( a j + α j u ) . Here: 1 ≤ m ≤ q , 0 ≤ n ≤ p , α j , β j > 0 and a j , b j ∈ C (see [11, 33, 46℄ for more details). Starting from Eq. (63 ) and skipping to the Mellin transform, it is easy to sho w that w e ha v e the follo wing relation: M β ( τ , t ) = t − β H 1 , 0 1 , 1  τ t − β    (1 − β , β ) (0 , 1)  , τ , t ≥ 0 , 0 < β < 1 . (68) 15 5. Using the represen tation Eq. (68) and Eq. (67 ) w e ha v e for an y η , β ∈ (0 , 1 ) , see also [29℄: M ν ( x, t ) = Z ∞ 0 M η ( x, τ ) M β ( τ , t ) dτ , ν = η β x ≥ 0 . (69) The expression (59 ) for the funtion h ( τ , t ) follo ws from Eq. (63), that is: h ( τ , t ) = M β ( τ , t ) , τ , t ≥ 0 . (70) Moreo v er, when β → 1 ,Eq. 64 ) giv es h ( τ , t ) = δ ( τ − t ) as exp eted (see Remark 7.2 ). Comparing Eq. (9) and Eq. (65 ) one observ es that: G ( x, t ) = 1 2 M 1 / 2 ( | x | , t ) . (71) Using Theorem 4.1 and Eq. (69) together with Eq. (70 ) and Eq. (71 ) w e reo v er the fundamen tal solution of the time-frational diusion equation [27℄: f ( x, t ) = Z ∞ 0 G ( x, τ ) h ( τ , t ) d τ = 1 2 Z ∞ 0 M 1 / 2 ( | x | , τ ) M β ( τ , t ) dτ = 1 2 M β / 2 ( | x | , t ) = 1 2 t − β / 2 M β / 2 ( | x | t − β / 2 ) . (72) Sev eral plots of the M -funtion are presen ted: in Figure 1 the funtion h ( τ , t ) = M β ( τ , t ) is dra wn at a xed time t = 1 and for dieren t v alues of the parameter β ; in Figure 2 is presen ted the plot of f ( x, t ) = 1 2 M β / 2 ( | x | , t ) at a xed time t = 1 and for dieren t v alues of β ; in Figure 3 is sho wn the time ev olution of f ( x, t ) for xed β = 1 / 2 . −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 x t=1 f(x,t) β =1/4 β =1/2 β =3/4 β =1 Figure 2: Plot of the densit y funtion f ( x, t ) giv en b y Eq. (72) at time t = 1 , for dieren t v alues of the parameter β = [1 / 4 , 1 / 2 , 3 / 4 , 1] . F or β = 1 one reo v ers the standard Gaussian densit y ( 71 ). 16 −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x β =1/2 f(x,t) t=0.1 t=1 t=10 t=10 2 Figure 3: Plot of the densit y funtion f ( x, t ) for xed β = 1 / 2 , at dieren t times t = [0 . 1 , 1 , 10 , 10 2 ] . 7.1.2 Sto  hasti in terpretations of the solution F rom a sto  hasti p oin t of view, the funtion h ( τ , t ) in Eq. ( 59) an b e regarded as the marginal distribution of l β ( t ) , t ≥ 0 , where l β ( t ) , t ≥ 0 , is an H -ss random time with H = β . W e ha v e that for ea h in teger m ≥ 0 : E ( l β ( t ) m ) = m ! Γ( β m + 1) t β m . (73) In fat, from Eq. (58), for ea h in teger m ≥ 0 , w e ha v e : Z ∞ 0 τ m s β − 1 e − τ s β dτ = m ! s − mβ − 1 , whi h, in v erting the Laplae transform, giv es Eq. (73 ). F or instane, with the suitable on v en tions [8℄, l β ( t ) , t ≥ 0 , an b e view ed as the lo al time in zero at time t of a d = 2(1 − β ) -dimensional Bessel pro ess [37℄. The funtion f ( x, t ) in Eq. ( 72 ) is then the marginal densit y funtion of D ( t ) = B ( l β ( t )) , whi h is self-similar with H = β / 2 . In this ase, b eause l β ( t ) is self-similar of order β , w e immediately ha v e an example of a dieren t pro ess with the same marginal distribution of D ( t ) (see Example 5.1 ). In fat, if w e onsider a standard frational Bro wnian motion B β / 2 of order β / 2 , then f ( x, t ) an also b e seen as the marginal distribution of Y ( t ) = q l β (1) B β / 2 ( t ) , (74) 17 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −2 −1 0 1 2 t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 t 10 −2 10 −1 10 0 10 0 log(t) B(l 1/2 (t)) l 1/2 (t) log σ 2 (t) β =1/2 Figure 4: T ra jetory of the pro ess B ( l β ( t )) (top panel), with 0 < t < 1 and β = 1 / 2 . The random time pro ess is  hosen to b e l 1 / 2 ( t ) = | b ( t ) | where b ( t ) is a standard Bro wnian motion (see Example 1.1 ). The orresp onding tra jetory of the random time pro ess is presen ted in the middle panel. The estimated v ariane, omputed on a sample of dimension N = 500 0 , is presen ted in logarithmi sale in the b ottom panel and ts p erfetly the theoretial urv e 2 t 1 / 2 / Γ(3 / 2) . where B β / 2 ( t ) is assumed to b e indep enden t of l β (1) (see Example 5.1 ). The pro ess Y ( t ) , t ≥ 0 , is alled gr ey Br ownian motion [43℄. F rom Eq. (73 ) one an deriv e immediately all the momen ts for the pro esses D ( t ) and Y ( t ) . F or an y in teger m ≥ 0      E ( D ( t ) 2 m +1 ) = E ( Y ( t ) 2 m +1 ) = 0; E ( D ( t ) 2 m ) = E ( Y ( t ) 2 m ) = 2 m ! Γ( β m + 1) t β m . (75) Beause 0 < β < 1 , the v ariane gro ws slo w er than linearly with resp et to time. In this ase one sp eaks ab out slow anomalous diusion . Moreo v er, the inremen ts of the frational Bro wnian motion B β / 2 ( t ) do not ha v e long-range dep endene. In on trast, the next example allo ws for the presene of long-range dep endene through the in tro dution of a saling funtion g ( t ) = t α/β (see also Example 1.2 ). 7.2 Stret hed time-frational diusion equation If in the setup of Setion 7.1 , where the k ernel K ( t ) is giv en b y Eq. ( 56 ), w e in tro due a saling time g ( t ) = t α/β 18 −6 −4 −2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 x f(x,t) β =1/2 t=1 Figure 5: Marginal densit y funtion f ( x, t ) = 1 2 M 1 / 4 ( | x | , t ) of the pro ess B ( l 1 / 2 ( t )) at time t = 1 and x ∈ [ − 5 , 5 ] . The histogram is ev aluated o v er N = 10 4 sim ulated tra jetories of the pro ess B ( | b ( t ) | ) (Figure 4). with α > 0 , then the in tegral equation (57 ) is replaed b y Eq. (37 ), namely u ( x, t ) = u 0 ( t ) + 1 Γ( β ) α β Z t 0 s α β − 1  t α β − s α β  β − 1 ∂ xx u ( x, s ) ds. (76) Therefore, using Eq. 59 ): h ( τ , g ( t )) = g ( t ) − β M β ( τ g ( t ) − β ) = t − α M β ( τ t − α ) , and, using Eq. (72), the fundamen tal solution f ( x, t ) of Eq. (76 ) reads: f ( x, t ) = f ( x, g ( t )) = 1 2 t − α/ 2 M β / 2 ( | x | t − α/ 2 ) , t ≥ 0 . (77) The funtion f ( x, t ) , t ≥ 0 , is the marginal distribution of the pro ess D ( t ) = B  l β ( t α/β ))  , t ≥ 0 . The time- hange pro ess l β ( t α/β ) is self-similar of order H = α and the pro ess D ( t ) is then self-similar with H = α/ 2 . In the ase 0 < α < 2 , the funtion f ( x, t ) is also the marginal densit y of Y ( t ) = q l β (1) B α/ 2 ( t ) , t ≥ 0 , 0 < α < 2 , (78) where B α/ 2 ( t ) is a standard fBm of order H = α/ 2 indep enden t of l β (1) . The pro ess Y ( t ) , t ≥ 0 , is alled gener alize d gr ey Br ownian motion [38℄. 19 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1.5 −1 −0.5 0 0.5 t 10 −1 10 0 10 −2 10 0 log(t) −5 −4 −3 −2 −1 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 x (l 1/2 (1)) 1/2 B 3/4 (t) σ 2 (t) f(x,t) β =1/2 α =3/2 t=1 Figure 6: T ra jetory of the pro ess p l β (1) B α/ 2 ( t ) (top panel), with 0 < t < 1 , β = 1 / 2 and α = 3 / 2 . The random v ariable l 1 / 2 (1) is Gaussian, see Eq. (65). The estimated v ariane, omputed on a sample of dimension N = 50 00 , is presen ted in logarithmi sale in the middle panel together with the theoretial urv e 2 t 3 / 2 / Γ(3 / 2) . In the b ottom panel the histogram, ev aluated o v er a sample of N = 10 4 tra jetories, ts the exat marginal densit y Eq. (77 ) at time t = 1 . In this ase, for an y in teger m ≥ 0 :      E ( D ( t ) 2 m +1 ) = E ( Y ( t ) 2 m +1 ) = 0; E ( D ( t ) 2 m ) = E ( Y ( t ) 2 m ) = 2 m ! Γ( β m + 1) t αm . (79) W e ha v e slo w diusion when 0 < α < 1 (the v ariane gro ws slo w er than linearly in time) and fast diusion when 1 < α < 2 (the v ariane gro ws faster than linearly in time). In this ase the inremen ts of the pro ess Y ( t ) exhibit long-range dep endene. 7.3 Exp onen tial-dea y k ernel Let g ( t ) = t . With the exp onen tial-dea y k ernel: K ( t ) = e xp( − at ) , a ≥ 0 , t ≥ 0 , (80) w e obtain the follo wing equation: u ( x, t ) = u 0 ( x ) + Z t 0 e − a ( t − s ) ∂ xx u ( x, s ) ds. (81) In this ase e K ( s ) = ( s + a ) − 1 and the marginal distribution of the random time pro ess l ( t ) , t ≥ 0 , is dened b y Eq. (26 ): L { f l ( τ , t ); t, s } = s + a s e − τ ( s + a ) , τ ≥ 0 . 20 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 τ 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 1.5 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 t=0.5 t=1 t=1.5 f l ( τ ,t) a=1 Figure 7: Plots of the marginal densit y of the random time f l ( τ , t ) Eq. 82 ) as a funtion of τ at times t = [0 . 5 , 1 , 1 . 5] , and with a = 1 . The v ertial line orresp onds to a p oin t mass (delta funtion). Therefore, f l ( τ , t ) = e − τ a ( δ ( τ − t ) + aθ ( t − τ )) = e − ta δ ( τ − t ) + ae − τ a θ ( t − τ ) , (82) where θ ( x ) is the step funtion (28). A graphial represen tation of the time ev olution of f l ( τ , t ) is presen ted in Figure 7. Remark 7.3. The funtion f l ( τ , t ) dened in Eq. 82 ) is the fundamen tal solution, in the sense of distributions, of the exp onential forwar d drift equation: u ( τ , t ) = u 0 ( τ ) − Z t 0 e − a ( t − s ) ∂ τ u ( τ , s ) ds. This follo ws from Prop osition 3.1 . T o  he k it diretly w e note that f l ( τ , 0) = δ ( τ ) and for an y t > 0 : − Z t 0 e − a ( t − s ) ∂ τ f l ( τ , s ) ds = − Z t 0 e − a ( t − s ) ∂ τ  e − as δ ( τ − s ) + ae − aτ θ ( s − τ )  ds = − Z t 0 e − a ( t − s )  − e − as δ ′ ( τ − s ) + ae − aτ δ ( s − τ ) − a 2 e − aτ θ ( s − τ )  ds = e − at δ ( τ − t ) + ae − at θ ( t − τ ) + ae − a ( t + τ ) θ ( t − τ )( e at − e aτ ) = e − at δ ( τ − t ) + ae − aτ θ ( t − τ ) = f l ( τ , t ) , where w e ha v e used the fat that: Z t 0 δ ′ ( τ − s ) ds = δ ( t − τ ) . W e observ e that when a → 0 w e reo v er the forw ard drift equation (5) and indeed f l ( τ , t ) = δ ( τ − t ) . 21 As noted in Example 3.2 , Eq. (82 ) atually denes a probabilit y densit y for an y t ≥ 0 . The follo wing prop osition pro vides its momen ts. Prop osition 7.1. F or e ah inte ger m ≥ 0 one has: E ( l ( t ) m ) = m ! a m  1 − e − at  + e − at t m − m X k =1 m ! k ! t k a k − m ! . (83) Pro of : for an y t ≥ 0 , w e m ust ev aluate: Z ∞ 0 τ m f l ( τ , t ) dτ = e − at t m + a Z t 0 τ m e − aτ dτ , where w e ha v e used Eq. (82 ). In order to ev aluate the exp onen tial in tegral in the ab o v e equation w e write: a Z t 0 τ m e − aτ dτ = ( − 1 ) m a∂ m a  (1 − e − at )( a − 1 )  = ( − 1) m a m X k =0  m k  ∂ k a (1 − e − at ) ∂ m − k a ( a − 1 ) = m X k =0 ( − 1) k m ! k ! a k − m ∂ k a (1 − e − at ) = m ! a m (1 − e − at ) − m X k =1 m ! k ! t k a k − m e − at th us one has Eq. 83).  The funtion f l ( τ , t ) an b e written: f l ( τ , t ) = e − at δ ( τ − t ) + (1 − e − at ) ϕ ( τ , t ) , τ , t ≥ 0 , a ≥ 0 , (84) where: ϕ ( τ , t ) = a e − aτ θ ( t − τ ) 1 − e − at , τ , t ≥ 0 , a ≥ 0 . (85) Beause f l ( τ , t ) is a probabilit y densit y , then so is ϕ ( τ , t ) . The orresp onding random time pro ess l ( t ) , t ≥ 0 , an then b e  hosen to b e: l ( t ) = b t t + (1 − b t ) j ( t ) , t ≥ 0 , (86) where b t , t ≥ 0 , is a sto  hasti pro ess su h that, for an y xed t ≥ 0 , b t is a Bernoulli random v ariable with P r ( b t = 1) = e − at and P r ( b t = 0) = 1 − e − at , and j ( t ) , t ≥ 0 , is a sto  hasti pro ess, indep enden t of b t , with marginal distribution giv en b y ϕ ( τ , t ) . Remark 7.4. The random time l ( t ) dened b y Eq. ( 86) annot b e inreasing ev erywhere. This is due to the fat that b t and j ( t ) are indep enden t and P r ( j ( t ) < t ) = 1 for an y t ≥ 0 . Indeed, supp ose that l ( t ) is inreasing. This implies that for an y t ≥ 0 and ǫ > 0 : 1 = P r ( l ( t + ǫ ) ≥ l ( t )   b t = 1 ) = P r ( l ( t + ǫ ) ≥ t ) = P r ( l ( t + ǫ ) ≥ t   b t + ǫ = 1 ) P r ( b t + ǫ = 1 ) + P r ( l ( t + ǫ ) ≥ t   b t + ǫ = 0) P r ( b t + ǫ = 0) = e − a ( t + ǫ ) +  1 − e − a ( t + ǫ )  P r ( j ( t + ǫ ) ≥ t ) = 1 −  1 − e − a ( t + ǫ )  P r ( j ( t + ǫ ) < t ) therefore taking ǫ → 0 w e get 1 = e − at with a, t ≥ 0 , whi h is a on tradition as so on as a 6 = 0 and t > 0 . On the other hand, a trivial example of an inreasing pro ess with marginal distribution giv en b y Eq. (82 ) is: l ( t ) = min( X , t ) , t ≥ 0 , (87) where X is an exp onen tially distributed random v ariable: X ∼ ae − aτ , τ ≥ 0 . W e no w turn to Eq. 81 ). W e ha v e the follo wing result: 22 Prop osition 7.2. The fundamental solution of Eq. ( 81 ) is: f ( x, t ) = e − at G ( x, t ) + (1 − e − at ) φ ( x, t ) , (88) with: φ ( x, t ) = √ a 4(1 − e − at )  e x √ a Erf  x 2 √ t + √ at  − e − x √ a Erf  x 2 √ t − √ at  − 2 sinh( | x | √ a )  , (89) wher e Erf ( x ) = 2 √ π R x 0 e − y 2 dy and wher e Erf ( − x ) = − Erf ( x ) . −4 −3 −2 −1 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x f(x,t) a=2 a=1 a=0.1 a=0 t=1 Figure 8: Plot of the fundamen tal solution f ( x, t ) , Eq. (88), at time t = 1 , for dieren t v alues of the parameter a = [0 , 0 . 1 , 1 , 2] . When a = 0 w e ha v e the standard Gaussian densit y . Pro of : b y Theorem 4.1 andEq. 85 ), the fundamen tal solution of Eq. ( 81 ) is: f ( x, t ) = Z ∞ 0 G ( x, τ ) f l ( τ , t ) dτ = e − at G ( x, t ) + (1 − e − at ) φ ( x, t ) , (90) where φ ( x, t ) = Z ∞ 0 G ( x, t ) ϕ ( τ , t ) dτ . W e ha v e that: φ ( x, t ) = a 1 − e − at Z ∞ 0 G ( x, τ ) e − aτ θ ( t − τ ) dτ . One has to ev aluate: χ ( x, t ) = Z t 0 e − x 2 / 4 τ e − aτ √ 4 π τ dτ , x ∈ R , t ≥ 0 . (91) 23 First w e observ e that: χ (0 , t ) = Z t 0 e − aτ √ 4 π τ dτ = 1 2 √ a 2 √ π Z √ at 0 e − y 2 dy = 1 2 √ a Erf ( √ at ) . after the  hange of v ariables y = √ aτ . Beause Erf ( − u ) = − Erf ( u ) w e an write: χ (0 , t ) = 1 4 √ a n Erf ( √ at ) − Erf ( − √ at ) o . (92) No w, for an y x ∈ R : χ ( x, t ) = 1 4 √ a  e x √ a Erf  x 2 √ t + √ at  − e − x √ a Erf  x 2 √ t − √ at  − 1 2 √ a sinh( | x | √ a ) , (93) b eause: d dτ  1 4 √ a  e x √ a Erf  x 2 √ τ + √ aτ  − e − x √ a Erf  x 2 √ τ − √ aτ  = 1 4 √ a ( 2 √ π e x √ a exp −  x 2 √ τ + √ aτ  2 !  − x 4 τ − 3 / 2 + √ a 2 √ τ  ) − 1 4 √ a ( 2 √ π e − x √ a exp −  x 2 √ τ − √ aτ  2 !  − x 4 τ − 3 / 2 − √ a 2 √ τ  ) = 1 √ 4 π τ exp  − x 2 4 τ − aτ  . Moreo v er, b eause Erf ( ±∞ ) = ± 1 , w e ha v eEq. 93 ), whi h atually redues to Eq. (92 ) when x = 0 . Therefore, the fundamen tal solution of Eq. (81 ) is: f ( x, t ) = e − at G ( x, t ) + √ a 4  e x √ a Erf  x 2 √ t + √ at  − e − x √ a Erf  x 2 √ t − √ at  − √ a 2 sinh( | x | √ a ) , (94) whi h an b e rewritten as Eq. (88).  Remark 7.5. With the  hoie (86) the pro ess: B ( l ( t )) = B ( b t t + (1 − b t ) j ( t )) , t ≥ 0 , has marginal densit y (88 ). Observ e that, for an y t ≥ 0 : B ( b t t + (1 − b t ) j ( t )) = d b t B ( t ) + (1 − b t ) B ( j ( t )) , whi h naturally orresp onds to Eq. 88 ). Remark 7.6. W e observ e that Eq. ( 88 ) redues to G ( x, t ) when a = 0 (see Figure 8), whi h is as exp eted b eause the memory k ernel disapp ears. F or small times, the non-lo al memory eets are negligible and the pro ess app ears Mark o vian. Fig. 8 displa ys the fundamen tal solution at xed t and v arious v alues of a , whereas Fig. 9 displa ys the fundamen tal solution at xed a and v arious v alues of t . F or large times w e ha v e: lim t →∞ f ( x, t ) = lim t →∞ φ ( x, t ) = φ ( x ) , where: φ ( x ) = √ a 2 (cosh( x √ a ) − sinh( | x | √ a )) = √ a 2 e −| x | √ a , x ∈ R . (95) 24 −4 −3 −2 −1 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 x t=0.5 t=1 t=1.5 φ (x) f(x,t) a=1 Figure 9: Plot of the fundamen tal solution f ( x, t ) , Eq. (88 ), at time t = [0 . 5 , 1 , 1 . 5] , and a = 1 . The dashed line represen ts the asymptoti distribution φ ( x ) , Eq. ( 95). Remark 7.7. In view of Eq. (82), it is alw a ys p ossible to  ho ose the random time pro ess l ( t ) , t ≥ 0 , su h that it b eomes stationary at large times, in the sense of nite-dimensional densities. With this  hoie, the sub ordinated pro ess B ( l ( t )) tends to a stationary pro ess with asymptoti marginal distribution giv en b y Eq. (95). F or instane, if w e lo ok at Eq. (87 ), as t → ∞ w e ha v e l ( t ) = X ∼ ae − aτ , τ ≥ 0 . A less trivial example an b e onstruted b y replaing the random v ariable X with a stationary pro ess X ( t ) , t ≥ 0 , su h that for ea h t ≥ 0 the random v ariable X ( t ) has an exp onen tial distribution with mean E ( X ( t )) = a − 1 . The resulting pro ess l ( t ) = min( X ( t ) , t ) is not inreasing, has marginal distribution dened b y Eq. ( 82) and tends to X ( t ) for large t . See Fig. 10. Remark 7.8. T o obtain an idea on ho w fast the stationary regime is rea hed, one an lo ok at the v ariane of the sub ordinated pro ess. Using Eq. ( 83 ) with m = 1 , w e nd: E ( B ( l ( t )) 2 ) = 2 a (1 − e − at ) , (96) whi h, for large times, tends exp onen tially to 2 /a (i.e. the v ariane of eq. 95 ). 7.4 Exp onen tial-dea y k ernel with logarithmi saling time What happ ens if w e  ho ose an exp onen tial k ernel K ( t ) = e − at and a logarithmi saling time? That is: g ( t ) = lo g( t + 1) , t ≥ 0 . (97) Sine g ′ ( t ) K ( g ( t ) − g ( s )) = ( t + 1) − a ( s + 1) a − 1 , w e get: u ( x, t ) = u 0 ( x ) + 1 ( t + 1) a Z t 0 ( s + 1) a − 1 ∂ xx u ( x, s ) ds. (98) 25 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 t t t sample true 0 1 2 3 4 5 6 7 8 9 10 −2 0 2 4 0 1 2 3 4 5 6 7 8 9 10 0 2 4 6 σ 2 (t) l(t) B(l(t)) a=1 Figure 10: T ra jetory of the pro ess B ( l ( t )) (top panel), with 0 < t < 10 , E ( B (1)) = 1 , l ( t ) = min( t, X ( t )) where X ( t ) is an exp onen tial White Noise with mean one. The orresp onding tra jetory of the random time l ( t ) pro ess is presen ted in the middle panel. The estimated v ariane is omputed on a sample of dimension N = 50 0 . The smo oth bla k line in the b ottom panel orresp onds to σ 2 ( t ) giv en b y Eq. (96 ) and the stationary v alue is lim t →∞ σ 2 ( t ) = 1 . Its fundamen tal solution is: f ( x, t ) = 1 ( t + 1) a G ( x, log ( t + 1 )) − √ a 2 sinh( | x | √ a ) + √ a 4 ( e x √ a Erf x 2 p log( t + 1) + p a log( t + 1) ! − e − x √ a Erf x 2 p log( t + 1) − p a log( t + 1) !) (99) Remark 7.9. As in Remark 7.7 , onsider a random time pro ess l ( t ) , t ≥ 0 , with marginal distribution dened b y Eq. (82 ), that b eomes stationary for large times. The sub ordinated pro ess B ( l (log ( t + 1))) , t ≥ 0 , has marginal densit y funtion dened b y f ( x, t ) of Eq. (99 ). Observ e that in this ase the random time pro ess l (log( t + 1)) is no longer asymptotially stationary . This is b eause the translational time-in v ariane is brok en b y the logarithmi transformation. Ho w ev er, w e an alw a ys onsider a random time pro ess l ∗ ( t ) , t ≥ 0 , with the same marginal distribution of l (lo g ( t + 1)) , whi h b eomes stationary for large times. Th us, the pro ess B ( l ∗ ( t )) still has a marginal densit y funtion dened b y f ( x, t ) but b eomes stationary as t → ∞ , in the sense of nite-dimensional distribution, with asymptoti marginal distribution giv en b y Eq. ( 95 ). See Fig. 11. Remark 7.10. While B ( l ( t )) , t ≥ 0 , satises Eq. ( 96 ) and th us has a v ariane whi h tends exp onen tially fast to the limit v alue 2 /a , here the stationary regime is rea hed more slo wly . Indeed, the v ariane of the sub ordinated 26 pro ess is: E ( B ( l ∗ ( t )) 2 ) = 2 a  1 − 1 ( t + 1) a  , (100) whi h, for large times, on v erges to the stationary v alue 2 /a with a p o w er-lik e b eha vior. 0 1 2 3 4 5 6 7 8 9 10 0 0.5 1 1.5 t t t 0 1 2 3 4 5 6 7 8 9 10 −1 −0.5 0 0.5 1 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 sample true σ 2 (log(t+1)) l(log(t+1)) B(l(log(t+1))) Figure 11: T ra jetory of the pro ess B ( l (log ( t + 1))) (top panel), with 0 < t < 10 , E ( B (1)) = 1 , l ( t ) = min( t, X ( t )) where X ( t ) is an exp onen tial White Noise with mean one. The orresp onding tra jetory of the random time pro ess l (log( t + 1)) is presen ted in the middle panel. The estimated v ariane is omputed on a sample of dimension N = 500 . The smo oth bla k line in the b ottom panel orresp onds to Eq. ( 96). The stationary v alue is lim t →∞ σ 2 (log( t + 1)) = 1 . The stationary regime is a hiev ed more slo wly than in the ase of Figure 10 . 8 Examples in v olving other diusions W e shall no w onsider examples of frational and stret hed F okk er-Plan k equations in v olving diusion op erators other than P x = ∂ xx whi h orresp onds to standard Bro wnian motion. W e will  ho ose K ( t ) = t β − 1 / Γ( β ) and g ( t ) = t α/β as in Setion 7.1 and onsider the partial in tegro-dieren tial equation: u ( x, t ) = u 0 ( x ) + 1 Γ( β ) α β Z t 0 s α β − 1  t α β − s α β  β − 1 P x u ( x, s ) ds, 0 < β ≤ 1 , α > 0 . (101) Its fundamen tal solution is the marginal densit y of the pro ess: D ( t ) = Q ( l β ( t α/β )) , (102) 27 where Q ( t ) , t ≥ 0 , is the sto  hasti diusion asso iated to P x and l β ( t ) , t ≥ 0 , is a suitable self-similar random time pro ess. One has the follo wing partiular ases: • When α = β and 0 < β ≤ 1 , Eq. (101 ) b eomes the time-fr ational F okker-Plank e quation : u ( x, t ) = u 0 ( x ) + 1 Γ( β ) Z t 0 ( t − s ) β − 1 P x u ( x, s ) ds, 0 < β ≤ 1 , (103) whose fundamen tal solutions are the marginal distributions of the pro ess: D ( t ) = Q ( l β ( t )) , and are giv en b y: f D ( x, t ) = Z ∞ 0 f Q ( x, τ ) f l β ( τ , t ) dτ , (104) where f l β ( τ , t ) = t − β M β ( τ t − β ) , τ , t ≥ 0 , 0 < β ≤ 1 (105) and f Q ( x, t ) is the probabilit y densit y of Q ( t ) . • When β = 1 and α > 0 w e get a time-str ethe d F okker-Plank e quation : u ( x, t ) = u 0 ( x ) + Z t 0 αs α − 1 P x u ( x, s ) ds, 0 < β ≤ 1 . (106) In this ase f l ( τ , t ) = δ ( τ − t α ) and w e get: f D ( x, t ) = f Q ( x, t α ) , whi h orresp onds to the stret hed diusion: D ( t ) = Q ( t α ) , α > 0 . • The ase α = β = 1 is trivial and orresp onds merely to the Mark o vian ase where the equation is: u ( x, t ) = u 0 ( x ) + Z t 0 P x u ( x, s ) ds whose fundamen tal solution is the densit y funtion of D ( t ) = Q ( t ) , namely the Mark o vian pro ess. In the follo wing subsetions w e study the ab o v e equations under partiular  hoies of the F okk er-Plan k op erator P x . In all the ases onsidered, w e also giv e the results in v olving the exp onen tial-dea y k ernel. 8.1 Bro wnian motion with drift Let µ ∈ R and σ 2 > 0 b e giv en. Consider a linear diusion B ( µ ) = B ( µ,σ ) on R satisfying the sto  hasti dieren tial equation: dB ( µ ) ( t ) = µdt + σ dB ( t ) , t ≥ 0 , (107) where B ( t ) , t ≥ 0 , is a standard Bro wnian motion. The pro ess B ( µ ) ( t ) , t ≥ 0 , is alled Br ownian motion with drift µ . It orresp onds merely to a Bro wnian motion plus a drift term, namely: B ( µ ) ( t ) = µt + σB ( t ) , t ≥ 0 . (108) The marginal densit y funtion of B ( µ ) (t), t ≥ 0 , is: f B ( µ ) ( x, t ) = 1 | σ | √ 4 π t exp  − ( x − µt ) 2 σ 2 4 t  , t ≥ 0 , x ∈ R , (109) whi h is the fundamen tal solution of the F okk er-Plan k equation: ∂ t u ( x, t ) = − µ∂ x u ( x, t ) + σ 2 ∂ xx u ( x, t ) , t ≥ 0 . (110) 28 8.1.1 The β -p o w er k ernel. W e onsider the fr ational F okker-Plank e quation , see Eq. (103 ) (see also [35℄): u ( x, t ) = u 0 ( x ) + 1 Γ( β ) Z t 0 ( t − s ) β − 1 ( − µ∂ x u ( x, s ) + σ 2 ∂ xx u ( x, s )) ds, 0 < β ≤ 1 . (111) Its fundamen tal solution an b e regarded as the marginal densit y funtion of the pro ess: D ( t ) = B ( µ ) ( l β ( t )) , t ≥ 0 , 0 < β ≤ 1 , (112) where the pro ess l β ( t ) , t ≥ 0 , is a self-similar random time pro ess with parameter H = β / 2 , indep enden t of B ( µ ) , su h that its marginal distribution is giv en b y Eq. (105 ). Prop osition 8.1. The fundamental solution of Eq. ( 111 ) is: f D ( x, t ) = Z ∞ 0 f B ( µ ) ( x, τ ) f l β ( τ , t ) dτ , t ≥ 0 , x ∈ R , i.e. f D ( x, t ) = Z ∞ 0 1 | σ | √ 4 π τ exp  − ( x − µτ ) 2 4 σ 2 τ  M β ( τ , t ) dτ , t ≥ 0 , x ∈ R , (113) whih is e qual to: f D ( x, t ) = e µx/ 2 σ 2 1 2 | σ | ∞ X k =0 ( − µ 2 t β / 4 σ 2 ) k k ! t − β / 2 H 2 , 0 2 , 2  | xσ − 1 | t − β / 2    (1 / 2 , 1 / 2) , (1 − β / 2 + β k , β / 2) (0 , 1) , ( k + 1 / 2 , 1 / 2)  , (114) wher e the F ox H -funtion is dene d by Eq. ( 67). Pro of : In order to ev aluate f D ( x, t ) w e write: f D ( x, t ) = | σ | − 1 e µx ′ / 2 σ Z ∞ 0 e − µ 2 τ / 4 σ 2 G ( x ′ , τ ) M β ( τ , t ) dτ , where G ( x, t ) is the standard Gaussian densit y , see Eq. (9 ) and x ′ = x/σ . In view of Eq. (71 ), w e ha v e to ev aluate an in tegral of the form: Φ( x, t ) = 1 2 Z ∞ 0 e − aτ M 1 / 2 ( | x | , τ ) M β ( τ , t ) dτ , x ∈ R , t ≥ 0 , a ≥ 0 . (115) One has: Φ( x, t ) = 1 2 Z ∞ 0 e − aτ τ − 1 / 2 M 1 / 2 ( | x | τ − 1 / 2 ) t − β M β ( τ t − β ) dτ = 1 2 Z ∞ 0 1 y M 1 / 2  | x | y  2 y e − ay 2 t − β M β ( y 2 t − β ) dy . after the  hange of v ariables y = √ τ . Beause of the symmetry , it is enough to onsider only the ase x ≥ 0 . W e get: Φ( x, t ) = 1 2 ( M 1 / 2 ⋆ Y t )( x ) , x ≥ 0 , where ( ϕ ⋆ φ )( x ) = Z ∞ 0 1 y ϕ  x y  φ ( y ) dy indiates the Mellin on v olution and where: Y t ( x ) = 2 xe − ax 2 t − β M β ( x 2 t − β ) , x ≥ 0 , t ≥ 0 . (116) 29 −6 −4 −2 0 2 4 6 0 0.1 0.2 0.3 0.4 x −6 −4 −2 0 2 4 6 0 0.1 0.2 0.3 0.4 x β =1/2 t=1 f(x,t) f(x,t) µ =1.5 σ =1 µ =2 σ =1 σ =1.5 σ =2 µ =1 µ =1 β =1/2 t=1 Figure 12: Plot of the fundamen tal solution f ( x, t ) , Eq. (114 ), with β = 1 / 2 , at time t = 1 , for dieren t v alues of the parameters µ = [1 , 1 . 5 , 2] and σ = [1 , 1 . 5 , 2] . Using the Mellin on v olution theorem w e get: M { 2Φ( x, t ); x, u } = M { M 1 / 2 ( x ); x, u } M { Y t ( x ); x, u } . (117) Beause ofEq. 68 ) andEq. 67 ), this an b e written as: M { 2Φ( x, t ); x, u } = Γ( u ) Γ(1 / 2 + u / 2) M { Y t ( x ); x, u } . (118) W e no w ev aluate: M { Y t ( x ); x, u } = Z ∞ 0 e − ax 2 2 xt − β M β ( x 2 t − β ) x u − 1 dx. After the  hange of v ariables x 2 t − β = z , w e get M { Y t ( x ); x, u } = Z ∞ 0 ( z t β ) 1 2 ( u − 1) e − az t β M β ( z ) dz = t β 2 ( u − 1) ∞ X k =0 ( − at β ) k k ! Z ∞ 0 z k − 1 2 + u 2 M β ( z ) dz = t β 2 ( u − 1) ∞ X k =0 ( − at β ) k k ! M { M β ( x ); x, k + 1 / 2 + u/ 2 } = t β 2 ( u − 1) ∞ X k =0 ( − at β ) k k ! Γ(1 / 2 + k + u/ 2 ) Γ(1 + β k − β / 2 + β u/ 2) , 30 −6 −4 −2 0 2 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 x β =1/2 t=0.1 f(x,t) σ =1 t=2 µ =1 t=1 Figure 13: Plot of the fundamen tal solution f ( x, t ) , Eq. (114 ) with β = 1 / 2 , µ = 1 , σ = 1 , at times t = [0 . 1 , 1 , 2] . where w e ha v e used Eq. (68 ) and Eq. (67 ). Th us: M { 2Φ( x, t ); x, u } = ∞ X k =0 ( − at β ) k k ! t β 2 ( u − 1) Γ( u )Γ(1 / 2 + k + u/ 2 ) Γ(1 / 2 + u / 2)Γ(1 + β k − β / 2 + β u/ 2) . In v erting the Mellin transform,Eq. 67 ) giv es: Φ( x, t ) = 1 2 ∞ X k =0 ( − at β ) k k ! t − β / 2 H 2 , 0 2 , 2  | x | t − β / 2    (1 / 2 , 1 / 2) , (1 − β / 2 + β k , β / 2 ) (0 , 1) , ( k + 1 / 2 , 1 / 2)  , (119) with x ∈ R and t ≥ 0 . Therefore, the fundamen tal solution of Eq. ( 103 ) an b e expressed as: f D ( x, t ) = e µx/ 2 σ 2 1 2 | σ | ∞ X k =0 ( − µ 2 t β / 4 σ 2 ) k k ! t − β / 2 H 2 , 0 2 , 2  | xσ − 1 | t − β / 2    (1 / 2 , 1 / 2) , (1 − β / 2 + β k , β / 2) (0 , 1) , ( k + 1 / 2 , 1 / 2)  , that is Eq. (114 ).  When µ = 0 and σ = 1 ,Eq. 114 ) redues to: f D ( x, t ) = 1 2 t − β / 2 H 2 , 0 2 , 2  | x | t − β / 2    (1 / 2 , 1 / 2) , (1 − β / 2 , β / 2) (0 , 1) , (1 / 2 , 1 / 2)  , that is, using the redution form ula for the F o x H -funtion [30℄, f D ( x, t ) = 1 2 t − β / 2 H 1 , 0 1 , 1  | x | t − β / 2    (1 − β / 2 , β / 2 ) (0 , 1)  = 1 2 M β / 2 ( | x | , t ) . 31 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 −1 −0.5 0 0.5 1 t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 t σ 2 (t) m(t) l 1/2 (t) D(t) Figure 14: T ra jetory of the pro ess D ( t ) = B ( µ ) ( l 1 / 2 ( t )) dened in Eq. 112 ) with β = 1 / 2 (top panel). The random time pro ess is l 1 / 2 ( t ) = | b ( t ) | , where b ( t ) is a standard Bro wnian motion (middle panel). The v ariane and the mean are ev aluated o v er a sample of size N = 5 · 10 4 and t the theoretial v alues (b ottom panel). As exp eted, w e reo v er in this ase the fundamen tal solution of the time-frational diusion equation ( 72 ). Moreo v er, if w e set β = 1 in Eq. ( 114 ) w e ha v e: f D ( x, t ) = e µx/ 2 σ 2 1 2 | σ | ∞ X k =0 ( − µ 2 t/ 4 σ 2 ) k k ! t − 1 / 2 H 2 , 0 2 , 2  | xσ − 1 | t − 1 / 2    (1 / 2 , 1 / 2) , (1 / 2 + k , 1 / 2) (0 , 1) , (1 / 2 + k , 1 / 2)  , = e µx/ 2 σ 2 ∞ X k =0 ( − µ 2 t/ 4 σ 2 ) k k ! 1 2 | σ | t − 1 / 2 H 1 , 0 1 , 1  | xσ − 1 | t − 1 / 2    (1 / 2 , 1 / 2) (0 , 1)  = 1 | σ | √ 4 π t exp  − ( x − µt ) 2 4 σ 2 t  , and w e reo v er f B ( µ ) ( x, t ) . In Figure 12 and Figure 13 w e ha v e used Eq. ( 113 ) to plot the fundamen tal solution (114 ) with β = 1 / 2 for dieren t v alues of the parameters µ and σ at xed time and, for xed parameters, at dieren t times t . As exp eted, the fundamen tal solution is not symmetri in spae with a time-gro wing sk ewness. Moreo v er, due to the presene of the p ositiv ely tak en drift term ( µ = 1 ), the probabilit y to nd the partile in the p ositiv e semi-axis inreases with time (g. 13 ). In Figure 14 is presen ted a tra jetory of the pro ess D ( t ) = B ( µ ) ( l β ( t )) with β = 1 / 2 . Using Eq. (73 ) it is 32 −5 −4 −3 −2 −1 0 1 2 3 4 5 6 7 8 9 10 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 x f(x,t) β =1/2 t=1 Figure 15: Marginal densit y funtion f ( x, t ) of the pro ess B ( µ ) ( l 1 / 2 ( t )) at time t = 1 Eq. 114 ). The histogram is ev aluated o v er N = 10 5 sim ulated tra jetories (see Figure 14 ). easy to write all the momen ts of the pro ess: E ( D ( t ) m ) = [ m/ 2] X k =0  m 2 k  2 k !( m − k )! k ! σ 2 k µ m − 2 k t β ( m − k ) Γ( β ( m − k ) + 1) , 0 < β ≤ 1 , (120) where m is an in teger greater than zero and [ a ] indiates the in teger part of a . Therefore, w e ha v e: m ( t ) = E ( D ( t )) = µ t β Γ( β + 1) , (121) and σ 2 ( t ) = E ( D ( t ) 2 ) − m ( t ) 2 = 2 µ 2 t 2 β Γ(2 β + 1) − µ 2 t 2 β Γ( β + 1) 2 + 2 σ 2 t β Γ( β + 1) . (122) In the b ottom panel of Figure 14 the mean and the v ariane ha v e b een estimated from a sample of tra jetories of the pro ess B ( µ ) ( l 1 / 2 )( t ) . Then, they ha v e b een ompared with the theoretial v alues giv en ab o v e. In Figure 15 w e ompare the theoretial densit y funtion f ( x, t ) giv en b y Eq. 114 ) at time t = 1 with an histogram ev aluated o v er a sample of N = 10 5 tra jetories. 8.1.2 Exp onen tial-dea y k ernel The exp onential-de  ay kernel ase is straigh tforw ard. The non-Mark o vian F okk er-Plan k equation is: u ( x, t ) = u 0 ( x ) + Z t 0 e − a ( t − s ) ( − µ∂ x u ( x, s ) + σ 2 ∂ xx u ( x, s )) ds, a ≥ 0 . (123) 33 If w e indiate b y G ( x, t ) the fundamen tal solution of the Mark o vian equation; i.e. Eq. ( 109 ) G ( x, t ) = 1 | σ | √ 4 π t exp  − ( x − µt ) 2 4 σ 2 t  , t ≥ 0 , x ∈ R , (124) then, using Eq. (82 ), the fundamen tal solution of Eq. ( 123 ) is: f ( x, t ) = e − at G ( x, t ) + (1 − e − at )Φ( x, t ) , (125) where: Φ( x, t ) = a 1 − e − at e µx/ 2 σ 2 Z t 0 e − x 2 / 4 σ 2 τ e − ( a + µ 2 / 4 σ 2 ) τ | σ | √ 4 π τ dτ . Using Eq. (91 ) and Eq. (93 ) w e ha v e: Prop osition 8.2. The fundamental solution of Eq. ( 123 ) is: f ( x, t ) = e − at G ( x, t ) − ae µ 2 σ 2 x 2 | σ | q a + µ 2 4 σ 2 sinh | xσ − 1 | r a + µ 2 4 σ 2 ! + + ae µ 2 σ 2 x 4 | σ | q a + µ 2 4 σ 2 ( exp x | σ | r a + µ 2 4 σ 2 ! Erf x 2 | σ | √ t + r  a + µ 2 4 σ 2  t ! − exp − x | σ | r a + µ 2 4 σ 2 ! Erf x 2 | σ | √ t − r  a + µ 2 4 σ 2  t !) . (126) When t → ∞ w e obtain the stationary distribution: φ ( x ) = ae µ 2 σ 2 x 2 | σ | q a + µ 2 4 σ 2 cosh x | σ − 1 | r a + µ 2 4 σ 2 ! − sinh | xσ − 1 | r a + µ 2 4 σ 2 !! that is: φ ( x ) = a 2 | σ | q a + µ 2 4 σ 2 exp µx/ 2 σ 2 − | xσ − 1 | r a + µ 2 4 σ 2 ! . (127) 8.2 Geometri Bro wnian motion Let µ ∈ R and σ 2 > 0 b e giv en. Consider a linear diusion S on R dened b y the sto  hasti dieren tial equation: dS ( t ) = µS ( t ) dt + σ S ( t ) dB ( t ) , t ≥ 0 , (128) where B ( t ) , t ≥ 0 , is a standard Bro wnian motion. The pro ess S ( t ) , t ≥ 0 , is alled Ge ometri Br ownian motion . If S starts in x 0 at time t = 0 (i.e. P ( S (0) = x 0 ) = 1 ), then a solution of Eq. (128 ) is: S ( t ) = x 0 exp  ( µ − σ 2 / 2) t + σ B ( t )  , t ≥ 0 , x 0 > 0 . (129) The marginal densit y funtion of S ( t ) is the log-normal distribution: f S ( x, t ) = 1 x | σ | √ 4 π t exp    −  log( x/x 0 ) − ( µ − σ 2 / 2) t  2 σ 2 4 t    , t ≥ 0 , x ≥ 0 . (130) The funtion f S ( x, t ) is a solution of the F okk er-Plan k equation: ∂ t u ( x, t ) =  (2 σ 2 − µ ) + (4 σ 2 − µ ) x∂ x + σ 2 x 2 ∂ xx  u ( x, t ) , x ≥ 0 , (131) with deterministi initial ondition u 0 ( x ) = δ ( x − x 0 ) , x ≥ 0 , x 0 > 0 . (132) 34 8.2.1 β -p o w er k ernel If w e in tro due the β -p o w er k ernel K ( t ) = Γ( β ) − 1 t β − 1 , 0 < β ≤ 1 , in this setting w e obtain the follo wing fr ational F okker-Plank equation: u ( x, t ) = u 0 ( x ) + 1 Γ( β ) Z t 0 ( t − s ) β − 1  (2 σ 2 − µ ) + (4 σ 2 − µ ) x∂ x + σ 2 x 2 ∂ xx  u ( x, s ) ds, x ≥ 0 . (133) A solution of the ab o v e equation with initial ondition giv en b y Eq. ( 132 ) is giv en b y (see Corollary 6.1 ): f D ( x, t ) = Z ∞ 0 f S ( x, τ ) M β ( τ , t ) dτ (134) whi h is the marginal distribution of the pro ess D ( t ) = S ( l β ( t )) , t ≥ 0 , 0 < β ≤ 1 , (135) starting almost surely in x 0 > 0 , where l β ( t ) , t ≥ 0 , is a self-similar random time pro ess with H = β / 2 , indep enden t of the geometri Bro wnian motion S ( t ) and with marginal densit y funtion giv en b y Eq. (105 ). It is easy to see that: f D ( x, t ) = 1 x | σ | exp  log( x/x 0 )( µ − σ 2 / 2) 4 σ 2  1 2 Z ∞ 0 e − aτ M 1 / 2 ( | x ′ | , τ ) M β ( τ , t ) dτ , where: a = ( µ − σ 2 / 2) 2 / 4 σ 2 , x ′ = log ( x/x 0 ) /σ . W e ha v e the same in tegral as in Eq. (115 ). Therefore: Prop osition 8.3. for e ah t ≥ 0 : f D ( x, t ) = 1 x | σ | exp  log( x/x 0 )( µ − σ 2 / 2) 4 σ 2  × × ∞ X k =0 1 k !  − ( µ − σ 2 / 2) 2 t β 4 σ 2  k t − β 2 H 2 , 0 2 , 2  | x ′ | t − β 2    (1 / 2 , 1 / 2) , (1 − β / 2 + β k , β / 2 ) (0 , 1) , ( k + 1 / 2 , 1 / 2)  . (136) This result an b e obtained diretly from Eq. (114 ) b eause our pro ess is: D ( t ) = x 0 exp( B ( µ ′ ) ( l β ( t ))) , where B ( µ ′ ) is a Bro wnian motion with drift µ ′ = ( µ − σ 2 / 2) . When β = 1 w e reo v er Eq. ( 130 ). Moreo v er, if µ = σ 2 / 2 (i.e. µ ′ = 0 ) w e ha v e (see Figure 16 ): f D ( x, t ) = 1 x | σ | t − β / 2 M β / 2      log( x/x 0 ) σ     t − β / 2  , x ≥ 0 , t ≥ 0 , (137) whi h is the marginal probabilit y densit y of: D ( t ) = x 0 e σB ( l β ( t )) , t ≥ 0 . In Figure 16 w e sho w the plot of the fundamen tal solution f ( x, t ) in the partiular ase giv en b y Eq. ( 137 ). Here w e an see the b eha vior of the solution v arying the parameter β . F or β = 1 w e reo v er the log-normal densit y (130 ) with µ = σ 2 / 2 . In Figure 17 w e p oin t out the dep endene of the solution with resp et to the drift parameter µ for xed β = 1 / 2 , t = 1 , σ = 1 and x 0 = 1 . In Figure 18 w e presen t the time ev olution of the fundamen tal solution with β = 1 / 2 and β = 1 / 4 . 35 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x f(x,t) β =1 β =3/4 β =1/2 β =1/4 µ = σ 2 /2=1/2 t=1 x 0 =1 Figure 16: Plot of the fundamen tal solution f ( x, t ) , Eq. (136 ) at time t = 1 , when µ = σ 2 / 2 = 1 Eq. 137 ), x 0 = 1 , for dieren t v alues of the parameter β = [1 / 4 , 1 / 2 , 3 / 4 , 1] . F or β = 1 f ( x, t ) redues to the log-normal densit y (130 ). The angular p oin t orresp onds to the initial v alue x 0 = 1 and is due to the presene of | log( x/x 0 ) | in the solution. In Figure 19 w e presen t a tra jetory of the pro ess D ( t ) = S ( l β ( t )) with β = 1 / 2 , Eq. ( 135 ). W e shall no w ompute the mean and the v ariane of the pro ess D ( t ) . W e ha v e that: E ( S ( t )) = E  x 0 exp  ( µ − σ 2 / 2) t + σB ( t ))  = x 0 exp  ( µ − σ 2 / 2) t  E  e σB ( t )  . Therefore, b eause: E ( e σB ( t ) ) = Z R e σx G ( x, t ) dx = e σ 2 t 1 √ 4 π t Z R e − ( x − 2 σt ) 2 4 t dx, w e ha v e: E ( S ( t )) = x 0 exp  ( µ + σ 2 / 2) t  . (138) In the same w a y one has: E ( S ( t ) 2 ) = x 2 0 exp  (2 µ + 3 σ 2 ) t  . (139) Using the ab o v e equations w e ha v e: E ( S ( l β ( t ))) = x 0 E  exp  ( µ + σ 2 / 2) l β ( t )  = x 0 ∞ X k =0 ( µ + σ 2 / 2) k k ! E ( l β ( t ) k ) , whi h, using Eq. (73), b eomes: E ( S ( l β ( t ))) = x 0 ∞ X k =0  ( µ + σ 2 / 2) t β  k Γ( β k + 1) = x 0 E β (( µ + σ 2 / 2) t β ) , 36 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 x f(x,t) σ =1 x 0 =1 t=1 µ =0.1 µ = σ 2 /2=1/2 µ =1 µ =2 β =1/2 Figure 17: Plot of the fundamen tal solution f ( x, t ) , Eq. ( 136 ), with β = 1 / 2 , σ = 1 , x 0 = 1 , at time t = 1 , for dieren t v alues of the parameter µ = [0 . 1 , 1 / 2 , 1 , 2] . F or µ = 1 / 2 w e ha v e Eq. ( 137 ), see also Figure 16 . where E β ( z ) = ∞ X k =0 z k / Γ( β k + 1) is the Mittag-Leer funtion of order β [26℄. Similarly: E ( S ( l β ( t )) 2 ) = x 2 0 E  exp  (2 µ + 3 σ 2 ) l β ( t )  = x 2 0 E β ((2 µ + 3 σ 2 ) t β ) . Finally one has:    m ( t ) = E ( D ( t )) = x 0 E β (( µ + σ 2 / 2) t β ) σ 2 ( t ) = E ( D ( t ) 2 ) − m ( t ) 2 = x 2 0  E β ((2 µ + 3 σ 2 ) t β ) − E β (( µ + σ 2 / 2) t β ) 2  (140) 8.2.2 Exp onen tial-dea y k ernel W e no w onsider the exp onential-de  ay kernel K ( t ) = e − at , a ≥ 0 . The non-Mark o vian F okk er-Plan k equation is: u ( x, t ) = u 0 ( x ) + Z t 0 e − a ( t − s )  (2 σ 2 − µ ) + (4 σ 2 − µ ) x∂ x + σ 2 x 2 ∂ xx  u ( x, s ) ds, a ≥ 0 . (141) W e denote b y G ( x, t ) the fundamen tal solution of the Mark o vian equation; namely Eq. ( 130 ) G ( x, t ) = 1 x | σ | √ 4 π t exp    −  log( x/x 0 ) − ( µ − σ 2 / 2) t  2 σ 2 4 t    , x, t ≥ 0 . (142) Then, using Eq. (82 ), the fundamen tal solution of Eq. (123 ) is: f ( x, t ) = e − at G ( x, t ) + (1 − e − at )Φ( x, t ) , (143) 37 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x 0 1 2 3 4 5 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 x β =1/4 t=0.5 β =1/2 t=0.5 t=1 t=2 t=10 t=1 t=2 t=10 f(x,t) f(x,t) Figure 18: Plot of the fundamen tal solution f ( x, t ) , Eq. (136 ), with β = 1 / 4 (left panel) and β = 1 / 2 (righ t panel), at dieren t times t = [0 . 5 , 1 , 2 , 10] . where: Φ( x, t ) = 1 1 − e − at  a | σ | x e log( x x 0 )( 2 µ − σ 2 4 σ 2 ) Z t 0 G ( x ′ , τ ) e − a ′ τ dτ  , x ≥ 0 and:    a ′ = ( µ − σ 2 / 2) 2 4 σ 2 + 4 a, a, µ ≥ 0 , σ > 0 , x ′ = log( x/x 0 ) /σ , x ≥ 0 . (144) Th us, as in Eq. (94 ), w e ha v e: Prop osition 8.4. f ( x, t ) = e − at G ( x, t ) + a | σ | x e log( x x 0 )( 2 µ − σ 2 4 σ 2 )  1 4 √ a ′ exp( x ′ √ a ′ ) Erf  x ′ 2 √ t + √ a ′ t  − exp( − x ′ √ a ′ ) Erf  x ′ 2 √ t − √ a ′ t  − a 4 | σ | x √ a ′ e log( x x 0 )( 2 µ − σ 2 4 σ 2 ) sinh( | x ′ | √ a ′ ) . (145) The stationary distribution, obtained as t → ∞ , is: Φ( x ) = a 4 | σ | x √ a ′ exp  log  x x 0  2 µ − σ 2 4 σ 2   (cosh( x ′ √ a ′ ) − sinh( | x ′ | √ a ′ ) . (146) 38 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 t 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 0.5 1 t 0 0.2 0.4 0.6 0.8 1 2 2.5 3 3.5 t 0 0.2 0.4 0.6 0.8 1 0 10 20 30 t σ 2 (t) m(t) l 1/2 (t) D(t) Figure 19: T ra jetory of the pro ess D ( t ) = S ( l 1 / 2 ( t )) dened in Eq. (135 ) with β = 1 / 2 (top panel). The random time pro ess is l 1 / 2 ( t ) = | b ( t ) | (middle panel). The v ariane and the mean are ev aluated o v er a sample of size N = 5 · 10 4 and are presen ted together with the theoretial funtions, Eq. (140 ), in the b ottom panels. 9 Conlusions Theorem 6.1 states that the fundamen tal solution f ( x, t ) of a non-Mark o vian diusion equation of the form (43 ) u ( x, t ) = u 0 ( t ) + Z t 0 g ′ ( s ) K ( g ( t ) − g ( s )) P x u ( x, s ) ds, x ∈ R , t ≥ 0 , (147) is f ( x, t ) = Z ∞ 0 G ( x, τ ) h ( τ , g ( t )) dτ , (148) where G ( x, t ) is the fundamen tal solution of the Mark o vian equation ( 41 ) and h ( τ , t ) is the fundamen tal solution of the non-Mark o vian forw ard drift equation u ( τ , t ) = u 0 ( τ ) − Z t 0 K ( t − s ) ∂ τ u ( τ , s ) ds, τ , t ≥ 0 , (149) If the memory k ernel K ( t ) is  hosen in a suitable w a y (see Setion 3), the solution f ( · , t ) preserv es non- negativit y and normalization for all t ≥ 0 . Th us, it an b e in terpreted as the marginal densit y funtion of a non-Mark o vian sto  hasti pro ess. In view of Eq. (148 ), this sto  hasti pro ess is naturally in terpreted as a sub ordinated pro ess Eq. (55 ). W e fo used on t w o kind of memory k ernels: the p o w er k ernel K ( t ) = t β − 1 / Γ( β ) , 0 < β ≤ 1 , and the exp onen tial-dea y k ernel K ( t ) = e − at , a ≥ 0 . 39 The rst pro vides the so-alled time-frational F okk er-Plan k equations (101 ). In partiular w e studied the ase P x = ∂ xx (see Setion 7.1), whi h orresp onds to the  hoie of a standard Bro wnian motion for the paren t Mark o v mo del. In this ase, the fundamen tal solution an b e written in terms of an en tire transenden tal funtion, see Eq. (72), and is related to a F o x H -funtion through Eq. ( 68 ). W e ha v e also onsidered more ompliated ases, namely Bro wnian motion with drift µ (see Setion 8.1 ) and Geometri Bro wnian motion (see Setion 8.2 ). In these ases the fundamen tal solutions an b e written in terms of a sup erp osition of F o x H -funtions, see Eq. (114 and Eq. (136 ). The exp onen tial-dea y k ernel orresp onds heuristially to a system in whi h the non-lo al memory eets are negligible for small times. In fat, the fundamen tal solution an alw a ys b e written in the form of Eq. ( 88 ), f ( x, t ) = e − at G ( x, t ) + (1 − e − at ) φ ( x, t ) , t ≥ 0 , where G ( x, t ) is the fundamen tal solution of the Mark o vian equation, and where the funtion φ ( x, t ) is a proba- bilit y densit y whi h b eomes stationary for large times. Therefore, it is alw a ys p ossible to nd sto  hasti mo dels that b eome stationary for large times and whose marginal densit y is giv en b y Eq. (148 ). Ho w ev er, see Subsetion 5 , the sto  hasti represen tation is not unique, that is, there are man y dieren t sto  hasti pro esses whose marginal densit y is f ( x, t ) . F or example, onsider the ase where P x = ∂ xx and g ( t ) = t . Then, f ( x, t ) is the marginal densit y of B ( l ( t )) , t ≥ 0 , where B ( t ) is astandard Bro wnian motion and where l ( t ) is a random time pro ess satisfying Eq. (149 ). If the random time l ( t ) is required to b e self-similar of order β , then in view of Theorem 3.1 , the memory k ernel m ust b e a p o w er funtion K ( t ) = t β − 1 / Γ( β ) with 0 < β ≤ 1 . The orresp onding non-Mark o vian diusion equation ( 147 ) is alled in this ase time-frational diu- sion equation of order β (see Subsetion 7.1 ). The orresp onding random time pro ess l ( t ) = l β ( t ) , an b e the lo- al time of a d = 2(1 − β ) -dimensional frational Bessel pro ess or, alternativ ely , the in v erse of the totally sk ew ed stritly β -stable pro ess. Ho w ev er, f ( x, t ) is also the marginal densit y of the pro ess Y ( t ) = p l β (1) B β / 2 ( t ) , where B β / 2 is a frational Bro wnian motion indep enden t of the random time l β ( t ) . In all the previous exam- ples, the self-similarit y parameter H = β / 2 is restrited to the region 0 < H ≤ 1 / 2 . W e an obtain sto  hasti pro esses with higher v alues of the self-similarit y parameter b y in tro duing the time-saling funtion g ( t ) . In this w a y , for example  ho osing g ( t ) = t α/β , 0 < α < 2 , w e obtain the pro ess D ( t ) = B ( l β ( t α/β )) , t ≥ 0 , and the pro ess Y ( t ) = p l β (1) B α/ 2 ( t ) , t ≥ 0 , whi h are self-similar with parameter H = α/ 2 so that 0 < H < 1 (see Subsetion 7.2 ). In on trast to D ( t ) the pro ess Y ( t ) has stationary inremen ts. The solution of the non-Mark o vian equation ( 147 ) an b e stated expliitly in all the ases onsidered. W e omputed it analytially and graphed it in partiular ases. This solution is a marginal (one-p oin t) densit y funtion. W e ha v e then presen ted v arious random pro esses whose marginal densit y funtion oinides with that solution. A  kno wledgmen ts This w ork has b een arried out in the framew ork of a resear h pro jet for F r ational Calulus Mo del ling (URL: www.fraalmo.org ). It w as pursued while An tonio Mura w as visiting Boston Univ ersit y as a reipien t of a fello wship of the Maro P olo pro jet of the Univ ersit y of Bologna. The authors appreiate partial supp ort b y the NSF Gran ts DMS-050547 and DMS-0706786 at Boston Univ ersit y , b y the Italian Ministry of Univ ersit y (M.I.U.R) through the Resear h Commission of the Univ ersit y of Bologna, and b y the National Institute of Nulear Ph ysis (INFN) through the Bologna bran h (Theoretial Group). Finally , the authors w ould lik e to thank the anon ymous referees for their ommen ts. 40 Referenes [1℄ E. Bark ai, CTR W path w a ys to the frational diusion equation, Chem. Phys. 284 (2002) 1327. [2℄ E. Bark ai, R. Metzler and J. Klafter, F rom on tin uous time random w alk to frational F okk er-Plan k equation, Phys. R ev. 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