یک نکته درباره ویژگی مارکوف از فرآیندهای تصادفی شرح داده شده توسط معادلات فوکر؛ پلانک غیر خطی

We study the Markov property of processes described by generalized Fokker–Planck equations that are nonlinear with respect to probability densities such as mean field Fokker–Planck equations and Fokker–Planck equations related to generalized thermostatistics. We show that their transient solutions describe non-Markov processes. In contrast, stationary solutions can describe Markov processes. As a result, nonlinear Fokker–Planck equations can be used to model transient non-Markov processes that converge to stationary Markov processes.

In general, stochastic processes can be characterizedby means of transition probability densities. In the trivial case, transition probability densities depend on a single time-point and we deal with pure random processes. In the simplest, nontrivial case transition probability densities depend on two time-points. Then, we deal with Markov processes [1]. In view of this property, Markov processes play an important role in the theory of stochastic processes. Moreover, it has been foundthat many stochastic processes observedin physics andother sciences can indeedbe regardedas Markov processes [2–5]. However, Markov processes describe an idealized situation [6]. In the general case, there is an eEect of the history of a system on its current behavior. Such long-term memory eEects cannot be taken into account by Markov processes and require a description in terms of non-Markov processes. Therefore, to diEerentiate between Markov andnon-Markov processes basically means to discuss the relevance of long-term memories of systems. Such a discussion can be carried out by means of appropriately de,ned stochastic models. To this end, however, we need to know whether or not the models describe Markov processes. Recently, there has been an increasing interest in modeling stochastic processes by means of Fokker–Planck equations that are nonlinear with respect to their probability densities. Such processes have been used, for example, to describe synchronization phenomena [3,7–12], muscular contractions [13,14], noise-induced phase transitions [15,16], andnonextensive systems [17–24] (see also Ref. [25]). In spite of an increasing number of applications, little attention has been directed towards the Markov property of processes described by nonlinear Fokker–Planck equations. In fact, in literature one can ,ndbrief comments on this issue. However, these comments are controversial [26–36]. Some authors have suggestedthat nonlinear Fokker–Planck equations describe Markov processes, others have mentioned that they describe non-Markov processes. In addition, some authors prefer to avoid using the terms Markov or non-Markov processes andhave calledstochastic processes describedby nonlinear Fokker–Planck equations “nonlinear Markov processes”.