Levy flights, dynamical duality and fractional quantum mechanics

L ´ evy fligh ts, dynamical du a li ty and fractional quan tum mec hanics Piot r Garbaczew ski ∗ Institut e of Ph ysics, Universit y of Op ole, 45-0 52 Op o le, Poland Octob er 29, 201 8 Abstract W e discuss dual time ev olution scenarios whic h , albeit run n ing according to the same r e al time clo c k, in eac h considered case ma y b e mapp ed among eac h other by means of a su itable analytic con tinuation in time pro cedure. This d y- namical dualit y is a ge n eric feature of diffusion-t yp e pr o cesses. T ec h nically that in vol ves a familiar transformation from a non-Hermitian F okk er-Planc k op erator to the Hermitian o p erator (e .g. Sc hr ¨ odinger Hamiltonian), whose n egativ e is kno wn to generate a d ynamical semigroup . Under suitable restrictions up on the generator, the semigroup admits an analyti c con tinuatio n in time and ultimately yields d ual motions. W e analyze an extension of the dualit y concept to L ´ evy fligh ts, free and with an external forcing, wh ile presumin g that the corresp ond- ing evolutio n rule (fractional dynamical semigroup) is a dual coun terpart of the quan tum motion (fractional unitary d ynamics). P A CS n umbers: 0 2.50.Ey , 05.20.-y , 05.40.Jc 1 Bro wnian motion inspirati o ns 1.1 Diffusion-t yp e pro cesses and dynamical semigroups The Langevin equation fo r a one-dimensional sto c hastic diffusion pro cess in an external conserv ativ e force field F = − ( ∇ V ): ˙ x = F ( x ) + √ 2 D b ( t ), where b ( t ) stands for the normalized white noise h b ( t ) i = 0, h b ( t ′ ) b ( t ) i = δ ( t − t ′ ), giv es rise to the corresp onding F okk er-Planck equation for the probability densit y ρ ( x, t ): ∂ t ρ = D ∆ ρ − ∇ ( F ρ ) . (1) By means of a standard substitution ρ ( x, t ) = Ψ ( x, t ) exp[ − V ( x ) / 2 D ], [1], w e pass to a generalized diffusion equation for an auxiliary function Ψ ( x, t ): ∂ t Ψ = D ∆Ψ − V ( x )Ψ (2) ∗ Presented at the 21 Marian Smolucho ws k i Symp o sium on Statistical Physics 1 where a compatibilit y condition V ( x ) = (1 / 2)[( F 2 / 2 D ) + ∇ F ] needs to b e resp ected. This transformation assigns the role of the dynamics generator to the Hermitian (even - tually self-adj o in t) op erator − ˆ H = D ∆ − V . Under suitable restrictions up on V ( x ), − ˆ H b ecomes a legitimate generator of a con tractiv e dynamical semigroup ex p ( − ˆ H t ), t ≥ 0. If additionally the dynamical semigroup is amenable to a n ana lytic con tin uatio n in time, the con tractive semigroup op erator exp( − ˆ H t ) can be related with the unitary op erator exp ( − i ˆ H t ) via so-called Wic k rotation t → it . This duality observ at ion underlies our f o rthcoming discussion and g eneralizations to L´ e vy fligh ts fra mework. 1.2 F ree propagation and its analytic con tin uation in time The standard theory of G aussian diffusion-type pro cesses tak es the Wiener pro cess as the ” free noise” mo del, with the Laplacian as the ”noise” generator. It is an elemen t of folk lore that the related dissipativ e semigroup dynamics exp( tD ∆) = exp( − t ˆ H 0 ) (and th us the heat equation) can b e mapp ed into the unitary dynamics exp ( itD ∆) = exp( − it ˆ H 0 ) (and thus the free Sc hr¨ odinger eq uatio n), by means of an analytic con- tin uation in time pro cedure, [2]. A para meter D may b e interpreted dimensionally as D = ~ / 2 m , or D = k B T /mβ ( Einstein’s fluctuation- dissipation statemen t). Quite often, this mapping is represen ted by a forma l it → t t ime transformation of the f r ee Sch r¨ odinger picture dynamics (o ne should b e a ware that to execute a mapping for concrete solutions, a prop er a djustmen t of the time in terv al b oundaries is necessary): i∂ t ψ = − D △ ψ − → ∂ t θ ∗ = D △ θ ∗ , (3) where the notation θ ∗ for solutio ns of t he heat equation has b een adopted, to sta y in conformit y with the forthcoming more general disc ussion, where θ ∗ ( x, t ) needs not to b e a probability densit y , [2]- [4 ]. The ma pping is usually exemplified in terms of integral kerne ls g and k as follows , c.f. also [5]: ψ ( x, t ) = Z dx ′ g ( x − x ′ , t ) ψ ( x ′ , 0) (4) g ( x − x ′ , t ) . = k ( x − x ′ , it ) = ( 4 π iD t ) − 1 / 2 exp [ − ( x − x ′ ) 2 4 iD t ] and θ ∗ ( x, t ) = Z dx ′ k ( x − x ′ , t ) θ ∗ ( x ′ , 0) (5) k ( x − x ′ , t ) . = g ( x − x ′ , − it ) = (4 π D t ) 1 / 2 exp [ − ( x − x ′ ) 2 4 D t ] , where the initial t = 0 data need to b e prop erly adjusted. Here, g ( x − x ′ , t ) is an in tegral k ernel of the unitary ev olution op erator : [exp( iD t ∆) ψ ]( x, 0) = ψ ( x, t ). The 2 heat k ernel k ( x − x ′ , y ) pla ys the same role with resp ect to the contractiv e se migr o up op erator: [exp( D t ∆) θ ∗ ]( x, 0). The sp ecial choice of ψ ( x, 0) = ( π α 2 ) − 1 / 4 exp  − x 2 2 α 2  (6) implies ψ ( x, t ) =  α 2 π  1 / 4 ( α 2 + 2 iD t ) − 1 / 2 exp  − x 2 2( α 2 + 2 iD t )  (7) and θ ∗ ( x, t ) . = ψ ( x, − it ) =  α 2 π  1 / 4 ( α 2 + 2 D t ) − 1 / 2 exp  − x 2 2( α 2 + 2 D t )  ′ (8) with θ ∗ ( x, 0) = ψ ( x, 0 ) . W e note that ρ = | ψ | 2 = ψ ψ ∗ is a quantum mec hanical probability densit y on R for all times ρ ( x, t ) =  α 2 π ( α 4 + 4 D 2 t 2 )  1 / 2 exp  − α 2 x 2 α 4 + 4 D 2 t 2  . (9) The real solution θ ∗ ( x, t ) of the heat equation is not a probabilit y densit y ρ ( x, t ) = θ ∗ ( x, t ) θ ( x, t ), unless m ultiplied b y an appropriate real function θ ( x, t ) whic h solv es t he time a dj o in t heat equation (that b ecomes an ill- p osed dynamical problem if considered carelessly). Case 1: Since ρ ( x, t ) = [2 π ( α 2 + 2 D t )] − 1 / 2 exp[ − x 2 / 2( α 2 + 2 D t )] actually is an example o f the free Brownian motion probabilit y densit y fo r all t ≥ 0, w e infer ρ ( x, t ) = (4 π α 2 ) 1 / 4 θ ∗ ( x, t ) . = ( θ θ ∗ )( x, t ) (10) where θ ( x, t ) ≡ θ = (4 π α 2 ) 1 / 4 is interpreted a s a trivial (constant) solution of the time adjoin t heat equation ∂ t θ = − D ∆ θ . W e stress that θ ∗ = (4 π α 2 ) − 1 / 4 ρ ∼ ρ . This , lo oking redundant observ ation, will prov e quite useful in a more general fra mework to b e in tro duced in b elo w. Case 2: A complex conjugate ψ ∗ ( x, t ) = ψ ( x, − t ) o f ψ ( x, t ), Eq. (7), solv es the time-adjoin t Sc hr¨ odinger equation i∂ t ψ ∗ = D ∆ ψ ∗ . Henc e a time-symmetric approac h to the analytic con tin uatio n in time mig ht lo o k more comp elling. Indeed θ ( x, t ) . = ψ ∗ ( x, it ) =  α 2 π  1 / 4 ( α 2 − 2 D t ) − 1 / 2 exp  − x 2 2( α 2 − 2 D t )  (11) is a legitimate solution of the time-a dj o in t heat equation ∂ t θ = − D ∆ θ as long as t ∈ [ − T / 2 , + T / 2 ] where T = α 2 /D . In the presen t case, b oth time adjoin t equations set we ll defined Cauc h y pro blems (at least in the just defined time in terv al). The subtle p oin t is that the w o uld b e 3 ”initial” da t a for the bac kw ard in time ev olution, in fact need to b e the terminal data, giv en at the end-p oint T / 2 o f the considered time-in terv a l. The only propagatio n to o l, we hav e in hands, is the heat kerne l (3): k ( x − x ′ , t → t − t ′ ) with t ≥ t ′ . There holds θ ∗ ( x, t ) = R k ( x − x ′ , t − t ′ ) θ ∗ ( x ′ , t ′ ) dx ′ and θ ( x ′ , t ′ ) = R θ ( x, t ) k ( x − x ′ , t − t ′ ) dx fo r an y t ′ < t ∈ [ − T / 2 , + T / 2]. The original quan tum mec ha nical probability density ρ = | ψ | 2 = ψ ψ ∗ , Eq. (7), is mapp ed into a Bro wnian bridge (pinned Brownian mot io n) probability densit y: ρ ( x, ± it ) . = ρ ( x, t ) = ( θ θ ∗ )( x, t ) =  α 2 π ( α 4 − 4 D 2 t 2 )  1 / 2 exp  − α 2 x 2 α 4 − 4 D 2 t 2  . (12) The price paid f or the time-symmetric a pp earance of this form ula is a limitation of the admissible time span to a finite time-in terv al of length T = α 2 /D . Case 3: T o make a direct comparison of Cas e 2 with the pr evious Case 1 , let us confine the time interv al of Case 2 to [0 , + T / 2]. No w, a conditional Bro wnian motio n connects ρ ( x, 0) = ρ ( x, 0) = ( α 2 π ) − 1 / 2 exp( − x 2 /α 2 ) with ρ ( x, t → + T / 2) of Eq. (10). Because of T = α 2 /D , a s t → T / 2 , instead of a regular function w e a rriv e at the linear functional (generalized function), here represen ted b y the D irac delta δ ( x ). Note t ha t δ ( x − x ′ ) is a standard initial t = 0 v alue of the heat k ernel k ( x − x ′ , t ). This behavior is faithfully repro duced by the time evolution of θ ∗ ( x, t ) and θ ( x, t ) that comp ose ρ ( x, t ) = ( θ ∗ θ )( x, t ) for t ∈ [0 , T / 2]. The initial v alue of θ ∗ ( x, 0) = ψ ( x, 0), Eq. (6), is propag ated forwar d in accordance with Eq. ( 8 ) to θ ∗ ( x, T / 2 ) = (4 π α 2 ) − 1 / 4 exp( − x 2 / 4 α 2 ). In parallel, θ ( x, t ) of (11) in terp olates b ackwar ds b etw een θ ( x, T / 2 ) ≡ ( 4 π α 2 ) 1 / 4 δ ( x ) and θ ( x, 0 ) = θ ∗ ( x, 0). W e hav e here emplo yed an iden tity δ ( ax ) = (1 / | a | ) δ ( x ). Because of f ( x ) δ ( x ) ≡ f (0) δ ( x ), w e arriv e at ρ ( x, T / 2) = ( θ ∗ θ )( x, T / 2) ≡ δ ( x ). 1.3 Sc hr¨ odinger’s b ound ary data problem The abov e discussion provides particular solutions to so-called Sc hr¨ odinger b o undary data problem, under an assumption t ha t a Mark ov sto c hastic pro cess which in terp olates b et w een t wo a priori given probability densities ρ ( x, 0) and ρ ( x, T / 2 ) can b e mo deled b y means of the G auss probability law (e.g. in terms of the heat k ernel). That incorp orates the free Bro wnian motion (Wiene r pro cess) and all its conditional v arian ts, Bro wnian bridges b eing included, [3, 4] and [6]-[8], c.f. also [2]. F or our purposes the relev ant info r ma t io n is that, if the in terp olating pro cess is to displa y the Mark o v prop erty , then it has to b e specified b y the join t probability measure ( A and B are Borel sets in R ): m ( A, B ) = Z A dx Z B dy m ( x, y ) (13) 4 where R R m ( x, y ) dy = ρ ( x, 0) , and R R m ( x, y ) dx = ρ ( y , T / 2). F rom the start, w e assign densities to all measures to b e dealt with, and w e assume the functiona l form of the densit y m ( x, y ) m ( x, y ) = f ( x ) k ( x, 0 , y , T / 2) g ( y ) (14) to inv olve tw o unknow n functions f ( x ) and g ( y ) whic h are of the same sign and nonzero, while k ( x, s, y , t ) is any b ounded strictly p o sitive (dynamical semigroup) k ernel defined for all times 0 ≤ s < t ≤ T / 2. F or eac h concrete c hoice of the k ernel, the abov e in tegral equations are kno wn to determine f unctions f ( x ) , g ( y ) uniquely ( up to constan t factors). By denoting θ ∗ ( x, t ) = R f ( z ) k ( z , 0 , x, t ) dz and θ ( x, t ) = R k ( x, t, z , T / 2) g ( z ) dz it follo ws that ρ ( x, t ) = θ ( x, t ) θ ∗ ( x, t ) = Z p ( y , s, x, t ) ρ ( y , s ) dy , (15) p ( y , s, x, t ) = k ( y , s, x, t ) θ ( x, t ) θ ( y , s ) , for all 0 ≤ s < t ≤ T / 2. The ab ov e p ( y , s, x, t ) is the tr ansition probability densit y o f the p ertinen t Mark ov pro cess that in terp olates b etw een ρ ( x, 0) and ρ ( x, T / 2). Cases 1 through 3 are particular examples of the ab o ve reasoning, o nce k ( x, s, y , t ) is sp ecified to be the heat k ernel (3) and the c or r espo nding b oundary densit y data are c hosen. Clearly , θ ∗ ( x, 0) = f ( x ) while θ ( x, T / 2) = g ( x ). W e recall that in case of the free ev olution, b y setting θ ( x, t ) = θ ≡ const , as in Case 1, w e effectiv ely transform an inte g r al kernel k of the L 1 ( R ) no r m- preserving semigroup in to a tr ansition pro ba bilit y densit y p of the Mark ov sto chastic pro cess. Then θ ∗ ∼ ρ . 2 F ree noi se mo de ls: L ´ evy fligh t s and fractio n al (L ´ evy) semigroup s The Sc hr¨ odinger b o undar y dat a problem is amenable to an imme dia t e generalization to infinitely divisible probabilit y laws whic h induce contractiv e semigroups (and their k ernels) for general Gaussian and non- Gaussian noise mo dels. They allo w for v ario us jump and jump-type sto c hastic pro cesses instead of a diffusion pro cess. A sub class of stable probabilit y law s con t ains a subset that is ass o ciated in the literature with the concept of L ´ evy fligh ts. At this po int let us in vok e a functional analytic lore, where con tractive semigroup op erators, their generators and the p ertinen t in tegral k ernels can b e directly deduced from the L´ evy-Khitc hine form ula, compare e.g. [8]. Let us consider semigroup generators (Hamiltonians, up to dimensional constan ts) of the form ˆ H = F ( ˆ p ), where ˆ p = − i ∇ stands for the momentum op erator (up to the 5 disregarded ~ or 2 mD factor) and for −∞ < k < + ∞ , the function F = F ( k ) is r eal v alued, b ounded from b elo w and lo cally integrable. Then, exp( − t ˆ H ) = Z + ∞ −∞ exp[ − tF ( k )] dE ( k ) (16) where t ≥ 0 and dE ( k ) is the sp ectral measure of ˆ p . Because of ( E ( k ) f )( x ) = 1 √ 2 π Z k −∞ exp( ipx ) ˜ f ( p ) d p (17) where ˜ f is the F ourier tra nsform of f , w e learn that [exp( − t ˆ H )] f ( x ) = [ Z + ∞ −∞ exp( − tF ( k )) dE ( k ) f ]( x ) = (18) 1 √ 2 π Z + ∞ −∞ exp[ − tF ( k )] d dk [ Z k −∞ exp( ipx ) ˜ f ( p ) d p ] dk = (22) 1 √ 2 π Z + ∞ −∞ exp( − tF ( k )) exp ( ik x ) ˜ f ( k ) dk = [exp( − tF ( p )) ˜ f ( p )] ∨ ( x ) where the sup erscript ∨ denotes the in ve rse F ourier transform. Let us set k t = 1 √ 2 π [exp( − tF ( p )] ∨ . (19) Then the action of exp( − t ˆ H ) can b e give n in terms of a con volution (i.e. b y means of an inte g r a l kerne l k t ≡ k ( x − y , t ) = k ( y , 0 , x, t )): exp( − t ˆ H ) f = [ exp ( − tF ( p )) ˜ f ( p )] ∨ = f ∗ k t (20) where ( f ∗ g )( x ) := Z R g ( x − z ) f ( z ) dz . (21) W e shall restrict considerations o nly to those F ( p ) whic h giv e rise to p o sitivit y preserving semigroups: if F ( p ) satisfies the ce lebrated L ´ evy-Khin tchine form ula, then k t is a positive m easure for all t ≥ 0. The most general case refers to a comb ined con tribution from three t yp es of pro cesses : deterministic, G aussian, and the jump- t yp e pro cess. W e recall that a characteris tic function of a random v ariable X completely deter- mines a pro ba bilit y distribution of that v ariable. If this distribution admits a densit y w e can write E [exp( ipX )] = R R ρ ( x ) exp( ipx ) dx whic h, for infinitely divisible probabilit y la ws, giv es rise to the L ´ evy-Khin tchine form ula E [exp( ipX )] = exp { iαp − ( σ 2 / 2) p 2 + Z + ∞ −∞ [ exp ( ipy ) − 1 − ipy 1 + y 2 ] ν ( dy ) } (22) 6 where ν ( d y ) stands for the so-called L´ evy measure. In terms of Marko v sto chastic pro cesses all that amoun ts to a decomp osition of X t in to X t = αt + σ B t + J t + M t (23) where B t stands for the free Bro wnian motion ( Wiener pro cess), J t is a Pois son pro cess while M t is a general jump-t yp e pro cess (more tec hnically , martingale with jumps). By disregarding the deterministic and j ump-t yp e con tributions in the ab ov e, we a r e left with the Wiener pro cess X t = σ B t . F or a Gaussian ρ ( x ) = (2 π σ 2 ) − 1 / 2 exp( − x 2 / 2 σ 2 ) w e directly ev aluate E [exp( ipx )] = exp( − σ 2 p 2 / 2). Let us set σ 2 = 2 Dt . W e get E [exp ( ipX t )] = exp( − tD p 2 ) and subse quen tly , b y emplo ying p → ˆ p = − i ∇ , we arriv e at the contractiv e semigroup op erator exp( tD ∆) where t he one-dimensional Laplacian ∆ = d 2 /dx 2 has b een in tro duced. That amounts to c ho osing a sp ecial v ersion of the previously in tro duced Hamiltonian ˆ H = F ( ˆ p ) = D ˆ p 2 . Note that we can get read of the constant D b y rescaling the time para meter in the ab o ve . Presen tly , w e shall concen trate on the in tegra l part of the L ´ evy-Khin tchine for - m ula, whic h is resp onsible for ar bitrary sto c hastic jump features. By disregarding the deterministic and Brownian motion en tries w e arriv e a t : F ( p ) = − Z + ∞ −∞ [ exp ( ipy ) − 1 − ipy 1 + y 2 ] ν ( dy ) (24) where ν ( dy ) stands for the appropriate L ´ evy measure. The correspo nding non-G aussian Mark ov pro cess is characterize d b y E [exp ( ipX t )] = exp[ − tF ( p )] (25) with F ( p ), (22). Accordingly , t he con tractiv e semigroup generator may b e define d as follo ws: F ( ˆ p ) = ˆ H . F or concreten ess we can men tion some explicit examples of non-G aussian Marko v semigroup generators. F ( p ) = γ | p | µ where µ < 2 and γ > 0 stands for the in tensity parameter of the L ´ evy process, up on p → ˆ p = − i ∇ giv es rise to a pseudo-differen tial op erator ˆ H = γ ∆ µ/ 2 often named the fra ctional Hamiltonian. No t e that, by construc- tion, it is a p ositiv e op erator (quite a lik e − D ∆). The corresp onding jump-ty p e dynamics is in terpreted in t erms of L ´ evy fligh ts. In particular F ( p ) = γ | p | → ˆ H = F ( ˆ p ) = γ | ∇| . = γ ( | ∆ | ) 1 / 2 (26) refers to the Cauc h y pro cess. Since we kno w that the proba bility densit y of the free Brownian motio n is a solution of the F okk er-Planck ( here, simply - heat) equation ∂ t ρ = D ∆ ρ (27) 7 it is instructiv e to set in comparison the pseudo-differen t ia l F okk er-Planck e quatio n whic h corresp onds to the fractional Hamiltonian and t he fr actional semigroup exp( − t ˆ H ) = exp( − γ | ∆ | µ/ 2 ) ∂ t ρ = − γ | ∆ | µ/ 2 ρ . (28) As men tioned in the discussion of Case 1, instead of ρ in the ab ov e w e can insert θ ∗ ∼ ρ , while r emem b ering that θ ≡ const . 3 F ree fractional Sc hr ¨ odinger equation F ractional Hamiltonians ˆ H = γ | ∆ | µ/ 2 with µ < 2 and γ > 0 a r e self-adjoin t o p erators in suitable L 2 ( R ) domains. They are a lso p ositive op erators, so that the resp ectiv e fractional semigroups are holomorphic, i. e. w e can r eplace the time parameter t b y a complex o ne σ = t + is, t > 0 so that exp( − σ ˆ H ) = Z R exp( − σ F ( k )) dE ( k ) . (29) Its action is defined by [exp( − σ ˆ H )] f = [( ˜ f exp( − σ F )] ∨ = f ∗ k σ . (30) Here, the integral k ernel r eads k σ = 1 √ 2 π [exp( − σ F )] ∨ . Since ˆ H is selfadjoin t, the limit t ↓ 0 lea v es us with the unitary group exp ( − is ˆ H ), acting in the same w ay: [ exp ( − is ˆ H ) ] f = [ ˜ f exp ( − isF )] ∨ , except that now k is := 1 √ 2 π [ exp ( − isF )] ∨ in general is not a probabilit y measure. In view of unitarit y , the unit ba ll in L 2 is an inv aria nt of the dynamics. Henc e probabilit y dens ities, in a standard form ρ = ψ ∗ ψ can b e asso ciated with solutions of the free fractional (pseudo diferen tial) Sc hr¨ odinger equations: i∂ t ψ ( x, t ) = γ | ∆ | µ/ 2 ψ ( x, t ) (31) with initial data ψ ( x, 0). A ttempts t ow ards form ulating the fractional quan tum me - c hanics can b e fo und in R efs. [8, 1 1, 12, 13]. All that amoun ts to an analytic con t inuation in time, in close affinit y with the Gaussian pat tern (1) : i∂ t ψ = γ | ∆ | µ/ 2 ψ ← → ∂ t θ ∗ = − γ | ∆ | µ/ 2 θ ∗ (32) W e assume that θ ∗ ∼ ρ and thence the corresp onding θ ≡ co nst . Stable sto c hastic pro cesses and their quantum coun terpart s are pla g ued by a com- mon disease: it is extremely hard, if p ossible at all, to pro duce insightful analytic solutions. T o get a flav or of intricacies to b e faced a nd the lev el of techn ical difficulties, 8 w e shall repro duce some observ ations in regard to the Cauc hy dynamical s emigroup and it s unitary (quan tum) partner. F or con ve nience w e scale out a parameter γ . F or the Cauc h y pro cess, whose generator is |∇| , we deal with a pr o babilistic classics: ρ ( x, t ) = 1 π t t 2 + x 2 = ⇒ k ( y , s, x, t ) = 1 π t − s ( t − s ) 2 + ( x − y ) 2 ] (33) where 0 < s < t . W e hav e h exp[ ipX ( t )] i := R R exp( ipx ) ρ ( x, t ) dx = exp[ − tF ( p )] = exp( −| p | t ) and ρ ( x, t ) = Z R k ( y , s, x, t ) ρ ( y , s ) dy (3 4) for all t > s ≥ 0. W e r ecall that lim t ↓ 0 t π ( x 2 + t 2 ) ≡ δ ( x ). The characteristic function of k ( y , s, x, t ) for y , s fixed, reads exp [ ipy − | p | ( t − s )], and t he L´ evy measure needed to ev aluate the L´ evy-Khin tchine in tegral reads: ν 0 ( dy ) := l im t ↓ 0 [ 1 t k (0 , 0 , y , t )] dy = dy π y 2 . (35) T o pass to a dual Cauc h y-Schr¨ odinger dynamics, w e need to p erform an a na lytic con tin ua t io n in time. W e deal with a holomorphic fractional se migro up exp( − σ t |∇| ), σ = t + is , (27 ). It is clear that exp ( − t |∇| ) and exp ( − is |∇| ) hav e a common, iden tit y op erator limit as t ↓ 0 and s ≡ t ↓ 0. An ana lytic con tinuation of the Cauc h y k ernel by means of (28) g iv es rise to: k t ( x ) = 1 π t x 2 + t 2 − → g s ( x ) . = k is ( x ) = 1 2 [ δ ( x − s ) + δ ( x + s )] + 1 π P is x 2 − s 2 , (36) where P indicates that a con v olutio n of the in tegral k ernel with a n y function should b e considered as a principal v alue of an improp er integral, [8]. T his should b e compared with an a lmost trivial o ut come of the previous mapping (2) → (3). He re, w e emplo y the usual no t ation for the Dira c delta functionals, and the new time lab el s is a remnan t of the limiting pro cedure t ↓ 0 in σ = t + is . The function denoted by is/π ( x 2 − s 2 ) comes from the inv erse F o urier transform of − i √ 2 π sg n ( p ) sin ( sp ). Because o f [ sg n ( p )] ∨ = i r 2 π P ( 1 x ) (37) where P ( 1 x ) stands for the functional defined in terms of a principal v alue of the in tegral. Using the notation δ ± s for the Dirac delta functional δ ( x ∓ s ): [ sin ( sp )] ∨ = i r π 2 ( δ s − δ − s ) (38) w e realize tha t 1 π is x 2 − s 2 = i 2 π ( δ s − δ − s ) ∗ P ( 1 x ) (39) 9 is given in t erms of the implicit conv olution of t wo generalized functions. Obviously , a propagation of an initial f unction ψ 0 ( x ) to time t > 0: ψ ( x, t ) = Z R g ( x − x ′ , t ) ψ 0 ( x ′ ) dx ′ (40) giv es a solution of the fra ctional ( Cauch y) Schr¨ odinger equation i∂ t ψ = −|∇| ψ . In compar ison with the Gaussian case of Section 1, one imp ortant difference m ust b e emphasized. The improp er in tegrals, whic h app ear while ev a luating v arious con volu- tions, need to b e handled b y means of their principal v alue. Therefore, a simple it → t transformation recip e no long er works on the leve l of in tegral k ernels and resp ectiv e ψ and θ ∗ functions. One explicit example is provide d by the incongruence of (3 1 ) and (34) with r espect to the formal t → − it mapping. Another is provided by considering sp ecific solutions of pseudo-differen tial equations (30). T o that end, let us consider θ ∗ 0 ( x ) = (2 /π ) 1 / 2 1 1+ x 2 , together w ith θ = (2 π ) − 1 / 2 . Then, θ θ ∗ ( x, 0) = 1 π (1+ x 2 ) is an L ( R ) normalized Cauch y densit y , while θ ∗ 0 ( x ) itself is the L 2 ( R ) no r malized f unction. Clearly: θ ∗ ( x, t ) = [exp( − t |∇| ) θ ∗ 0 ]( x ) = Z k ( y , 0 , x, t ) θ ∗ ( y , 0) dy =  2 π  1 / 2 1 + t x 2 + (1 + t ) 2 (41) while t he corresp onding ψ ( x, t ) with ψ 0 ( x ) = θ ∗ 0 ( x ) reads (for details see e.g. [8]): ψ ( x, s ) = [exp( − is |∇| ) ψ 0 ]( x ) = 1 2 [ ψ 0 ( x + s ) + ψ 0 ( x − s )]+ (42) i 2 [( x − s ) ψ 0 ( x − s ) − ( x + s ) ψ 0 ( x + s )] . 4 Dynamical duality in external p oten tials : frac- tional Sc hr¨ odinger semig r o ups and L ´ evy flig h ts 4.1 Sc hr¨ odinger semigroups for Smoluc ho wski pro cesses Considerations of Section 1, where the fr ee quan tum dynamics and free Brow nian mo- tion w ere considered as dual dyn a mical scenarios, can be generalized to an externally p erturb ed dynamics, [2]. Namely , one kno ws that the Sc hr¨ odinger equation for a quan- tum pa r t icle in an external p o ten tial V ( x ), and t he generalized heat equation are con- nected by analytic con tinuation in time, know n t o take the F eynman-Kac (ho lo morphic semigroup) k ernel in to the Green function of the corresp onding quantum mec hanical problem. i∂ t ψ = − D ∆ ψ + V ψ ← → ∂ t θ ∗ = D ∆ θ ∗ − V θ ∗ . (43) 10 Here V . = V ( x ) / 2 mD . F or V = V ( x ) , x ∈ R , b ounded from below, the generator ˆ H = − 2 mD 2 △ + V is essen tially selfadjoin t on a natural dense subset of L 2 , and the kerne l k ( x, s, y , t ) = [exp[ − ( t − s ) ˆ H ]]( x, y ) of t he relat ed dynamical semigroup exp( − t ˆ H ) is strictly p ositiv e. The quan tum unitary dynamics exp( − i ˆ H t ) is the an ob vious res ult of the analytic con tin ua t io n in time of a dynamical semigroup. Assumptions concerning the admissible p oten tial ma y b e relaxed. The necessary demands are that ˆ H is self-adjoint and bounded from b elow. Then the respectiv e dynamical semigroup is holomorphic. The key role of an in tegra l k ernel of t he dynamical semigroup op erato r has b een elucidated in form ulas ( 1 1)-(13), w here an explicit form of a transition probabilit y densit y of the Mark ov diffusion pro cess w as giv en. W e hav e determined as w ell the time dev elopmen t of θ ∗ ( x, t ) and θ ( x, t ), so that ρ ( x, t ) = ( θ θ ∗ )( x, t ) is a proba bility densit y of the p ertinen t pro cess. If we a priori consider θ ( x, t ) in the functional f o rm θ ( x, t ) . = exp Φ( x, t ), so that θ ∗ ( x, t ) . = ρ ( x, t ) exp[ − Φ( x )], and prop erly define the forward drif t b ( x, t ) . = 2 D ∇ Φ( x, t ) in the p ertinen t F okke r- Planc k equation: ∂ t ρ = D ∆ ρ − ∇ ( b ρ ) (44) w e can recast a diffusion problem in terms o f a pa ir of time adjoin t generalized heat equations ∂ t θ ∗ = D ∆ θ ∗ − V θ ∗ (45) and ∂ t θ = − D ∆ θ + V θ , (4 6) i. e. as t he Sc hr¨ odinger b oundary data problem, where an interpola t ing sto c hastic pro cess is uniquely determined by a con tinuous and p ositiv e F eynman-Kac k ernel of the Schr¨ odinger semigroup exp( − t ˆ H ) , where ˆ H = − D ∆ + V . If our departure p oin t is the F okk er-Planc k (or Langevin) equation with the a priori prescribed p oten tial func tio n Φ( x, t ) for the f orw ard drift b ( x , t ), then the backw ard equation (4 4) b ecomes an iden tity from whic h V directly f ollo ws, in terms of Φ and its deriv ativ es, [6, 7 ]: V ( x, t ) =  ∂ t Φ + 1 2 ( b 2 2 D + ∇ b )  (47) F or the time-indep enden t drift p otential, whic h is the case for standard Smoluc hows ki diffusion pro cesses , w e g et ( c.f. also [1 ], where the a transformation of the F okk er- Planc k equations (42) into a n a sso ciated Hermitian problem (43) is describ ed in detail): V ( x ) =  1 2 ( b 2 2 D + ∇ b )  . (48) 11 Notice that Φ( x ) is defined up to an additiv e constan t. T o giv e an example of a p edestrian reasoning based on the ab o v e pro cedure in case of a c o ncrete Smoluc hows ki diffusion pro cesses, let us begin f rom the Lang evin equation for the one-dimensional sto chastic pro cess in the external conserv ative fo r ce field F ( x ) = − ( ∇ V )( x ) (to k eep in touch with the previous notations, note that Φ ≡ − V ): dx dt = F ( x ) + √ 2 D B ( t ) (49) where B ( t ) stands for t he normalized white noise: h B ( t ) i = 0, h B ( t ′ ) B ( t ) i = δ ( t − t ′ ). The corresp onding F okk er- Planc k equation f or the probability densit y ρ ( x, t ) reads: ∂ t ρ = D ∆ ρ − ∇ ( F ρ ) (50) and b y means of a s ubstitution ρ ( x, t ) = θ ∗ ( x, t ) exp[ − V ( x ) / 2 D ], [1], can b e trans- formed into the generalized diffusion equation for an auxiliary function θ ∗ ( x, t ): ∂ t θ ∗ = D ∆ θ ∗ − V θ ∗ (51) where the consistency condition (reconciling the functional form of V with this for F ) V = 1 2  F 2 2 D + ∇ F  . (52) directly comes from the time-a djoin t equation ∂ t θ ≡ 0 = − D ∆ θ + V θ (53) with θ ( x ) = exp[ − V ( x ) / 2 D ]. F or t he Ornstein-Uhlen b ec k process b ( x ) = F ( x ) = − κx and accordingly V ( x ) = κ 2 x 2 4 D − κ 2 . (54) is an explicit functional form of the p otential V , presen t in previous f orm ulas (41)-(44) . 4.2 F ractional semigroups and p erturb ed L ´ evy fligh ts External p erturbations in the additiv e form: i∂ t ψ ( x, t ) = γ | ∆ | µ/ 2 ψ ( x, t ) + V ( x ) ψ ( x, t ) (55) w ere considered in the fra mew ork of fractional quantum mec hanics, [11]-[13], c.f. also [8, 9]. With the dual dynamics concept in mind, Eq. (30), we exp ect that an anlytic con tin ua t io n in time (if admitted) tak es us from the fractional Sc hr¨ odinger equation t o the fra ctional analog of the generalized diffusion equation: ∂ t θ ∗ = − γ | ∆ | µ/ 2 θ ∗ − V θ ∗ . (56) 12 The t ime-adjoin t equation has the form ∂ t θ = γ | ∆ | µ/ 2 θ + V θ , (57) W e shall b e particularly in terested in the t ime-indep endent θ ( x, t ) ≡ θ ( x ), an assump- tion affine to that inv olve d in the passage fro m (44 )-(46). Hermitian fractional pro blems of the form (48) and/or (49) hav e also b een studied in Refs. [14, 15, 1 6]. Ho wev er, the ma jor (alb eit implicit, nev er o p enly stated) assumption of Refs. [14, 15, 16] w as to consider the so-called step L ´ evy pro cess instead of the jump- t yp e L ´ evy pro cess prop er. This amoun ts t o in tro ducing a lo w er b ound on the length of admissible jumps: arbitrarily small jumps are then excluded. That allo ws to b y-pa ss a serious tec hnical obstacle. Indeed, for a pseudo-differen tial op erator γ ∆ µ/ 2 , the action o n a function from its domain can b e greatly simplified b y disregarding jumps of length not excee ding a fixed ǫ > 0, see e.g. R efs. [8 , 9]: γ | ∆ | µ/ 2 f )( x ) = − Z R [ f ( x + y ) − f ( x ) − y ∇ f ( x ) 1 + y 2 ] ν ( d y ) (58) ⇓ γ | ∆ | µ/ 2 ǫ f )( x ) = − Z | y | >ǫ [ f ( x + y ) − f ( x )] ν ( dy ) . Compare e.g. Eq. (2) in [15] and Eq. (6) in [16]. Note that these Autho r s ha v e skipp ed the minus sign that must app ear on the righ t-ha nd-side of b oth fo rm ulas (50). As a side commen t, let us p o int out that the principal in tegral v a lue issues of Section 3 would not arise in our previous discuss ion of Cauc h y fligh ts and their generators, if arbitrarily small j umps w ere eliminated from the start. Nonetheless , if t he ǫ ↓ 0 limit is under con trol, the step pro cess can b e considered as a meaningful appro ximation of the fully-fledged (p erturb ed) jump-type L´ evy pro cess. This approx imation problem has b een in ves tig ated in detail, in the construction of the p erturb ed Cauc h y pro cess, go ve rned by the Hermitian dynamical problem (53), with the input (55), under suitable restrictions on the b ehav ior of V , [9]. Let us come back to time-adjoint fr a ctional equations (54) and (55). W e ha v e ρ ( x, t ) = ( θ θ ∗ )( x, t ) and emplo y the trial ansatz of Section 4.2: θ ( x, t ) ≡ θ ( x ) = exp[Φ( x )] (59) θ ∗ ( x, t ) = ρ ( x, t ) exp[ − Φ( x )] . Accordingly (55) implies, compare e.g. [14] for an indep enden t ar g umen t: V = − γ exp( − Φ) | ∆ | µ/ 2 exp(Φ) (60) 13 to b e compared with Eq. (8) in Ref. [15]. In view of (5 4) w e hav e ∂ t ρ = θ ∂ t θ ∗ = − γ exp( φ )[ | ∆ | µ/ 2 exp( − Φ) ρ ] + V ρ . = −∇ j . (61) Langevin-st yle description of p erturb ed L´ evy fligh ts ( deterministic comp onen t plus the L ´ evy noise con tribution) are kno wn, [17, 18, 19 ], to generate fractional F okk er- Planc k equations of the form ∂ t ρ = −∇ ( F ρ ) − γ | ∆ | µ/ 2 ρ . = −∇ j (62) where F = −∇ V ≡ ∇ Φ, w e face problems which are left unsettled at the presen t stage of our inv estigation: (i) May the sto c hastic pro cesses driving (58) and/or (5 9) coincide under an y cir- cumstances, o r basically not at all ? (ii) Giv e an insigh tful/useful definition of the probability c urrent j ( x, t ) in b oth considered cases, while remem b ering that for fractional deriv ativ es the comp osition rule fo r consecutiv e (R iesz) deriv atives typic a lly breaks down. Both problems (i) and (ii) ha ve ha ve an immediate resolution in case of diffusion- t yp e pro cesses, whe re b y departing from the Langevin equation one infers F okke r- Planc k and con tinu ity equations. In turn, these equations can b e alternativ ely deriv ed b y means o f the Sc hr¨ odinger b oundary data problem,pro vided its integral k ernel stems from thee Sc hr¨ odinger semigroup, b o th in the free and p erturb ed cases. The sto c hastic diffusion pro cess (corresp onding to tha t asso ciated with the Langevin equation) is then reconstructed as w ell. Then ce, the Sc hr¨ odinger lo op gets closed. While passing to L´ evy pro cesses, w e hav e demonstrated that, with suitable reserv a- tions, this Schr¨ odinger ”lo op” can b e completed in case of free L ´ evy fligh ts. Ho w ev er, the ”lo op” remains incomplete (neither definitely prov ed or dispro v ed) f or p erturb ed L ´ evy fligh ts. A t this p oint w e should mention clear indications [1 4 ] that, once discussing L´ evy fligh ts, w e actually encounte r tw o differen t classes of pro cesses with incompatible dy- namical prop erties. 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