title: Lyapunov exponent for Whitney's problem with random drive
authors: Nikolai A. Stepanov Mikhail A. Skvortsov
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      Skolkovo Institute of Science and Technology, Moscow 121205, RussiaL. D. Landau Institute for Theoretical Physics, Chernogolovka 142432, RussiaSkolkovo Institute of Science and Technology, Moscow 121205, RussiaL. D. Landau Institute for Theoretical Physics, Chernogolovka 142432, Russia
      
      
      
      
      We consider the statistical properties of a non-falling trajectory in the
      Whitney problem of an inverted pendulum excited by an external force. In the
      case when the external force is white noise, we recently found the
      instantaneous distribution function of the pendulum angle and velocity over
      an infinite time interval using a transfer-matrix analysis of the
      supersymmetric field theory. Here, we generalize our approach to the case of
      finite time intervals and multipoint correlation functions. Using the
      developed formalism, we calculate the Lyapunov exponent, which determines
      the decay rate of correlations on a non-falling trajectory.
      Lyapunov exponent for Whitney's problem with random drive
          Mikhail A. Skvortsov
          August 27, 2020
      =========================================================
      
      1.
      Balancing an inverted pendulum under a given time-dependent horizontal force
      f(t) is a famous mathematical problem formulated by Courant and Robbins in
      their book What is Mathematics? (first edition in 1941) <cit.>,
      where Whitney was credited as the author of the problem. Using fairly
      general mathematical arguments based on the intermediate value theorem, they showed
      that for any force f(t) acting during a finite time interval [0,T], an
      initial position of the pendulum in the upper half-plane can be chosen such
      that it will remain in the upper half-plane during the further evolution for
      all t∈[0,T]. The existence of a non-falling trajectory (non-FT) in
      the Whitney problem has been the subject of an ongoing debate in the
      mathematical literature <cit.>, resulting in a critical
      analysis and refinement of the original arguments of Courant and Robbins.
      Fresh interest in the problem of an inverted pendulum is associated with
      Arnold, whose view in 2002 was that this problem still awaits a rigorous
      solution <cit.>. In 2014, Polekhin presented a proof of the existence
      of the non-FT using the Wazewski topological principle <cit.>. This work
      provoked several publications generalizing his approach and proposing new
      topological methods <cit.> (see Refs.<cit.> for good reviews of the history of the Whitney
      problem).
      
      
      
      Recently we developed a theory of the statistical description of a
      never falling trajectory (NFT) of an inverted pendulum under the
      action of a random force <cit.>. An NFT can be regarded as the limit of
      non-FTs in the Whitney problem as the length T of the time interval tends
      to infinity. The NFT concept is illustrated in Fig. <ref>, which shows 
      numerical solutions to the boundary value problem for the pendulum equation
      (the angle θ is measured from the vertical)
      θ̈= ω^2 sinθ + f(t) cosθ
      with different initial and final values θ(0)=θ_1 and
      θ(T)=θ_2 and a sufficiently rapidly varying force f(t). For
      any θ_1,2 in the strip -π/2<θ_1,2<π/2, a non-falling
      solution (-π/2<θ(T)<π/2) of this boundary value problem exists
      and is unique <cit.>. As θ_1 and θ_2 run through all
      possible values in the strip, the set of corresponding non-FTs form a
      bundle, shown in color in Fig. <ref>. This bundle shrinks as one
      moves away from the boundary, becoming exponentially thin in the middle of
      the interval for large T. In the limit T→∞, when the pendulum
      must be balanced on the entire real axis, the non-FT bundle for the Whitney
      problem on a finite time interval becomes infinitely thin and defines a
      unique never falling trajectory, which is a functional of the given force f(t).
      
      In Ref. <cit.>, we studied the statical properties of an NFT in the case
      when the driving force is Gaussian white noise with the correlator
      f(t)f(t') = 2 αδ(t-t'),
      
      and calculated the instantaneous distribution function P(θ,p) of
      the angle θ and its velocity p=θ̇. Our approach is based on
      the supersymmetric field theory formulation of stochastic dynamics proposed
      by Parisi and Sourlas <cit.>, which allows averaging over the
      random force at the very beginning of the calculations. It is essential that
      for the considered problem, the Parisi–Sourlas method is free from the
      problem of the sign of the fermionic determinant because of the uniqueness of the non-FT. 
      Using the idea of reducing the one-dimensional functional integral
      to an effective quantum mechanics <cit.>, we were able to express the
      distribution function P(θ,p) in terms of the zero mode of the
      transfer-matrix Hamiltonian, which reduces to the Fokker–Planck operator
      with a special type of boundary conditions ensuring that the trajectories do
      not leave the strip.
      
      Here, we extend the ideas of Ref. <cit.> and consider a range of issues
      related to the Lyapunov exponent for a non-FT. The Lyapunov exponent
      determines both the law of the convergence of a non-FT on a finite time
      interval to the NFT on an infinite time interval (see Fig. <ref>)
      and the decay of different-time correlators on the NFT. From the
      technical standpoint, our result consists in describing the entire spectrum
      of the transfer-matrix Hamiltonian, whose zero mode was studied in Ref. <cit.>. 
      In this language, the Lyapunov exponent is determined by the energy
      of the first excited state. The developed theory allows calculating any
      correlation functions for a non-FT on infinite, semi-infinite, and finite
      time intervals.
      
      We show that the Lyapunov exponent in the Whitney problem with white-noise 
      driving (<ref>) can be written as
      λ = ω g(α/ω^3) ,
      
      where the function g(x) has the asymptotic behavior
      
        g(x) =
        
          1
          + 3/8 x
          - 525/1024 x^2
      
      
          + … ,     x≪1 , 
      
           0.66    x^1/3 + 0.26 + 0.30  x^-1/3…,     x≫1 .
        
      In the absence of driving (α=0), the Lyapunov exponent
      λ=ω is determined by the exponential instability of the
      trajectories near the upper pendulum position. For weak driving
      (α/ω^3≪1), the typical non-FT angle is of the order
      θ∼(α/ω^3)^1/2 <cit.>, and the nonlinearity of
      Eq. (<ref>) leads to an increase in the Lyapunov exponent, which can be
      expanded in an asymptotic series in powers of the small parameter
      α/ω^3. Finally, under strong driving (α/ω^3≫1),
      the Lyapunov exponent becomes independent of ω, reaching the limiting
      value λ≈0.66 α^1/3. We show the numerically found
      dependence of the Lyapunov exponent on α/ω^3 in Fig. <ref>.
      
      2.
      According to the approach developed in Ref. <cit.>, the statistical properties
      of a non-FT are expressed in terms of the two-component “wave function”Ψ̂(θ,p)=(Ψ,Φ)^T, whose evolution is governed by the imaginary-time
      Schrödinger equation with the corresponding transfer-matrix Hamiltonian:
      ∂/∂ t[ Ψ; Φ ]
        =
        -
         H[ Ψ; Φ ] ,
           H
        =
        [   L  -1; V_2   L ] ,
      
      where L is the Fokker–Planck operator for the Kramers
      problem <cit.>,
      
        L = p∂_θ
        + ω^2sinθ ∂_p
        - αcos^2θ ∂_p^2 ,
      
      and the potential V_2 has the form
      
        V_2
        =
      - ω^2 cosθ
       - αsin2θ ∂_p .
      
      
      In Ref. <cit.>, we studied the one-point correlation function of the NFT on
      the infinite time interval, which is determined by the zero mode
      Ψ̂_0 of the Hamiltonian H. Finding the zero mode is
      significantly simplified due to the presence of the Becchi-Rouet-
      Stora-Tuytin (BRST) symmetry of the action in the Parisi–Sourlas
      representation of stochastic dynamics <cit.>, 
      which allows expressing both components of
      Ψ̂(θ,p) in terms of a scalar superpotential ψ(θ,p)
      via
      Ψ =  ∂_p ψ ,
          Φ = -  ∂_θψ .
      
      The time evolution of ψ is determined by the Fokker–Planck
      operator L:
      
          ∂ψ/∂ t
            =
            -
          L ψ
          .
      
      But reduction (<ref>) works neither for calculating multi-time
      correlation functions of the NFT nor for describing the statistics of a
      non-FT on bounded intervals. In the former case, the BRST symmetry is
      broken by the operators of physically observable quantities acting
      identically on the wave function components Ψ and Φ. In the latter
      case, the BRST symmetry is broken by the BRST-asymmetric initial condition at
      the boundary of the interval [see Eq. (<ref>) below]. In both cases,
      to describe the non-FT statistics, one must work with the two-components 
      wave function (Ψ,Φ) and understand the properties of the Hamiltonian
      H.
      
      We start with discussing the initial condition for the wave function at the boundary 
      of an interval. To ensure that the non-FT is unique, we must fix the value of
      θ at the boundary. (Generally speaking, one can fix the value of
      θ̇ or even a linear combination of θ and θ̇, but
      for simplicity, we assume that the angle is given.) By construction, the
      wave function Ψ̂ is closely related to the partition function of the
      supersymmetric functional integral <cit.>. Right at the boundary, it
      cannot contain Grassmann variables, which leads to the component Φ
      vanishing. Hence, the wave function at the interval boundary with the fixed value
      θ=θ_0 has the form
      Ψ̂_θ_0^(b)
        =
        [ δ(θ-θ_0);        0 ]
      .
      
      
      Consider the boundary value problem [T_L,T_R] with the
      boundary conditions θ(t_L)=θ_L  and
      θ(t_R)=θ_R. The essence of the reduction of the
      Parisi–Sourlas integral to the quantum mechanics (<ref>) is that the
      correlation function O_1(t_1)O_2(t_2)… of physical quantities
      O_i at the instants t_i (t_1<t_2<…) can be represented as the matrix
      element
      Ψ̂_θ_L^(b) | …
          O_2(t_2) e^- H(t_2-t_1)
          O_1(t_1) e^- H(t_1-t_L)
        | Ψ̂_θ_L^(b) ,
      
      where the scalar product of two wave functions is defined as <cit.>Ψ̂| Ψ̂'
        =
        ∫ dθ  dp 
        [
          Ψ(θ,p)Φ'(θ,-p)
        + Φ(θ,p)Ψ'(θ,-p)
        ]
        .
      
      In Ref. <cit.>, we studied the instantaneous joint distribution function
      P(θ,p) of the angle and velocity on the NFT corresponding to the
      operator O=δ(θ-θ_0)δ(p-p_0). Replacing
      Ψ̂_L,R with the zero mode and using Eq. (<ref>), one
      can express P(θ,p) in terms of the Poisson bracket of the
      superpotential ψ:
      
          P(θ,p)
            =
            {ψ(θ,p), ψ(θ,-p)
            }_θ,p .
      
      
      Both the Hamiltonian in Eq. (<ref>) and the Fokker–Planck operator (<ref>) 
      are non-Hermitian. Generally speaking, such operators can lack
      a complete system of eigenfunctions. However it is known that in the presence 
      of friction the Fokker–Planck operator can be diagonalized <cit.>, 
      which makes it possible to construct a system of biorthogonal eigenfunctions 
      and work with them practically as with eigenfunctions of a Hermitian 
      operator <cit.>. But there is no friction in our case, 
      and we should therefore expect that the operators H and
      L reduce to the Jordan normal form. This results not in a simple
      exponential decay of correlators as t→∞ but in the appearance of
      additional powers of time [e.g., as can be seen in
      expression (<ref>)].
      
      3.
      To illustrate the developed approach, we consider the case of a weak
      noise (α/ω^3≪1) in detail, where the Jordan structure of the
      operators H and L can be studied analytically. We start with the
      Fokker–Planck operator. In the considered limit, the deviation of the
      pendulum from the vertical is small (θ≪1), and the operator (<ref>) 
      can be replaced with 
      
        L = p∂_θ
        + ω^2θ∂_p
        - α∂_p^2 .
      
      The zero mode of this operator corresponding to the NFT has the form
      ψ_0(θ,p)
        =
        (z)/2 ,
      
      where we introduce “holomorphic” and “antiholomorphic” coordinates with
      different signs of the momentum,
      
        z = κ (p-ωθ)
      ,   z = - κ (p+ωθ) ,
      
      where κ=√(ω/2α). The spectrum of the
      operator (<ref>) can be found using the identity [L,∂_z]=
      ω∂_z, which allows generating the eigenfunctions by
      consecutively differentiating the zero mode with respect to z. We thus
      find the eigenfunction of the nth excited state (n=1,2,3,…) with the
      energy ϵ_n=nω:
      ψ_n
        =
        1/√(π)
        H_n-1(z) e^-z^2 ,
      
      where H_n(z)=(-1)^ne^z^2d^ne^-z^2/dz^n is the Hermite polynomial (in
      the physical definition). But the functions ψ_n(θ,p) thus
      constructed depend only on the difference p-ωθ (do not
      contain z) and therefore do not form a complete basis. This
      circumstance is related to the fact that the non-Hermitian operator (<ref>)
      can be brought to the Jordan normal form and in addition to the eigenfunction has
      several generalized eigenfunction corresponding to the same eigennumber _n. 
      It is easy to verify that the eigenfunction ψ_n has n-1 generalized 
      eigenfunctions, which we choose in the form
      ψ_n,k
        =
        (-1)^k/2^kk!√(π)
        H_k(z) H_n-k-1(z) e^-z^2 ,
      
      where the index k ranges from 1 to n-1. Together with ψ_n,0=ψ_n,
      they form the basis of a Jordan block of dimension n corresponding to the
      energy ϵ_n=nω:
      
        L ψ_n,k
        =
        ϵ_n ψ_n,k + ωψ_n,k-1
      (to truncate the chain at the eigenfunction ψ_n,0, we set
      ψ_n,-1=0).
      
      The constructed system of functions is complete. An arbitrary function can
      be decomposed with respect to the basis ψ_n,k using the orthogonality
      relation
      ψ_n,k|ψ_n',k'_z
        =
        (-1)^n-1δ_n,n'δ_k+k'+1,n ,
      
      where the scalar product · | · _z is defined as
      ψ|ψ'_z
        =
        ∫ dz   dz ψ(z,z)
        ψ'(z,z) ,
      
      and exchanging the arguments in one of the functions thus agrees with the
      sign change for p in Eq. (<ref>). We note that the integration
      measures in Eqs. (<ref>) and (<ref>) are related by
      dz dz=2ωκ^2dθ dp.
      
      During the evolution of the wave function ψ_n,k under the action of
      the operator L, other states of the Jordan block corresponding to the same
      energy are mixed into it, which leads to the appearance of powers of t on
      top of the exponential decay:
      
        e^-Ltψ_n,k
        =
        e^-n ω t∑_m=0^k
        (-ω t)^m/m!ψ_n,k-m .
      
      
      We now turn to studying the spectral properties of the Hamiltonian H
      in Eq. (<ref>). In the considered case of weak noise, Eq. (<ref>)
      gives V_2=-ω^2, which partitions the state space of H into
      even and odd sectors with the wave functions Ψ̂_e,o=(Ψ,±ωΨ)^T
      evolving independently with the Hamiltonians H_e,o=L∓ω. The
      system of eigenfunctions and generalized eigenfunctions of the operator L 
      constructed above thus allows completely describing the evolution of the doublet
      Ψ̂ under the action of the Hamiltonian H.
      
      Consider the evolution of the wave function (<ref>) 
      away from the boundary in the limit α/ω^3≪1.
      Decomposing it into even and odd components, we obtain
      
        e^- HtΨ̂_θ_0^(b)
        =
        [  coshω t; ωsinhω t ]
        e^-Ltδ(θ-θ_0) .
      
      To calculate the evolution of the delta function, we expand it in the basis
      ψ_n,k:
      δ(θ-θ_0)
        =
        ∑_n=1^∞∑_k=0^n-1 c_n,kψ_n,k .
      
      The coefficients c_n,k can be obtained using orthogonality
      relations (<ref>) and the properties of Hermite polynomials
      
        H_n(x+y)
        =
        ∑_m=0^n n  k (2y)^n-k H_k(x)
      
      following from the Taylor expansion, and are given by
      
        c_n,k
        =
        (-1)^n-1
        2κω(θ_0/2κω)^n-2k-1/(n-2k-1)! .
      
      The evolution of the delta function in Eq. (<ref>) follows from
      expansion (<ref>) and relations (<ref>). The memory of the
      boundary is lost in the characteristic time ω^-1 (the inverse
      Lyapunov exponent). During this time, the difference between the two
      components of the wave function Ψ̂ is lost, and they both take the value
      determined by the state ψ_1,0 with the minimum energy
      ϵ_1=ω:
      lim_t→∞
        e^-HtΨ̂_θ_0^(b)
        =
        Ψ̂_0
        =
        [ 1; ω ]κψ_1,0 ,
      
      which is just the zero mode of (<ref>) in the limit
      α/ω^3≪1.
      
      4.
      We show how the developed spectral theory of the operators H and L
      allows systematically calculating various correlation functions of the
      non-FT in the case of weak noise. The results in this section can
      also be obtained directly by using the explicit expression for the non-FT in
      terms of f(t) with subsequent averaging over Gaussian white
      noise (<ref>)<cit.>, but deriving them using the transfer-matrix
      formalism is important methodologically because it illustrates the general
      scheme and allows verifying its workability.
      
      We begin by considering the calculation of the pair correlator for the NFT angle
      on the entire real axis. Substituting the zero mode (<ref>)
      into the general formula (<ref>) and taking into account that only
      the even sector of the theory does contribute, we can express the 
      correlator in terms of the scalar product (<ref>) in the
      z-representation as
      θ(0)θ(t)
        =
        ⟨ψ_1,0 | θ e^-(L-ω)tθ | ψ_1,0⟩_z .
      
      Using Eqs. (<ref>)  and (<ref>), we can express
      θψ_1,0 in terms of the functions ψ_2,0 and ψ_2,1.
      Then using the evolution law (<ref>), we obtain
      
        e^-(L-ω)tθψ_1,0
        =
        e^-ω tψ_2,1-(1/2+ω t)ψ_2,0/2κω .
      
      Calculating the matrix element (<ref>) as the overlap between the
      states e^-(L-ω)tθψ_1,0  and θψ_1,0 with the help of
      relations (<ref>), we find the sought pair correlator:
      θ(0)θ(t)
        =
        θ^2
        (1+ω t)
        e^-ω t
       ,
      
      where, as obtained in Ref. <cit.>,
      θ^2=α/2ω^3 .
      
      The appearance on the background e^-ω t of a contribution linearly
      increasing with time is related to excitation of the states ψ_2,0 and
      ψ_2,1 corresponding to the Jordan block of dimension 2.
      
      In a similar way one can calculate more complicated correlators of the NFT. For
      example,
      θ^2(0)θ^2(t)
        =
        θ^2^2
        [
        1+ 2(1+ω t)^2 e^-2ω t] .
      
      Formally, the operator θ^2 here applied to ψ_1,0 excites the
      Jordan triplet ψ_3,0, ψ_3,1, ψ_3,2, which leads to the
      appearance of terms up to t^2 on the background of the exponential
      decay. But the structure of the correlator (<ref>) is related to the
      Gaussian statistics of θ on the NFT <cit.>, which allows expressing
      it in terms of pair correlator (<ref>) using the Wick theorem.
      Generalizing the developed formalism to multipoint correlators is also straightforward.
      
      As the next example, we consider the calculation of the average angle
      θ(t)_θ_0 for the non-FT on the semi-infinite time
      interval t>0 with the boundary condition θ(0)=θ_0. According
      to Eq. (<ref>), the average angle is given by the matrix element
      θ(t)_θ_0=Ψ̂_0|θ e^- Ht|
      Ψ̂_θ_0^(b). It is easiest to calculate by convoluting
      expression (<ref>) with the wave function (<ref>) at the
      boundary. Integrating over the momentum, we see that the contribution from
      ψ_2,0=2ze^-z^2/√(π) disappears because it is odd in z, and we
      obtain the simple exponential decay
      θ(t)_θ_0
        =
        θ_0
        e^-ω t .
      
      One can derive the same expression differently by calculating the matrix element
      θ between the zero mode ψ_1,0 and evolved boundary wave
      function (<ref>). Such matrix elements are nonzero only with the
      Jordan doublet ψ_2,0 and ψ_2,1. But according
      to (<ref>), ψ_2,1 is not included in the expansion of
      the delta function, while ψ_2,0 is an eigenfunction and does not generate 
      a linear term during evolution. As a result, we again come to
      expression (<ref>).
      
      A comparison of expressions (<ref>) and (<ref>)
      shows that despite the presence of the additional factor ω t
      in Eq. (<ref>), the Lyapunov exponent can be standardly determined
      from either of the correlators at large times:
      λ
        =
        - lim_t→∞∂lnθ(0)θ(t)/∂ t
        =
        - lim_t→∞∂lnθ(t)_θ_0/∂ t .
      5.
      We now proceed to calculating the Lyapunov exponent for the non-FT for
      arbitrary values of the parameterα/ω^3. The Lyapunov exponent,
      which governs the decay of the correlations at large times, is determined
      by the energy of the first excited state. As shown above, in the case of
      weak driving, λ=ω. As the parameter α/ω^3
      increases, the anharmonicity of the pendulum leads to a deviation of
      λ from ω.
      
      For a small value of the parameter α/ω^3≪1, the nonlinear
      terms in Eq. (<ref>) can be taken into account perturbatively, 
      which allows obtaining both a correction to the
      eigenfunction ψ_n, which becomes dependent on the “antiholomorphic”
      coordinate z, and a correction to the eigenvalue ϵ_n.
      This procedure looks especially simple for the first excited state, which
      is nondegenerate and has no generalized eigenfunctions. For this, we represent
      the eigenfunction and the corresponding energy as power series in the small
      parameter x=α/ω^3:
      
          ψ_1 = [1 + h_1(z,z) x + h_2(z,z) x^2 + …] e^-z^2 ,
          
          ϵ_1 = ω(1 + γ_1 x + γ_2 x^2 + …) ,
      
      where h_m(z,z) is a polynomial of a degree not exceeding 4m.
      Substituting these expressions in the equation Lψ_1=ϵ_1ψ_1
      and solving sequentially in each order in x, we can calculate the first
      few polynomials h_m(z,z) and the coefficients γ_m. The
      result for ϵ_1 defining the Lyapunov exponent is given in
      Eq. (<ref>).
      
      A similar approach allows also finding corrections to the zero
      mode (<ref>) of the superpotential ψ_0 in powers of
      α/ω^3. As anticipated from the supersymmetry of the theory, 
      its energy remains zero. The found corrections allow obtaining an
      analytic expansion for the one-point statistics of the NFT, calculated
      numerically in <cit.>. In particular, they allow refining
      formula (<ref>) for θ^2,
      θ^2
        =
        x/2
        -13/16 x^2
          +26989/12288x^3
        + … ,
      
      and also describing the non-Gaussianity of the distribution function
      P(θ) characterized by the fourth cumulant θ^4=
      θ^4-3θ^2^2:
      θ^4
        =
      	-241/256 x^3+64725/8192x^4
      	
      	+…
      Note that the difference from the normal distribution measured by the
      kurtosis θ^4/θ^2^2 occurs only in the first
      order in x=α/ω^3. A negative value of θ^4
      is related to suppression of the tails of P(θ) due to the finiteness
      of the interval (-π/2,π/2).
      
      In the case of an arbitrary noise strength, the excited states of
      operator (<ref>) can be constructed only numerically. To determine the
      Lyapunov exponent λ=ϵ_1, we must find the first excited state
      by solving the equation Lψ=ϵ_1ψ with the boundary conditions
      
          ψ(π/2,p<0)=ψ(θ,-∞)=0,
          
          ψ(-π/2,p>0)=ψ(θ,∞)=0.
      
      These boundary conditions are similar to the boundary conditions for the
      zero mode of the superpotential derived in Ref. <cit.>, with the only
      difference that in the part of the boundary where the wave function is
      specified, its value is zero and not ±1/2.
      
      
      
      In Fig. <ref>, we show the first excited state 
      determined numerically for various values of the parameter α/ω^3.
      For small α/ω^3, the function ψ_1(θ,p) is close to the
      Gaussian ψ_1,0(z), slightly increasing near θ=±π/2. As
      α/ω^3 increases, the maximum of ψ_1(θ,p) near the
      boundaries of the interval become more pronounced, and at
      α/ω^3→∞, the first mode has two humps localized near the
      boundaries. In Fig. <ref>, we plot the energy of the first mode
      (which determines the Lyapunov exponent) as a function of the parameter
      α/ω^3. For small α/ω^3, the numerical calculation
      agrees with expression (<ref>) obtained using the perturbation
      theory up to the values α/ω^3≈0.25. For large a/ω^3, 
      the Lyapunov exponent in units of ω can be expanded in
      powers of (α/ω^3)^1/3 with the leading term λ≈
      0.66 α^1/3.
      
      In conclusion, we note that the developed theory is a generalization
      of the supersymmetric approach proposed in Ref. <cit.> to the case of 
      a non-FT on finite time intervals and to multipoint correlation functions.
      The suggested classification of the excited states of the transfer-matrix
      Hamiltonian completes the construction of the theory of the statistical
      properties of a non-FT in the Whitney problem with random short-range
      driving. The developed formalism allows finding any correlation functions on
      a non-FT by solving partial differential equations of the Fokker–Planck
      type with specific boundary conditions.
      
      The authors thank A. V. Khvalyuk and I. V. Poboiko for the help with
      numerical calculations. This work was supported by a grant from the Russian
      Science Foundation (Project No. 20-12-00361).
      
      99CR-book
      R. Courant and H. Robbins,
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      BolKoz
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      we
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      EL83
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      Risken-FPE
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      Jordan
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