Local time path integrals and their application to Lévy random walks

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1 Local time path integrals and their application to Lévy random walks Václav Zatloukal ( Faculty of Nuclear Sciences and Physical Engineering Czech Technical University in Prague talk given at Path Integration in Complex Dynamical Systems 8 February 27 Leiden V. Zatloukal Local time path integrals /33

2 Motivation TIME ZONES: V. Zatloukal Local time path integrals 2/33

3 Motivation TIME ZONES: NOT INCLUDED IN THIS PRESENTATION V. Zatloukal Local time path integrals 2/33

4 Motivation Local time of a stochastic process: xhtl l=2 l=.5 t - l= -2 V. Zatloukal Local time path integrals 3/33

5 Motivation Local time of a stochastic process: xhtl x x x l=2 l=.5 t LHxL LHxL LHxL l= l=2 l=.5 l= Stochastic trajectory x( )! Local time profile L(x) V. Zatloukal Local time path integrals 3/33

6 Outline Preliminaries: di usion equation and path integral Introduction of local time of a stochastic process Correlation functions and functionals of the local time Time-independent systems and the resolvent method Local time of Lévy random walks (Lévy flights) Local-time representation for Gaussian path integrals In collaboration with Petr Jizba (Czech Technical University in Prague) and Angel Alastuey (ENS Lyon). V. Zatloukal Local time path integrals 4/33

7 Di usion equation and path integral Di usion (or heat, or Fokker-Planck) equation t + H( i@ x, x, t)] P(x, t) = () where H(p, x, t) is a generic Hamiltonian, and x 2 R. [~ =] Solution for initial condition P(x, t a )= (x x a ) represented by phase-space path integral (x b t b x a t a ) P(x b, t b )= x(tz b )=x b x(t a )=x a Dx Z Dp 2 ea[p,x] (2) where the action A[p, x] = Z tb t a d ip( )ẋ( ) H(p( ), x( ), ) (3) V. Zatloukal Local time path integrals 5/33

8 Di usion equation and path integral We will gradually specify the Hamiltonian: H(p, x, t) generic, possibly time-dependent [ H(p, x) time-independent [ H(p) time and position-independent [ H (p) =D (p 2 ) /2 Lévy Hamiltonian In the end, special attention to H(p, x) = p2 2M + V (x) usual quantum Hamiltonian V. Zatloukal Local time path integrals 6/33

9 Di usion equation and path integral Significance of the heat equation: Di usion processes (or continuous random walks) P(x, t) is the probability of transition to point x after time t Quantum mechanics: by rotation to imaginary times t! it P are transition amplitudes Quantum statistical physics: identifying t = ~, where = k B T P are matrix elements of the equilibrium density matrix... In addition, corresponding path integrals describe: Polymers or other line-like objects Stock prices on financial markets... V. Zatloukal Local time path integrals 7/33

10 Local time of a stochastic process xhtl x x x l=2 l=.5 t LHxL LHxL LHxL l= l=2 l=.5 l= For stochastic trajectory x( ) define L(x; t a, t b, x( )) = Z tb t a d (x x( )) (4) P. Lévy, Sur certains processus stochastiques homogènes, Compos. Math. 7, 283(939). V. Zatloukal Local time path integrals 8/33

11 Local time of a stochastic process L is functional of x( ) and function of x: L(x) R R L(x)dx = t b t a L(x) has compact support L(x) is continuous L(x)... trajectories of a new stochastic process V. Zatloukal Local time path integrals 9/33

12 Correlation functions of local time hl(x )...L(x n )i = Z x(t b )=x b x(t a )=x a Dx = X 2S n Z t a <t <...<t n <t b dt...dt n Z Dp 2 ea[p,x] L(x )...L(x n ) ny (x (k+) t k+ x (k) t k ) (5) where are permutations of indices {,...,n}, with () = and (n + ) = n +, and we have denoted x = x a, t = t a, x n+ = x b, t n+ = t b. k= V. Zatloukal Local time path integrals / 33

13 Generic functionals of local time L(x )...L(x n ) is an example of functional of the local time F [L(x)]. In general, hf [L(x)]i = = F x(tz b )=x b x(t a )=x a Dx apple U(x) Z Dp 2 ea[p,x] F [L(x)] x(tz b )=x b x(t a )=x a Dx Z Dp 2 ea U[p,x] U= (6) where A U [p, x] corresponds to H U (p, x, t) =H(p, x, t)+u(x). We have employed the key identity Z tb t a U(x( )) d = Z U(x)L(x) dx (7) V. Zatloukal Local time path integrals / 33

14 Time-independent systems and the resolvent method Assume H(p, x) [ independent of time t ] ) (x b t b x a t a )=hx b e (t b t a )Ĥ x a i! take t a =, t b t from now on. Laplace representation hf [L(x)]i E = R dt e te hf [L(x)]i Correlation functions: hl(x )...L(x n )i E = X 2S n ny k= R(x (k), x (k+), E) (8) [ e.g. hl(x)i E = R(x a, x)r(x, x b ) ] Here R(x, x, E) hx (Ĥ + E) xi (9) denotes matrix elements of the resolvent operator ˆR(E) =(Ĥ E). V. Zatloukal Local time path integrals 2 / 33

15 Time-independent systems and the resolvent method Generic functionals: apple hf [L(x)]i E = F R U (x a, x b, E) U(x) U= where ˆR U (E) =(ĤU E) is the resolvent corresponding to the extended Hamiltonian H U = H + U., () One-point distribution: take U(x )=u (x x)! ) R U (x a, x b )=R(x a, x b ) u R(x a, x)r(x, x b ) +ur(x, x) (L L(x)) E = (L)R(x a, x)r(x, x b ) R(x, x) 2 e apple + (L) R(x a, x b ) L R(x,x) R(x a, x)r(x, x b ) R(x, x) () (2) V. Zatloukal Local time path integrals 3 / 33

16 *Multipointresolvents Resolvent identity for reads (ĤU + E) =(Ĥ + E) (ĤU + E) U(ˆx)(Ĥ + E) (3) U(x) = R U (x a, x b )=R(x a, x b ) Setting x a = x k for k =,...,n, where M jk R U (x j, x b )= nx u j (x x j ) (4) j= nx k= nx u j R(x a, x j )R U (x j, x b ) (5) j= are elements of the inverse of the matrix M jk R(x k, x b ) (6) M jk = jk + u k R(x j, x k ) (7) V. Zatloukal Local time path integrals 4 / 33

17 Distribution functions and moments of local time Invert the Laplace transform to return to the time domain: h...i E!h...i Moments of local time: µ(x,...,x n )= hl(x )...L(x n )i hi (8) n-point distribution functions of local time: Q n j= W (L,...,L n ; x,...,x n )= (L j L(x j )) hi (9) The normalization factor hi equals (x b t b x a t a ) for averaging over paths with fixed initial point x a and final point x b. Averaging over paths with arbitrary x b : h...i R dx b h...i!hi,µ, W,... V. Zatloukal Local time path integrals 5 / 33

18 Lévy random walks Hamiltonians H(p) only dependent on momentum form now on: P(x, t) (x b t x a ) = Z dp 2 e th(p) e ipx, x = x b x a (2) For the Lévy Hamiltonian H (p) =D (p 2 ) /2 [ we consider 2 [, 2] ]! symmetric Lévy stable distribution Heavy tails for For =2Gaussian: P =2 (x, t) = e x 2 4D 2 t p 4 D2 t <2: P (x, t) x! c x P l HxL l=2 l=.5 l= For =Cauchy(-Lorentz): P = (x, t) = D t (D t) 2 + x x V. Zatloukal Local time path integrals 6 / 33

19 Lévy stable distributions Stability: X Levy, X 2 Levy ) X + X 2 Levy Characteristic function: h exp ipµ cp i sgn(p) i, = ( tan( /2) 6= (2/ ) log p =...tail power ( x ), c...width, µ...shift of origin,...asymmetry Generalized central limit theorem: i.i.d. X,...,X n distribution with possibly infinite variance lim n! X X n N n Levy Applications in financial markets (fluctuation of stock prices), physics (e.g. François Bardou et al., Lévy Statistics and Laser Cooling: How Rare Events Bring Atoms to Rest, CUP(22).,...),... V. Zatloukal Local time path integrals 7 / 33

20 Lévy random walks xhtl x x x l=2 l=.5 t LHxL LHxL LHxL l= l=2 l=.5 l= For decreasing : Longer jumps (Lévy flights) Separation of local time L(x) into peaks V. Zatloukal Local time path integrals 8 / 33

21 Resolvent of Lévy Hamiltonian R (x, x, E) = Z dp 2... Linnik (or geometric stable) distribution e ip(x x) D (p 2 ) /2 + E (2) For =2 Gaussian resolvent: R l HxL.8 R 2 (x, E) = e p E/D2 x p 4D2 E.6 l=2 l=.5 For = Cauchy resolvent: apple sin E x D R (x, E) = D + cos E x D si E x D ci E x D l= divergent for = at x = x V. Zatloukal Local time path integrals 9 / 33

22 One-point distributions of local time at the origin ) x b = x a : (L L(x a )) E = e L R () =exp h LE i, D / sin W (L; x a )= = (D t)/ ( ) (L Z hi L(xa )) de e Et exp h LE cos i sin hle sin i (22) Gaussian case: For W =2 (L; x a )= 2D 2L t =(Cauchy case): singularity R () = + e D 2L 2 /t! flattening of the distribution W l HLL l=2 l=.5 l= L V. Zatloukal Local time path integrals 2 / 33

23 One-point distributions of local time at the origin 2) free x b : (L L(x a )) E = ER () e L R () = E / exp h LE i (L W (L; x a )= Z = de e Et E / L(xa )) hi (23) h exp LE cos i sin hle sin + i W l * HLL Gaussian case: W =2 (L; x a)= r 4D2 t e D 2L 2 /t l=2 l=.5 l= L V. Zatloukal Local time path integrals 2 / 33

24 *One-pointdistributionofBrownianlocaltime Gaussian case =2(Brownian motion): generic x b and x W 2 (L; x) = (L) x a x + x x b +2D 2 L apple t ( xa x + x x b +2D 2 L) 2 (x b x a ) 2 exp 4D 2 t apple ( xa x + x x b ) 2 (x b x a ) 2 + (L) exp 4D 2 t (24) r apple W2 4D2 (L; x) = (L) t exp ( xa x +2D 2 L) 2 4D 2 t + (L) erf apple xa x p 4D2 t (25) The -term vanishes for x between x a and x b. V. Zatloukal Local time path integrals 22 / 33

25 First moments of local time [ x b = x a ] [free x b ] m l HxL m l * HxL.5 l=2.5 l=2 l=.5 l=.5. l=. l= x x µ =2 (x) = s (x t b xa) 2 e 4D 2 t 4D 2 erfc " xa x + x x b p 4D2 t # µ =2 (x) = s t D 2 e (xa x) 2 4D 2 t x a x 2D 2 " # xa x erfc p 4D2 t µ = (x) = n 2 D 2 h t (D t) 2 +(xa+x b 2x) 2i h 2(x x a ) h +2(x b x) (D t) 2 +(x b x) 2 (x x a ) 2i arctan D t x b x h +D t (D t) 2 +(x x a ) 2 +(x b x) 2i ln (D t) 2 +(x x a ) 2 (x b x) 2i arctan D t x xa (x xa) 2 (x b x) h 2 (D t) 2 +(x xa) 2ih (D t) 2 +(x b x) 2 i ) ) µ = (x) = " ln + 2 D x D t x a 2 # V. Zatloukal Local time path integrals 23 / 33

26 *SecondmomentsofBrownianlocaltime ) fixed x b µ 2 (x, x 2 )= t 2D 2 +( 2 $ 2 ) apple 2 exp 2 (x b x a ) 2 4D 2 t p apple 2 p 4D2 t erfc 2 p 4D2 t apple (xb x a ) 2 exp 4D 2 t where 2 x a x + x x 2 + x 2 x b, 2 x a x 2 + x 2 x + x x b 2) free x b µ 2 (x, x 2 )= 4D 2 p apple t p 2 ( exp 2 ) 2 D2 4D 2 t + apple ( 2 ) 2 2D 2 + t erfc apple 2 p 4D2 t +( 2 $ 2) where 2 x a x + x x 2, 2 x a x 2 + x 2 x V. Zatloukal Local time path integrals 24 / 33

27 Path integrals in quantum statistical physics From now on H(p, x) = p2 2M + V (x) (26) Motivation in quantum statistical physics: Gibbs operator: e! partition function: Tr (e! thermal density matrix: e Ĥ Ĥ ), =/k B T Ĥ /Tr (e Ĥ ) (x a, x b, ) hx b e Ĥ x a i... Heat kernel for di usion generated by H, with time variable + H( i~@x, x) (x a, x, )= (27) V. Zatloukal Local time path integrals 25 / 33

28 Path integrals in quantum statistical physics ) Feynman path-integral representation For (x a, x b, )= Z x( ~)=x b x()=x a Dx( )exp!... high temperatures: ~ Z ~ d apple M 2 ẋ 2 ( )+V (x( )) (28) V (x( )) = V (x a )+V (x a )(x( ) x a )+...) e V (x a ) p 2 ~ 2 /M P. Jizba and V. Zatloukal, Path-integral approach to the Wigner-Kirkwood expansion, Phys. Rev. E 89, 235(24) 2) Spectral representation (x a, x b, )= X n e E n n(x a ) n (x b ) where Ĥ n i = E n n i (29) For!... low temperatures: e E gs gs(x a ) gs (x b ) V. Zatloukal Local time path integrals 26 / 33

29 Local-time path integral 3) Local-time representation? path integral over local time profiles! s = / ~ L(x) = Z ~ d (x x( )) (3) small large V. Zatloukal Local time path integrals 27 / 33

30 Local-time path integral (x a, x b, )= Z x( ~)=xb x()=x a Dx( )exp ( ~ Z ~ M 2 ẋ 2 ( )d ~ Z R V (X )L(X )dx ) + change of variables: x( )! L(x) (x a, x b, )= Z DL(x)W [L(x); + ~, x a, x b ]exp Z R L(x)V (x)dx where L(x) and W [L(x)] = if R R L(x)dx 6= ~! identify the weights W [L(x)] V. Zatloukal Local time path integrals 28 / 33

31 Local-time path integral: glimpses of the derivation Di usion equation in Laplace picture: E + H( i~@x, x) e (x a, x, E) = (x a x) Quantum-field-theoretic representation: e (x a, x b, E) =2 R D (x) (x a ) (x b )e h E+Ĥ i / R D (x)e h E+Ĥ i Replica trick: a/b =lim D! ab D R PD o 2 e =lim D! D D D (x) (x a ) (x b )expn = h E + Ĥ i Spherical coordinates in -space: radial part = q ~ ~ Inverse Laplace transform ) Local-time representation of (x a, x b, ) V. Zatloukal Local time path integrals 29 / 33

32 Local-time path integral (x a, x b, ) hx b e Z (X+ )= + (X )= Ĥ x a i = D (x) lim lim X ±!± Z X+ X D! 2 D 2 2 (x)dx lim ( +) ±!! D 2 (x a ) (x b )exp{ A [ (x)]} where (x) 2 (x) $ L(x)/~ The action of a radial harmonic oscillator ($ Bessel process indexed by x) A [ (x)] Z X+ X dx apple ~ 2 2M (x) 2 + V (x) 2 (x)+ M ~ 2 (x) 2 2 (x) (3) V. Zatloukal Local time path integrals 3 / 33

33 Local-time path integral at low temperatures Rescaling! p : A [ p (x)] = Z X+ X dx apple ~ 2 2M (x) 2 + V (x) 2 (x)+ M ~ 2 (x) (x)!: Saddle-point approximation of the path integral (! neglect the last term) $ Minimization of the functional: Z X+ X dx apple ~ 2 2M (x) 2 + V (x) 2 (x) = h Ĥ i under the constraint h i = ) Rayleigh-Ritz variational principle for the ground state V. Zatloukal Local time path integrals 3 / 33

34 Generic functionals of the local time Average value of generic functional F [L(x)]: Z x( ~)=xb x()=x a = lim Dx( )F [L(X )] exp lim X ±!± D! Z (X+ )= + (X )= 2 D 2 ( ~ Z ~ lim ( +) ±! D (x)f [~ 2 (x)] d D 2 apple M 2 ẋ 2 ( )+V (x( )) Z X+ X 2 (x)dx! ) (x a ) (x b )e A [ (x)] V. Zatloukal Local time path integrals 32 / 33

35 Summary We discussed local times of stochastic processes for generic Hamiltonian Calculated moments and distributions of local time In particular, for Lévy random walks For Gaussian path integrals derived local-time path-integral representation xhtl x x x l=2 l=.5 t LHxL LHxL LHxL l= -2-2 l=2-2 l=.5-2 l= P. Jizba and VZ, Local-time representation of path integrals, Phys. Rev. E 92, 6237(25)[arXiv:56.888]. VZ, Local time of Lévy random walks: a path integral approach, [arxiv: ]. V. Zatloukal Local time path integrals 33 / 33

Green Function of the Double-Fractional Fokker-Planck Equation

Green Function of the Double-Fractional Fokker-Planck Equation Hagen Kleinert, Václav Zatloukal Max Planck Institute for the History of Science, Berlin and Faculty of Nuclear Sciences and Physical Engineering,