IEP Kosice, 9 May 2007
Lévy Flights
and
Fractional Kinetic Equations
A. Chechkin
Institute for Theoretical Physics of National Science Center «Kharkov Institute of
Physics and Technology», Kharkov, UKRAINE
In collaboration
with:
 V. Gonchar, L. Tanatarov
(ITP Kharkov)
 J. Klafter (TAU)
 I. Sokolov (HU Berlin)
 R. Metzler (University of
Ottawa)
 A. Sliusarenko (Kharkov
University)
T.Koren (TAU)
A. Chechkin, V. Gonchar, J. Klafter and R. Metzler, Fundamentals of Lévy Flight
Processes. Adv. Chem. Phys. 133B, 439 (2006).
LÉVY motion or “LÉVY fLights”
(B. Mandelbrot): paradigm of nonbrownian random motion
Mathematical Foundations:
Brownian Motion
Lévy Motion
1. Generalized Central
1. Central Limit Theorem
Limit Theorem
2. Properties of Gaussian 2. Properties of Lévy
probability laws and
stable
laws
and
processes
processes
Paul Pierre Lévy (1886-1971)
Lévy stable distributions in Mathematical
Monographs:
Lévy (1925, 1937)
Хинчин (1938)
Gnedenko & Kolmogorov (1949-53)
Feller (1966-71)
Lukacs (1960-70)
Zolotarev (1983-86, 1997)
Janiki & Weron (1994)
Samorodnitski & Taqqu (1994)
B.V. Gnedenko, A.N. Kolmogorov, “Limit
Distributions for Sums of Independent
Random Variables“ (1949)
The prophecy: “All these distribution laws, called stable,
deserve the most serious attention. It is probable that the
scope of applied problems in which they play an essential
role will become in due course rather wide.”
The guidance for those who write books on statistical
physics: “… “normal” convergence to abnormal (that is,
different from Gaussian –A. Ch.) stable laws … ,
undoubtedly, has to be considered in every large textbook,
which intends to equip well enough the scientist in the
field of statistical physics.”
First remarkable property
of stable distributions
1. Stability: distribution of sum of independent identically distributed stable random
variables = distribution of each variable (up to scaling factor)
n
d
X 1  X 2  . . .  X n   X i  cn X
i 1
cn = n1/a , 0 <   2,  is Lèvy index
Gauss:  = 2, cn = n1/2
Corollary 1. Statistical self-similarity, random fractal
Mandelbrot: When does the whole look like its parts  description of random
fractal ptocesses
n
12
12

X
X

n

t

i
Corollary 2. Normal diffusion law, Brownian motion
i 1
or the rule of summing variances
Corollary 3. Superdiffusion, Lévy
Question: Xi are not stable r.v.
 X 
2
 X2 t
X
n
X
i 1
i
 n1   t 1  , 1/   1/ 2
second remarkable property
of stable distributions
Property 2. Generalized Central Limit Theorem: stable distributions are the
limit ones for distributions of sums of random variables (have domain of
attraction)  appear, when evolution of the system or the result of experiment is determined by the sum of a large number of random quantities
Gauss: attracts f(x) with finite
variance

x
2


x 2 f ( x )dx  

Lévy: attract f(x) with infinite
variance

x
2



x 2 f ( x )dx  
third remarkable property
of stable distributions
Property 3. Power law asymptotics 
x (-1-)


(“heavy tails”)
pˆ X (k ; , D )  exp(ikX )  exp  D | k | , 0    2, D  0  p X ( x,  , D) 

x2   , 0    2
Particular cases
1. Gauss,  = 2
All moments are finite
2. Cauchy,  = 1
Moments of order q < 1 are finite
x
q
x | x |1
0 2
 , 0  q 
 x2 
1
p ( x, 2, D ) 
exp  

 4D 
4 D


p( x,1, D ) 

D
 D2  x2
Corollary: description of highly non-equilibrium processes
possessing large bursts and/or outliers
1

_1
Due to the three remarkable properties of
stable distributions it is now believed that the
Lévy statistics provides a framework for the
description of many natural phenomena in
physical, chemical, biological, economical, …
systems from a general common point of view
Electric Field Distribution in an Ionized
Gas (Holtzmark 1919)
Application: spectral lines broadening in plasmas
eri
Ei  3
ri
N
n   const
V
E  E1  E2  ...  En

3  dim ensional
   2 W  k  e
 
W E 
dk
3

 
W k e
a k
3
2
 ikE
 
W E ~
symmetric
1
5
E
2
stable
distribution
  32
Related examples
 gravity field of stars (=3/2)

electric fields of dipoles (= 1) and quadrupoles (= 3/4)
 velocity field of point vortices in fully developed
turbulence (=3/2) ……
(to name only a few)
 also: asymmetric Levy stable distributions in the first
passage theory, single molecule line shape cumulants in
glasses …
Normal vs Anomalous Diffusion
Normal
diffusion
 R  R0 
2
Anomalous diffusion
1
 cdim Dt  t
Bachelier (1900), Einstein (1905)
 R  R0 
Superdiffusion
 1
Gauss
n  1000
 turbulent media
( gases, fluids, plasmas )
 Hamiltonian chaotic
systems
 deterministic maps
Cauchy
n  3000

foraging

financial markets
movement
2
 t ,  1
Subdiffusion
 1
 turbulent plasmas
 transport on fractals
 contamimants in
underground water
 amorphous solids
 convective patterns
 polymeric systems
 deterministic maps
Self-Similar Structure of the Brownian and the
Lévy Motions
Normal diffusion
X 2 (t ) µ t
Levy Superdiffusion
X 2 (t ) µ t m,
m = 1.66 > 1
x100
x100
“If, veritably, one took the position from
second to second, each of these rectilinear
segments would be replaced by a polygonal
contour of 30 edges, each being as complicated
as the reproduced design, and so forth.”
J.B. Perren
x100
x100
“The stochastic process which we call linear
Brownian motion is a schematic representation which describes well the properties of real
Brownian motion, observable on a small
enough, but not infinitely small scale, and
which assumes that the same properties exist
on whatever scale.” P.P. Lévy
NORMAL vs ANOMALOUS DIFFUSION
Brownian motion of a small grain of
putty
Foraging movement of spider
monkeys
(Jean Baptiste Perrin: 1909)
Gabriel Ramos-Fernandes et al. (2004)
R 2 (t )  t
R 2 (t )  t1,7
Classical example of superdiffusion
Richardson: diffusion of passive admixture in locally isotropic turbulence (1927)
l 2 ( )   3
a) Frenkiel and Katz (1956),
b) Seneca (1955),
c) Kellogg (1956)
l 2 ( )
versus  (Gifford,
1957)
Monin: space fractional diffusion equation (1955, 1956)

p(l , )   D()1/ 3 p(l , )

1/ 3 linear integral operator with
()
2
2
2
complicated kernel (Monin)



1/ 3



, D  c
2
2
2
x
y
z
Diffusion of tracers in fluid flows.
Large scale structures (eddies, jets or convection rolls) dominate the transport.
Example. Experiments
in a rapidly rotating
annulus (Swinney et al.).
Ordered flow:
Levy diffusion
(flights and traps)
Weakly turbulent
flow:
Gaussian diffusion
Anomalous Diffusion in Channeling
Fig.1. NORMAL DIFFUSION:
randomly distributed atom
strings
Fig.2. SUPERDIFFUSION: positively and negatively
charged particles in a periodic field of atom strings along
the axis <100> in a silicon crystal.
Anomalous diffusion in Lorentz gases with infinite «horizon»
(Bouchaud and Le Doussal; Zacherl et al.)
 Distribution of the length the paths
 The typical displacement
p (l ) 
X  X
1 R
, l 
l3
2 1/ 2
 t1/ 2 (ln t )1/ 2
Polymer adsorption and self - avoiding Levy flights
(de Gennes; Bouchaud)
 Distribution of the size of the loops decays as a power law
 the projection of the chain’s conformation on the wall: self avoiding Levy flight (two adsorbed monomers cannot occupy the
same site)
1
p
(
l
)

,  1
 at the adsorption threshold
1 
l
Anomalous diffusion in a linear array of convection rolls
(Pomeau et al.; Cardoso and Tabeling)
each roll acts as a trap with a release
time distribution decaying as
X  t 1/ 3
1
 (  )  1 

Impulsive Noises Modeled with the Lévy
Stable Distributions
Naturally serve for the description of the processes
with large outliers, far from equilibrium
Examples include :
 economic stock prices and current exchange rates (1963)
 radio frequency electromagnetic noise
 underwater acoustic noise
 noise in telephone networks
 biomedical signals
 stochastic climate dynamics
 turbulence in the edge plasmas of thermonuclear devices (Uragan
2M, ADITYA, 2003; Heliotron J, 2005 ...)
Lévy noises and Lévy motion
LÉVY MOTION:
LÉVY NOISES
successive additions of the noise values
Lévy index   outliers 
Lévy index   “flights” become
longer
EXPERIMENTAL OBSERVATION OF LÉVY
FLIGHTS IN WIND VELOCITY
(Lammefjord, denmark, 2006)
Distribution of increments
25 min
250 min
Velocity increments
2.5 min
Measuring masts
Lévy Noise vs Gaussian Noise
Lévy turbulence in „Uragan 3M“
V.Gonchar
et.al., 2003
DEM/USD exchange rate
V.Gonchar, A.
Chechkin,
2001
«Lévy Turbulence» in boundary plasma of
stellarator «Uragan 3М» (IPP KIPT)
l  3, m  9, R0  1 m, a  0.1 m
Bφ  0.7 T,  (a ) / 2π  0.3,
PRF  200 kW,   60 ms,
ne  1018 m -3 , Te (0)  500 eV, Ti  100 eV
Helical magnetic coils
(from above)
Poincare section of magnetic line
, ЛЗ – Langmuir prove
Ion saturation current
at different probe positions:
а) R = 111 сm ; б) R = 112 сm ;
в) R = 112.5 сm ; г) R = 113 сm.
«Lévy Turbulence» in boundary plasma of
stellarator «Uragan 3М» (IPP KIPT)
1. Kurtosis and “chi-square” criterium: evidence of non-Gaussianity.
2. Modified method of percentiles: Lévy index of turbulent fluctuations
Ion saturation current  n
Floating potential  
_2
 Fluctuations of density and potential measured in boundary plasma of stellarator
“Uragan 3M” obey Lévy statistics (September 2002)
 similar conclusions about the Lévy statistics of plasma fluctuations have been
drawn for
 ADITYA (March 2003)
 L-2M ( October 2003)
 LHD, TJ-II (December 2003)
 Heliotron J (March 2004)
“bursty” character of fluctuations in other devices (DIII-D, TCABR, …)
“Levy turbulence” is a widely spread phenomenon 
Models are strongly needed !
problem:
KINETIC
THEORY
FOR NON-GAUSSIAN LÉVY WORLD
???
 KINETIC EQUATIONS WITH
FRACTIONAL DERIVATIVES …
Fractional Kinetics  “strange”
Kinetics
M.F. Schlesinger, G.M. Zaslavsky, J. Klafter, Nature (1993)) : connected with deviations
“Normal” Kinetics
Diffusion law
Fractional Kinetics
x 2 (t )  t 
x 2 (t )  t
,  1
x 2 (t )  ln t ,   0
Relaxation
Exponential
Non-exponential
“Lévy flights in time”
Stationarity
Maxwell-Boltzmann
equilibrium
Confined Lévy flights
as non-Boltzmann
stationarity
Types of “Fractionalisation”
 time – fractional



t
t 
f
  dU 
2 f

f D 2

t x  dx 
x
, 0   1
FOKKERPLANCK EQ
Caputo/Riemann-Liouville derivative
 space / velocity – fractional
2
x
2


x

, 0   2
 multi-dimensional / anisotropic
Question : Derivation ?
Answer:
Generalized Master Equation,
Generalized Langevin Equation
2
v
2


v

Riesz derivative
 time-space / velocity fractional
BUT:
OUT OF THE SCOPE
OF THE TALK !
Kinetic equations for the stochastic
systems driven by white Lévy noise
(assumptions: overdamped case, or strong friction limit, 1-dim, D = intensity of the
noise = const: no Ito-Stratonovich dilemma!)
Langevin description, x(t)
U(x) : potential energy, Y(t) : white noise,  : the Lévy index
dx
dU
 = 2 : white Gaussian noise

 Y (t )
dt
dx
0 <  < 2 : white Lévy noise
Kinetic description, f(x,t)
f
  dU 
 f
 
f D

t x  dx 
x
 /  x  :Riesz fractional derivative:
integrodifferential operator
 = 2 : Fokker - Planck equation (FPE)
0 <  < 2 : Fractional FPE (FFPE)
Definition of the Riesz fractional derivative via its Fourier representation

ˆ(k ) 
 dx e
 ( x)
ikx
FT
  ( x)

d
dx
Indeed,
Example:
d 2


FT

dk
d 2
2 ˆ
 ikx
 
 k  ( k )e
 2
2
2

dx
d x

 
 ( x) 
1
1  x2

 k ˆ( k )
.But:
d FT
  k ˆ(k ) 
d x
d FT
  ikˆ(k )
dx
, ˆ(k )   exp( k )
d     dk exp(ikx) k  ˆ(k )  ( 1) cos  1 arctan x 



 2
( 1)/2



dx


2
 1 x




Space-fractional diffusion equation, U=0:
free Lévy flights
f
 f
D
, D  const , f ( x,0)   ( x )

t
x
After Fourier transform:
fˆ  D k  fˆ , fˆ (k ,0)  1
t

fˆ (k , t )  exp  D k  t
Solution in Fourier - space:
characteristic function
of free Lévy flights
Characteristic diffusion displacement
x
1/
 Dt 




 superdiffusion:
faster than t1/2
(0<<2)

Free Lévy flights vs Brownian motion in
semi-infinite domain
Brownian Motion,  = 2
First passage time
PDF
p(t ) 
a
4 Dt 3
e
 a 2 / 4 Dt
 t 3/ 2
Lévy Motion, 0<<2
p(t )  t 3/ 2
Sparre Andersen universality
Method of images
fim ( x, t )  W ( x  a, t )  W ( x  a, t )
pim (t )  t 11/ 
 t 3/ 2
Method of images breaks down for Lévy flights (A. Chechkin et. al., 2004)
Free Lévy flights vs Brownian motion in
semi-infinite domain: Leapover
(T. Koren. A. Chechkin, Y. Klafter, 2007)
Brownian Motion,  = 2
Leapover: how large
is the leap  over the
boundary by the
Lévy stable process ?
No leapover
Lévy Motion, 0<<2
f ( a )   a1 / 2
Leapover PDF decays slower than Lévy flight PDF
Reminder: Stationary solution of the FPE
in external field,  = 2
Equation for the stationary PDF:
d  dU 
d2 f
f  D 2  0

dx  dx 
dx
(1)
Boltzmann’ solution:
 U ( x) 
f ( x)  C exp 
 ,
D 

ax 2 bx 4 bx 2m 2
U ( x) 
;
;
2
4 2m  2

 dxf ( x)  1

, a  0, b  0, m  0,1,2,...
Two properties of stationary PDF:
1. Unimodality (one hump at the origin).
2. Exponential decay at large values of x.
(2)
Lévy flights in a harmonic potential, 1  < 2
FFPE for the stationary PDF (dimensionless units):
d  dU  d  f
f 
0

dx  dx  d x 
x2
d  FT

U ( x) 
, (Remind: pass to the Fourier space
 k )

2
d x
FT
f ( x)  fˆ (k )
Equation for the characteristic function:
dfˆ
 1 ˆ
  sgn( k ) k
f (k )
dk
 k 
 : symmetric stable law
fˆ (k )  exp 
  


Two properties of stationary PDF:
1. Unimodality (one hump at the origin).
sin  / 2  ( )


2. Slowly decaying tails: f ( x  )  C
,
C

,
1

x


x2


 dx x
Harmonic force is not “strong” enough to “confine” Lévy flights
2f
( x)  
Confined Lévy flights. Quartic
potential well, U  x4
(A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2002)
Langevin equation:
dx
  x3  Y (t )
dt
 Fractional FPE for the stationary PDF:
d 3
d f
x f 
0

dx
d x
 
 FT


d

 reminder :
k



dx


(1)
FT
f ( x)  fˆ (k )
Equation for the characteristic function:
d 3 fˆ
dk
3
 sgn( k ) k  1 fˆ (k )
+ normalization + symmetry + boundary conditions …
(2)
Confined Lévy flights. Quartic
potential, U  x4. Cauchy case,  = 1
Equation for the characteristic function:
d 3 fˆ  sgn(k ) fˆ (k ) 
dk 3
PDF:
f ( x) 
fˆ (k ) 

2 exp  

3











k   3k  
 cos 
 


2  2 6
1
 (1  x 2  x 4 )
Two properties of stationary PDF
1. Bimodality: local minimum at xmin  0, two maxima at xmax  1/ 2
2. Steep power law asymptotics with finite variance:
x2 



dx x 2 f ( x )  1
f ( x)  x 4

Two propositions
(A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2004)
Proposition 1. Stationary PDF for the Lévy flights in external field U  x c , c  2
is non-unimodal. Proved with the use of the “hypersingular” representation of the
Riesz derivative.
c
Proposition 2. Stationary PDF for the Lévy flights in external field U  x , c  2
has power-law asymptotics,
C
f ( x) 
x
  c 1
,
x 
C     1 sin  2  is a “universal constant”, i.e., it does not depend on c.
Critical exponent ccr: ccr  4  
c  ccr  x 2  
c  ccr  x 2   : “confined”
Proved with the use of the representation of the Riesz derivative in terms of left- and
right Liouville-Weyl derivatives.
Numerical simulation
Langevin equation with a white Lévy noise,
dx
dU

 Y (t ),
dt
dx
U  x 2m  2 , m  0,1, 2,...
Illustration: Typical sample paths
Potential wells
Brownian motion
m   walls are steeper 
Lévy flights are shorter 
confined Lévy flights
Lévy motion
Quartic potential U  x4,  = 1.2. Time
evolution of the PDF
Potential well U  x5.5,  = 1.2. Trimodal
transient state in time evolution of the PDF
Potential well U  x5.5,  = 1.2. Trimodal
transient state in a finer scale
Confined Lévy flights in bistable potential:
Lévy noise induced transitions
(stochastic climate dynamics)
(A. Sliusarenko et al., 2007)
dx
dU

 Y (t )
dt
dx
U  x    x2 / 2  x4 / 4
 1 
TGauss  D   exp 
 , D  1
 4D 
Fig.1. Escape of the trajectory over
the barrier, schematic view.
Fig.2. Typical trajectories for different .
Confined Lévy flights in bistable
potentiaL: Kramers’ probLem
Completely unusual
behavior
of T at small D’s:
5













1.5
2.0
2.5
3.0
1.10
1.5
1.06
1.04
3.5


0.5
0.0
0.0
0.5
1.0
1.5
2.0

1.00
Escape probability p(t) as function
of walking time t:
-5
0.98




-6
-7
-8
0.5
1.0
1.5

A.V. Chechkin et al., 2007
-9
-10
-11
-12
D
  
1.0
1.02
-lg D
p t   1 exp   t T 


T
2.0
1.08
1
1.0
1.12
lg C()
2
C  
1.14

3
ln p(t)
lg Tesc
4
T
0
500
1000
t
1500
2000
2.0
Damped Lévy flights in velocity space.
Damping by dissipative non-linearity
(posed by B. West, V. Seshadri (1982), Chechkin et al. 2005)
Langevin equation V (dt )   (V )V (t )dt  L (dt )
FFPE
P V , t 
t

 P

V  V  P   D 
V
V
0 = 1.0, 1 = 0.0001, 2 = 0 (curve 1)
 (V )   (V )   0   1V   2V ,  n  0  = 1.0,  = 0,  = 0.000001 (curve 2),
0
1
2
 = 1.0, slopes = -1, -3, -5
2
4
Damped Lévy flights:
2nd Moments
4th Moments
Power-law truncated Lévy flights
(I. Sokolov, A. Chechkin, Y. Klafter, 2004)
 “PLT Lévy flights” : PDFs resembles Lévy stable distribution in the central part
 at greater scales the asymptotics decay in a power-law way, but faster, than the Lévy
stable ones, therefore, x 2    the Central Limit Theorem is applied 
 at large times the PDF tends to Gaussian, however, sometimes very slowly
Particular case of distributed order space fractional diffusion equation:

 2
1  C

 | x |2

 f ( x, t )
 2 f ( x, t )
D

 t
x 2

f ( x, t )
0<<2
(5   )sin  / 2  DC t

x
5
2
x (t )  
 2 fˆ
2
k k 0
 2 Dt
, x
The Lévy distribution is truncated by a power law with a power between 3 and 5. Due to
the finiteness of the second moment the PDF f(x,t) slowly converges to a Gaussian
 Probability of return to the origin:
f (0, t ) 



dk |k | t
e
 t 1/ 
2
, 0   2
The slope –1/ in the double logarithmic scale changes into -1/2 for the Gaussian
Power-law truncated Lévy flights:
Theory and experiment
Theory
Experiment on the ADITYA tokamak
Experiment on the ADITYA: probabilities of return to the origin for different radial positions of the
probes [R.Jha et al. Phys. Plasmas.2003.Vol.10.No.3.PP.699-704]. Insets: rescaled PDFs.
Evolution of the PDF of the PLT Lévy
process (I. Sokolov et al., in press)
PDF at small time = 0.001
PDF at large time = 1000
Intermediate zÁver
Scale-free models provide an efficient description of a broad variety of
processes in complex systems
This phenomenological fact is corroborated by the observation that the
power-law properties of Lévy processes strongly persist, mathematically, by
the existence of the Generalized Central Limit Theorem due to which Lévy
stable laws become fundamental
All naturally occurring power laws in fractal or dynamic patterns are finite
Here we present examples of regularisation of Lévy processes in presence
of
― non-dissipative non-linearity : confined Lévy flights,
― dissipative non-linearity : damped Lévy flights,
― power law truncation of free Lévy flights
These models might be helpful when discussing applicability of the Lévy
random processes in different physical, chemical, bio..... , financial, … systems
From flights in space – to flights in time
Harvey Scher and Elliott W. Montroll, Anomalous transit-time dispersion in
amorphous solids. Physical Review B, vol.12, No.6, 2455-2477 (1975).
Measuring photocurrent
Idealized transient current trace
Experimental and theoretical plots, log-log scale
Lévy flights in time
Model: jumps between traps, waiting time  distributed with w().
Two possibilities:
1. Duration of experiment >> typical waiting time in a trap = <>  normal
diffusion.
2. Slow decaying asymptotics:
w( ) const  (1  ) ,0    1,   
 SLOW anomalous diffusion.
Propagator for normal diffusion
Propagator for subdiffusion
Time-fractional Fokker-Planck equation
for Lévy flights in time
 f
  dU 
2 f

f D 2


x  dx 
t
x
w( )
1
 1
, 0   1,
 
Fractional Caputo derivative:

 (t )   ( s)   dt (t )e
0
 st


t 

 s  ( s)  s
 1
 (0)
Solution: separation of variables
  1:

 s   (0)
t
f n ( x, t )  Tn (t )n ( x )
Time evolution: Mittag – Leffler relaxation


n t  

exp

,

t
 1



n





1




Tn (t )  E  nt  

 1 1

, n t   1

 n  (1   )t


Stretched exponential law
Slow power law relaxation
General ZÁver
Theory of Lévy stable probability laws :
Mathematical basis for the description of a broad range of non-Brownian
random phenomena
Kinetic theory of Lévy random motion:
Based on diffusion-kinetic equations with fractional derivatives in time, space
and/or velocity
Approach based on Lévy random motion and fractional kinetic
equations:
already proved its effectiveness in different physical, bio, … financial
problems
BUT: THEORY IS ONLY AT THE BEGINNING
A. Chechkin, V. Gonchar, J. Klafter and R. Metzler, Fundamentals of Lévy Flight
Processes. Adv. Chem. Phys. 133B, (2006).
Б.В. Гнеденко, А.Н. Колмогоров
“Предельные распределения для сумм независимых
случайных величин” (1949):
“Все эти законы распределения, называемые “устойчивыми”
…заслуживают, как нам кажется, самого серьезного
внимания. Вероятно, круг реальных прикладных задач, в
которых они будут играть существенную роль, окажется со
временем достаточно широким.”
“ … “нормальная” сходимость к ненормальным (т.е. отличным
от гауссовского –А.Ч.) устойчивым законам … , несомненно,
уже сейчас должна быть рассмотрена во всяком большом
руководстве, рассчитывающем достаточно хорошо
вооружить, например, работника в области
статистической физики.”
Ю.Л. Климонтович (2002), R. Balescu (2007)
Dyakuem za pozornost’ !