One-Dimensional Fokker Planck Equation Invariant under Four- and Six-Parametrical Group

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1 Proceedings of Institute of Mathematics of NAS of Ukraine 2000, Vol. 30, Part, One-Dimensional Fokker Planck Equation Invariant under Four- and Six-Parametrical Group Stanislav SPICHAK and Valerii STOGNII Institute of Mathematics of NAS of Ukraine,3 Tereshchenkivska Street,Kyiv,Ukraine National Technical University of Ukraine Kyiv Polytechnic Institute, Peremohy Ave.,37,Kyiv 56,Ukraine Symmetry properties of the one-dimensional Fokker Planck equations with arbitrary coefficients of drift and diffusion are investigated. It is proved that the group symmetry of these equations can be one-, two-, four- or six-parametric and corresponding criteria are obtained. The changes of the variables reducing Fokker Planck equations to the heat and Schrödinger equations with certain potential are determined. Introduction Fokker Planck equation FPE) is a basic equation in the theory of continuous Markovian processes. In an one-dimensional case FPE has the form [, 2] L = u t + x [At, x)u] 2 [Bt, x)u] =0, ) 2 x2 where u = ut, x) is the probability density, At, x) andbt, x) are differentiable functions meaning coefficients of drift and diffusion correspondingly. We investigated symmetry properties of the equation ) under the infinitesimal basis operators [3 5] X = ξ 0 t, x, u) t + ξ t, x, u) + ηt, x, u) x u. 2) The symmetry operators are defined from the invariance condition ˆX L L=0 =0, 3) 2 where ˆX is the second prolongation of the operator X, which is constructed according to the 2 formulae [3 5]. From the condition of invariance 3), equating coefficients by a function u and its derivatives u x, u tt, u tx, u xx u t can be expressed from equation )) to zero it is possible to determine the following system of equations on functions ξ 0, ξ, η: ξ 0 = ξ 0 t), ξ = ξ t, x), η = χt, x)u, 2ξxB ξ B ξ B x ξ 0 B t =0, ξ A B x )ξt + ξ 0 A t B tx )+ξ A x B xx ) ξxa B x )+ 2 Bξ xx = Bχ x, χ t + ξt 0 A x ) 2 B xx + ξ 0 A tx ) 2 B txx +ξ A xx ) 2 B xxx + χ x A B x ) 2 Bχ xx =0. 4) Here lower indexes t, x mean differentiation on corresponding variables. Let us also introduce the following notations t = t, x = x, u = u.

2 One-Dimensional Fokker Planck Equation Invariant under Criterion ofinvariance FPE under fourand six-parametrical group ofsymmetry In [6] following Theorem was proved: Theorem. If there is a symmetry operator 2) Q u u for FPE ) then there exists a transformation of a form t = T t), x = Xt, x), u = vt, x)ũ, which reduces it to equation ) with coefficients of drift and diffusion à = A x), B = B x). And,if ξ 0 0 then t = T t), x = ω, u = vt, x)ũ, 5) dt where T t) = ξ 0,and the functions ω = ωt, x), vt, x) satisfy the equations: t) ξ 0 ω t + ξ ω x =0, ξ 0 v t + ξ v x = χv, 6) where ω constis further meant as any fixed solution of the equation 6). The consequence of this theorem is Theorem 2. The dimension of an invariance algebra of FPE ) can be equal to,2,4,6. Proof. If dimension of algebra more than then equation ) is reduced to the equation with à = à x), B = B x), but classification of such equations is known: dimension of their invariance algebra is either 2 or 4 or 6 [7]. In work [8] it is shown that any diffusion process with coefficient of drift At, x) and diffusion Bt, x) can be reduced to a process with appropriate coefficienct Ãt, x) =At, x)/bt, x) and Bt, x) = through random replacement of time τt). Using result of the theorem we carry out symmetry classification of FPE for the coefficient Bt, x) = and any At, x) just as it was made in [7] for a case A = Ax) homogeneous process). So puting in the equations 4) B = it is easy to show that ξ 0 = τt), ξ = 2 xτ + ϕt), 3 2 τ M + τm t + 2 τ x + ϕ)m x = 2 τ x + ϕ, χ = 2 τ xat, x) 4 x2 τ ϕ x + ϕat, x)+τ x x 0 At, ξ) dξ + θt), t 7) where M = A t + 2 A xx + AA x, x 0 and θt) are arbitrary point and function correspondently. Let us find a condition on M under wich there existes at least two the linearly independent solutions τt) of the equations 7). In this case from the Theorem 2 it is followed that there exists either 3 or 5 operators of symmetry besides trivial u u ). Let s assume that M xx 0. After differentiating twice on x both parts 7) we have: 5 2 τ M xx + τm txx + 2 τ x + ϕ ) M xxx =0. 8)

3 206 S. Spichak and V. Stognii Now if we assume that M xxx = 0, i.e. M xx = F t), then the following condition takes place: 5 2 τ F + τf =0. 9) For this equation there is only one linearly independent solution, therefore M xxx 0. Then from 8): ϕt) = 5M xx + xm xxx τ + M txx τ = ht, x)τ + rt, x)τ. 2M xxx M xxx So if τ,ϕ ), τ 2,ϕ 2 ) are linearly independent then τ, τ 2 are linearly independent, and also h x τ + r x τ =0. Thus h x τ + r x τ =0, h x τ + r x τ =0. As Wronskian τ τ τ 2 τ 2 0, then from this system it is followed that h x 0, r x 0, i.e. 5M xx + xm xxx 2M xxx = ht), From conditions 0) it is easy to deduce that M xxt M xxx = rt). 0) M = λx Ht)) 3 + F t)x + Gt), ) where λ =const 0,H, F, G are arbitrary functions. Now notice that if M xx =0,M has form ) with λ = 0. Thus the condition ) is necessary for the invariance algebra to have dimension either 4 or 6. Substituting ) in 8) and equating zero factors at x H, x H) 4 and we obtain the following conditions: 2τ F + τf = 2 τ, λ τh ) 2 τ H ϕ =0, ) 3 2 τ FH + G)+τF H + G )+F 2 τ H + ϕ = 2) 2 τ H + ϕ. ) Let λ 0. Then expressing from the second equation ϕt) =τh 2 τ H and substituting it in the third equation we have 3 2 τ FH + G H )+τfh + G H ) =0. Condition of existence of at least 2 independent solutions τ, τ 2 results in the equation FH + G H = 0. In this case the number of the fundamental solutions of system 2) is three. Really, there are three linear independent solutions τ, τ 2, τ 3 of the first equation 2). From the second equation 2) ϕ i is expressed through τ i, i =, 2, 3. 2) If λ = 0 the system of the equations 2) has 5 linearly independent solution τ i,ϕ i ), i =, 5. So the following theorem is proved. Theorem 3. ) The class FPE ) with B =admitting four-dimentional algebra of invariance is described by the condition A t + 2 A xx + AA x = λx Ht)) 3 + F t)x + Gt), 3)

4 One-Dimensional Fokker Planck Equation Invariant under 207 where λ =const 0, G satisfies the condition G = H FH, 4) F t), Ht) are arbitrary functions. 2) The class FPE ) with B =admitting six-dimensional invariance algebra invariance is described by condition 3) in which λ =0, F, G are arbitrary functions. Remark. In particular, if the coefficient At, x) satisfies the Burgers equation then FPE ) is reduced to the heat equation see [9]). 3 Transformation of the Fokker Planck equations to homogeneous equations ) It turns out that FPE ) B = ), 3) at λ = 0 is reduced to the heat equation [9]. We find the appropriate transformation 5), 6). Let τ be any solution of system 2) and τ>0 evidently that it is always possible to choose a solution τt) > 0 on some interval). From the formulae 6), 7) it is easy to prove that ωt, x) =τ /2 x ϕξ)τ 3/2 ξ)dt, where is arbitrary fixed point. Let us consider the transformation: t = 2 dt τ, x = ωt, x) =τ /2 x ϕξ)τ 3/2 ξ)dξ, 5) ut, x) =vt, x)ũ t, x). Having substitued into ), 3) the replacement variable 5) we come to the equation: ) ũ ũ t = 2τ vt v + A x + A v x 2 v xx v v 2 2 τ /2 τ x ϕτ /2 + Aτ /2 v x v τ /2 ) ũ x +ũ x x. 6) Equating zero factor at ũ x, we shall get: v = exp 4 τ τ x 2 τ ϕx + x x 0 At, ξ)dξ + ht), 7) where ht) is an arbitrary function, x 0 is some fixed point. Substituting 7) into the expression v t v + A x + A vx v v xx 2 v factor at ũ in 6)) and equating its to zero we get: h t) = [ τ 2 ϕ 2 ] 2 2 τ τ A x t, x 0 ) A 2 t, x 0 ), 8) 2 τ τ 4 τ2 τ ) 2 = F, τ ϕ 2 τ 2 τ ϕ = G. 9)

5 208 S. Spichak and V. Stognii Itiseasytoprovethatifτ 0,ϕ) is some solution of system 9) then it satisfies to system 2) λ =0,M = 0). Then we have the transformation 5), where funcitons vt, x), τt), ϕt) can be found from 7) 9), resulting FPE ), 3) λ = 0) to the heat equation ũ t =ũ x x. 20) Let us notice, that the system 9) is reduced to following: 2y + y 2 =4F, y = τ τ, ϕ = τ /2 τ /2 Gdt. 2) 2) We consider now FPE ), 3) with λ 0. As in the case of ) the transformation 5) reduces this equation to the equation 6). The conditions for 6) to be FPE are the following: à = Ãω) = τ /2 τ x 2ϕτ /2 +2Aτ /2 2τ /2 v x v, vt à ω =2τ v + A x + A v x v ) v xx, 2 v where ω is given in 5). The first condition is equivalent to the equation ) ] tã [τ = t + 2 τ x + ϕ x τ /2 τ x 2ϕτ /2 +2Aτ /2 2τ /2 v ) x =0. 23) v Omitting intermediate calculations we give the general solution vt, x) of equation 23): x vt, x) = exp At, ξ)dξ 4 τ τ x 2 τ ϕx + kω), 24) x 0 where kω) is an arbitrary function, x 0 is some fixed point. ) Substituting 24) into the first equation 22) one can prove that à = k ω) k ω) = dkω) dω. Let us substitute Ãω) = k ω), vt, x) 24) in the second equation 22). Under chosen conditions 22) τ /2 ϕτ 3/2 dt = H, k k 2 = λω 2, 2 τ τ 4 τ 2 τ 2 = F, τ ϕ 2 τ 2 τ ϕ = G, 25) 26) the second equaiton 22) is satisfied. It is possible to choose the condition 25) because, as it is easy to prove, any solution τ 0,ϕ of the given system is a particular solution of th system equations 2), 4) that it is enough for construction of the transformation 5). System 25) taking into account 4)) is equivalent to: 2y + y 2 =4F, y = τ τ, ϕ = τ 3/2 τ /2 H). 27) Thus we have proved Theorem 4. FP equation ),3),4) with λ 0,invariant under four-parameter algebra of invariance,through transformations t = T t), x = τ /2 x τ /2 Ht), u = vt, x)ũ t, x),

6 One-Dimensional Fokker Planck Equation Invariant under 209 where T = dt, vt, x) has the form 24), τ 0is any solution of the first equation 27), 2 τt) kω) is a solution of the equation 26),is reduced to the equation ũ t =2k ω)ũ +2k ω)ũ ω +ũ ωω. Remark. Making the replacement in last equation t = t, x = ω, ũ = expkω))ū, and taking into account the condition 26), we can reduce this equation to the following Schrödinger equation: ū t =ū x x + λ x 2 ū. Thus in the case FPE with four-parametrical group of symmetry there exists an initial equation, to which they are reduced; though it is not FPE as it is in the case of the sixparametrical group. Acknowlegments Authors is grateful to Prof. R.Z. Zhdanov for valuable discusions. References [] Gardiner K.V., Handbook of Stochastic Methods, Springer, Berlin, 985. [2] Risken H., The Fokker Planck Equation, Springer, Berlin, 989. [3] Ovsiannikov L.V., Group Analysis of Differential Equations, Academic, New York, 982. [4] Olver P., Applications of Lie Groups of Differential Equations, Springer, New York, 986. [5] Fushchych W., Shtelen W. and Serov N., Symmetry Analysis and Exact Solutions of Equations of Nonlinear Mathematical, Kyiv, Naukova Dumka, 989 in Russian). [6] Spichak S. and Stognii V., Symmetry classification and exact solutions of the one-dimensional Fokker Planck equation with arbitrary coefficients of drift and diffusion, J. Math. Phys., 999, V.32, N 47, [7] Cicogna G. and Vitali D., Classification on the extended symmetries of Fokker Planck equations, J. Phys. A: Math. Gen., 990, V.23, L85 L88. [8] Dynkin E.B., Markov Processes, Moscow, Fizmatgiz, 963. [9] Zhdanov R.Z. and Lagno V.I., On separability criteria for a time-invariant Fokker Planck equaiton, Dokl. Akad. Nauk of Ukraine, 993, N 2, 8 2.