Symmetry Analysis of the Fokker Planck Equation

In this work, the infinitesimal criterion of invariance for determining symmetries of partial differential equations is applied to the Fokker Planck equation. The maximum rang condition being satisfied, we determine the Lie point symmetries of this equation. Due to the nature of infinitesimal generators of these symmetries and the stability of Lie brackets, we obtain an infinite number of solutions from which we find examples of solutions for the Fokker Planck equation: other solutions are generated given a particular solution of the equation. Then, the Fokker Planck equation admits a conserved form, hence there is an auxiliary system associated to this equation. We show that this system admits six and an infinite number of infinitesimal generators of point symmetries giving rise to two potential symmetries of the Fokker Planck equation. We then use those potential symmetries to determine solutions of the associated system and therefore provide other solutions of the Fokker Planck equation. Note that these are essentially obtained on the basis of the invariant surface conditions. With respect to these conditions and from the potential symmetries that we have found, we finally show that in particular, some solutions of the considered Fokker Planck equation reduced to the trivial solution (solutions that are zero).


Introduction
The Fokker-Planck equation (FPE, for short) is a linear PDE that describes the transition probability density of a Markov process. It is also known as the Kolmogorov diffusion equation and is used to model many situations such as evolution of the distribution function of a particle, finance, turbulence, population dynamics, protein kinetics (see [3,4,6,10,12,21]).
We state the Fokker-Planck equation (FPE) in the following form: where a 1 and a 2 are real numbers; u (x, t) is a function that depends on the variables x and t, to be determined; and u α denotes differentiation of u with respect to the variable α.
Like most PDEs, it gives explicit solutions only in very specific cases related both to the form of the equation and the shape of the area where it is studied. Many techniques are used to solve particular cases of the FPE: quantum mechanics technique ( [2]), Fourier transform method ( [19]), differential transform method ( [7]), numerical method (e.g. [3,4,8,22]). Powerful means used in the study of DEs and PDEs are the Lie symmetries. Since their introduction by Sophus Lie ( [11]), Lie symmetries are experiencing a rapid development as a wonderful tool for the classification of invariant solutions of DEs and PDEs. Point symmetries are local symmetries as their infinitesimals depend on independent variables x's, dependent variables u (x)'s, and derivatives of dependent variables; and are determined if u (x) is sufficiently smooth in some neighbourhood of x. Potential symmetries when with them are nonlocal symmetries whose infinitesimals, at any point x, depend on the global behavior of u (x). Potential symmetries are very useful as they lead to the construction of solutions of a given system of PDEs which cannot be obtained as invariant solutions of its local symmetries. See Section 3 for wider discussion on potential symmetries. See also Chap 7 of [1] for more about potential symmetries.
The FPE is considered in his standard expression [15,18] in the form which is different from (1). The authors of the papers quoted above have determined the Lie point symmetries of the FPE, as well as the potential symmetries. They also have provided families of solutions of the FPE.
In this paper we consider the FPE (1) with the condition ≠ 0. We adopt the same approach as in [15] and determine the Lie point symmetries of the FPE in Section 2. Some of its solutions are also determined. In Section 3, we show that the FPE can be written in a conserved form. A conserved form leads to auxiliary dependent variables (which are potentials) and then to an auxiliary system of PDEs whose local symmetries are the potential symmetries of the FPE. We determine such symmetries in Section 3 and use them to construct other solutions of the Fokker-Planck equation.

Some basics about Lie point symmetries
Consider a general system of n th order DEs admitting p independent variables , … , ) in ≃ ℝ and q dependent variables = ( , … , ) in ≃ ℝ , Δ ( , ( ) ) = 0, = 1, … , , with u (n) denoting the derivatives of the u's with respect to the x's up to order n. The system (2) is thus defined by the vanishing of a collection of differentiable functions Δ ( ⟶ ℝ) defined on the n th jet space = ! = × ( ) , where E is the total space ! = × (see [14]). The points in the vertical space U (n) are denoted by u (n) and consist of all the dependent variables and their derivatives up to order n. The system (2) can therefore be viewed as defining (or defined by) a variety S $ = {( , ( ) )/ Δ ( , ( ) ) = 0, = 1, … , } contained in the n th order jet space, and consisting of all points ( , ( ) ) ∈ satisfying the system. The defining functions Δ are assumed to be regular in a neighbourhood of ) $ ; in particular, this is the case if the Jacobian matrix of the functions Δ with respect to the jet variables ( , ( ) ) has maximal rank m everywhere on ) $ . In the case of point transformations, the infinitesimal generators form a Lie algebra * consisting of vector fields 1 1 ( , ) ( , ) on the space of independent and dependent variables. Let + ( ) denote the n th prolongation of + to the jet space ([13, p. 117]): For any unordered multi-index The fundamental infinitesimal symmetry criterion for the system (2) is stated in the following: Theorem 2.1 ( [14]). A connected group of transformations G is a symmetry group of the fully regular system of DEs (2) if and only if the infinitesimal symmetry conditions hold for every infinitesimal generator V of the Lie algebra g of G .

Lie Point Symmetries of the FPE
To investigate the Lie point symmetries of the FPE, we have to check the maximal rank condition for the map x xx x t u u x t a u x t a x a u x t u x t ∆ + + + − ֏ whose kernel equation is (1) on a subset (2) M of the 2 nd jetspace (2) X U × of the manifold X U × . The independent variables ( , ) x t and the dependent variable leave on the spaces ≃ ℝ and ≃ ℝ , respectively. The expression ( 2) ( , , , , , ) x t xx xt tt u u u u u u u = represents the various partial derivatives up to the second order of u , and leaves on the second prolongation (2) U of the set U . The set (2) M is the corresponding 2 nd prolongation of the subspace M X U ⊂ × .  (2) M . Then, ∆ is of maximal rank. Let ξ τ and η are smooth functions. The second prolongation of V reads Where , , , x t xx xt η η η η and tt η are given by the formulae (see [14]): ( ) Proposition 2.3.: Point symmetries of the FPE are generated by the operators and an infinite number of generators ( , ) is any solution of the FPE. Proof. We make the assumption whenever, and check the corresponding conditions on , ξ τ andη . Those conditions lead to Now replace , x t η η and xx η in (16) by their expressions given in (7), (8) and (9) respectively, and eliminate t u by substituting it by the right hand side of (1) any time when it occurs. Then the derivatives of u with right to t disappear.
So, the resolution of the corresponding system of PDEs is equivalent to solving the following system: 2 0 Equation (17) implies that η is linear in u . So, it writes A and B being smooth functions depending only on x and t. From (18), we get where k is a smooth function of t . Substituting ξ and η by their expressions in (19) and differentiating the resulting expression with respect to x , we get Thus, where 1 A and 2 A are smooth functions of the variable t . Using Equation (20), we find that Note that (22) is nothing but the FPE (1). Now (19), (20) and (21) is any solution of the FPE. The rest of the proof is straightforward.

Examples of solutions of the FPE
In the sequel, we provide a family of solutions of the Fokker-Planck equation (1).
x t α be any solution of the FPE. Then the functions [ ]   As mentioned in [15], using the Lie brackets in Table 1, one can construct a family of solutions from a trivial solution. Consider e.g.

Preliminaries on Potential Symmetries
A partial differential equation of order n in the unknown function ( , ) is written in a conserved form if it has the following form: Since the PDE (43) is in a conserved form, a potential v considered as a new variable is in-traduced. A system of PDEs denoted by ( 1) ( , , One says that ( 1) ( , , , ) holds. In this case, the symmetry ( , , , ) ( , , , ) ( , , ; ) called a potential symmetry of Equation (42). Potential symmetries can also be used in the study of a boundary value problem posed for a given system of PDEs and for the study of ODEs. For a scalar ODE, a potential symmetry reduces the order (see [1]).
We are now going to explain how, from potential symmetries, one obtains solutions of the PDE (42) which admits a conserved form (43). See [18] for wider discussion. x t The associated characteristic system yields to the following independent integrals 1 1 ( , , , ) , we obtain from (47): 1 2 ( , , , ( ), ( )) 0 G x t z h z h z = . (50) The invariant solutions of (43) are given by (48) and (49), where ( ) i h z are the solutions of the ordinary system obtained by substitution in (43). Since (42) is a differential consequence of (43), the solution of (43) give those solutions of (42), which verify the differential relation obtained by eliminating v between (45) and 0 T X ξ τ ϕ + − = .

Potential symmetries of the FPE
The conserved form of the FPE can be written as Then, the corresponding system writes as follows: where the potential variable v has been introduced as a new dependent variable.
The coefficient functions ( , , , , , , , , , ) x t x t xx xt tt xx xt tt η η ϕ ϕ η η η ϕ ϕ ϕ in (2) W are given as follows: Hence, the criterion ( where L is a smooth function of t . Relations (70) imply that ϕ is independent from u and is linear with right to v . That is there exists functions D and E depending only on x and t such that Then, substituting ξ and ϕ by their expressions in (73) and differentiating the resulting expression with respect to x , one obtains the equation where 1 B and 2 B are smooth functions of t only. Coming back to Equation (74), we find that Here again, (81) is equivalent to (52). From Equations (78) and (80) 2( ) 0 x v a x a v − + = . (92) The system below admits the following solutions: where 1 q and 2 q are smooth functions of the variable t . If we replace the expression of ( , ) u x t given by (93) in (1) where a and b are constants.
Let us now deal with the symmetry generator 5 W which provides the potential symmetry 2 Y . The invariant surface conditions for this symmetry write where f is a smooth function. Replacing the expression of ( , ) v x t given by (98) where g is another smooth function. Now, setting (1 ) ( ) ( ) 0 a zf z a f z

Conclusions
The approach we used to determine the point symmetries of the Fokker Planck equation (1) already exists in the literature. Given a trivial solution, one has constructed a family of solutions for this equation. The symmetry analysis of the equation (1) that we have performed highlights the fact that the potential symmetries constitute the powerful tool for solving partial differential equations provided that these are written in their conserved form. Exact solutions can be found from these types of symmetries as it is the case in the present work with the invariant surface conditions associated with the equation (1).