Solutions of a class of non-Markovian Fokker-Planck equations

Abstract

We show that a formal solution of a rather general non-Markovian Fokker-Planck equation can be represented in a form of an integral decomposition and thus can be expressed through the solution of the Markovian equation with the same Fokker-Planck operator. This allows us to classify memory kernels into safe ones, for which the solution is always a probability density, and dangerous ones, when this is not guaranteed. The first situation describes random processes subordinated to a Wiener process, while the second one typically corresponds to random processes showing a strong ballistic component. In this case the non-Markovian Fokker-Planck equation is only valid in a restricted range of parameters, initial and boundary conditions. Many physical phenomena related to relaxation in complex systems are described by non-Markovian Fokker-Planck equations in a formwhere K(t) is a memory kernel and where L is a linear operator acting on variable(s) x. Such equations are often postulated on the basis of linear-response considerations for different physical situations and in several cases can be more or less rigorously derived based on a microscopic description. In the symmetric case, the usual form of the operator L reads:

Relaxation to stationary states for anomalous diffusion

“…The alternative approach is based on the subordination transformations [19][20][21] , see [19,20]. In this case T(s, t) is defined by its Laplace transform in its second variable (clock time) which reads T(s, s) = s 1Àm exp (Àss m ).…”

Area and perimeter covered by anomalous diffusion processes

“…Note that p α (x, t) can be considered as the solution of a non-Markovian diffusion equation, connected to its standard Markovian counterpart, p(x, s), through (3). This equation is valid in general as long as the two functions in the integrand remain non-negative [22].…”

“…One might argue that the results in (13) and (20) could have been obtained by applying the subordination transformation directly to the mean perimeter and area relative to the Brownian case. However, when dealing with the subordination method, the only way to be sure of obtaining meaningful results is to work with probability densities [22].…”

From power law intermittence to macroscopic coherent regime

“…͑10͒, which is the generalization of Lindblad equation proposed in this article. Adopting the language of subordination theory, [31][32][33][34] we refer to Eq. ͑41͒ as a transformation from the natural time, where the Lindblad equation applies, to the physical time, corresponding to the experimental observation.…”