On a class of exact solutions to the Fokker-Planck equations

In this paper we study under which circumstances there exists a general change of gross variables that transforms any Fokker-Planck equation into another of the Ornstein-Uhlenbeck class that, therefore, has an exact solution. We find that any Fokker-Planck equation will be exactly solvable by means of a change of gross variables if and only if the curvature tensor and the torsion tensor associated with the diffusion is zero and the transformed drift is linear. We apply our criteria to the Kubo and Gompertz models.

It has been possible to find an exact solution to the FPE when the diffusion is constant and when the drift is linear.I There exists also a class of FPE's that have exact solution and whose model is an FPE with linear drift and diffusion 8 !,". 2 Any FPE of this class can be reduced to its model by means of a change in the gross variables.We may ask ourselves immediately the following question: Under which conditions does there exist a change of gross variables that transforms any FPE into another FPE with linear drift and diffusion 8 "V, i.e., that has an exact solution?This last question has a complete answer and constitutes the central result of this paper, in which we intend to give the necessary and sufficient conditions to determine if any FPE has an exact solution related with the exact solution of the Ornstein-Vhlenbeck process by means of a general change of gross variables.
In Sec.II we show how any FPE can be transformed into another one with exact solution.We use here the covariant formulation of FPE 3 • 4 by means of which we obtain a clear and rigorous method for such a transformation.In Sec.III we give the necessary and sufficient conditions to be satisfied by our original FPE so that such a transformation of gross variables exists: the curvature and torsion tensors have to be zero.These criteria characterized with precision a class ofFPE's that have exact solutions. 2In Sec.IV we study some subclasses of FPE that have physical importance, among which we find the Kubo s and Gompertz 6 models.
In this paper we suppose the sum over repeated indices except those that are within a parenthesis.tf'v   Let M be the manifold formed by the physical states of the system. 7This manifold is characterized by two sets of gross variables, !qI' land!q'" l, related among themselves by means of continuous and differentiable functions and which conserve the number of gross variables.The diffusion tensor D""(q) is the metric tensor of the manifold.

III. NECESSARY AND SUFFICIENT CONDITIONS TO SOLVE EXACTLY AN FPE BY MEANS OF A CHANGE OF VARIABLES
There will not always exist a change of variables q = q(q') such that D 'JlV = 8 JlV.In the Appendix we show that the necessary and sufficient conditions, in order that D 'JlV = 8 JlV, are that the curvature tensor RJlvaf3 and torsion tensor T~a 8 associated with the diffusion matrix be zero.Therefore the conditions Tv"Jl = r':.a guarantee the existence of a change of variables determined by the matrix J).Jl such that in the new variables the diffusion matrix D 'JlV becomes 8 JlV.If we work with Riemann's connection D JlV, the Christoffel symbols are given by Eq. (2.3), and condition (3.2) is satisfied identically.
In the same appendix (Eq.(A3)l we show that the matrix J).Jl for the variables transformations must satisfy the relation aJlJ V ).= r:vJa )..
The diffusion matrix in the variables I q,vJ becomes a constant matrix; therefore, the general solution of (3.4) should satisfy: J Jl-1"J v-lpDJlV(q) =A "P or, equivalently, J;J~A"p = DJlv(q), (3.6) whereA"p is a constant matrix.Concretely, we could take A"p = 8"p; in this case the transformation matrix satisfies DJlAq) =J;J~8"p. (3.8) 2 to characterize a whole class of FPE that have exact solutions.
In order that the transformed FPE represent an Omstein-Uhlenbeck process, which has an exact solution, it is also necessary that the transformed driftj'Jl(q') be linear, i.e., a~).j'Jl(q') = O. (3.9) In the coordinates q' the covariant derivation V' coincides with the ordinary derivatives and besides the driftj'Jl is the covariant drift h 'Jl.Therefore, the covariant expression for (3.9) is From what has been said above, we can check whether or not any FPE (1.1) represents, in a certain set of gross variables, an Omstein-Uhlenbeck process that has an exact solution.We could test if the diffusion matrix DJlV satisfies the condition (3.1) and if the drift satisfies the condition (3.11).If the answer is affirmative, we can integrate the change of variables by means of a matrix J; of the type (3.7).
The solution of the FPE is the one transformed from Eq. (2.14).

IV. APPLICATIONS
An important case are those processes whose diffusion is diagonal: The only elements of curvature tensor that are not zero are R and R .As it is easy to check a diffusion matrix of with tp{Jl) (qJl) > 0 satisfies simultaneously the conditions With the metric (4.2) the covariant drift in its original variables is hI' = J I' + !tp (;) lal 'tpl/.Li' (4.8) and the condition that the original driftJ I'(q) should satisfy in order that the transformed driftf'V(q') be linear is To the subclass of models represented by Eq. (4.2) belong the Kubo and Gompertz models.Let the function tpl/.L) (ql') be of the form (4.10) if the transformed drift is constant, we have n monodimensional Kubo models.
Similarly, if we have when the transform drift is constant, we obtain n monodimensional Gompertz models. 2 • 6 Another example is the case when the diffusion matrix is conJormally flat: (4.12) with D (q) > O.In such a case there will exist a change of variables such that D 'I'V = <51''' if the function D (q) satisfies the equations A solution of these equations is Choosing as the metric tensor of the physical space the diffusion matrix, the corresponding FPE will be exactly solvable by means of a change of gross variables if and only if the curvature tensor and the torsion tensor associated with the diffusion is zero, and the transformed drift is linear.
To write the solution of the FPE in the original variables, we have to know the functions q' = q'(q) of the change of variables and substitute them in the solution (2.14).Therefore, with the method presented in this paper the solution of an FPE is reduced to integrating a change of gross variables.Anyway, with the method presented above we can test any FPE to see whether or not it has an exact solution by means of a change of coordinates: The diffusion should satisfy Eqs.(3.1) and (3.2) and the drift equation (3.11).
And so to write the solution of the FPE in the original variables, we should know the functions of such a change of variables q' = q'(q) and enter into the solution (2.14).Therefore, the integration of an FPE equation is reduced with the method presented here to the integration of a change of variables.
that the condition (3.11) does not require a knowledge of the change of gross variables: it is sufficient to know the Christoffel symbols r':.a that can be evaluated differentiating the diffusion D JlV.
with an associated transformation matrix J/ =A 1/2exp!~(ql' + qv) + cIM~.methods of the differential geometry we have found a class ofFPE that are exactly solvable by means of change of variables.