Abstract
In this paper we focus on strong solutions of some heat-like problems with a non-local derivative in time induced by a Bernstein function and an elliptic operator given by the generator or the Fokker–Planck operator of a Pearson diffusion, covering a large class of important stochastic processes. Such kind of time-non-local equations naturally arise in the treatment of particle motion in heterogeneous media. In particular, we use spectral decomposition results for the usual Pearson diffusions to exploit explicit solutions of the aforementioned equations. Moreover, we provide stochastic representation of such solutions in terms of time-changed Pearson diffusions. Finally, we exploit some further properties of these processes, such as limit distributions and long/short-range dependence.
Highlights
Pearson diffusions [29] constitute a family of stochastic processes X (t) that are solutions of Stochastic Differential Equations (SDEs) of the form, d X (t) = μ(X (t))dt + σ (X (t))dW (t), where W (t) is a standard Brownian motion, μ(x) is a polynomial of degree at most 1, σ 2(x) is a polynomial of degree at most 2 and the equation has to be considered in the context of Itô calculus
In the third spectral category we only have Student diffusions, whose generator admits a simple discrete spectrum and an absolutely continuous spectrum of multiplicity two separated by a cut-off value
Let us recall the spectral decomposition theorem for Pearson diffusions of spectral category II as given in [11,45]
Summary
Different inter-jump times in continuous time random walks and different relaxation patterns lead to a wider class of anomalous diffusions obtained via subordination of a random process (see [70]) In this case, one has to consider a more general non-local derivative in place of the Caputo one in Fokker–Planck equations. For instance, in [23,72] a link between abstract generalized fractional differential equations and time-changed semigroups is established, while in [25] properties of the Green measures of time-changed Markov properties are exploited Such non-local derivatives have been used for instance to define a class of time-non-local birth-death processes (see [8]) whose stationary distributions fall into the Katz family, which are discrete analogous of Pearson diffusions of the first spectral category.
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