Stochastic representation of subdiffusion processes with time-dependent drift. (English) Zbl 1180.60051
The paper deals with a class of real valued stochastic processes \(X(t)\), \(t\geq 0\) whose variance has the power-law form, namely, \(\text{Var}[X(t)]\sim ct^\alpha\) as \(t\to\infty\) with some constant \(\alpha\in(0,1)\). If \(\alpha= 1\), this reminds the behavior of a process like the Brownian motion. Thus the terms subdiffusion processes, or slow diffusions, are natural. The author has used fractional Fokker-Planck equations to describe the probability density functions of subdiffusion processes. Since these equations usually do not have explicit solutions, a well converging approximations are constructed. Other interesting results are derived for the moments of stochastic integrals driven by the inverse \(\alpha\)-stable subordinator. These moments determine uniquely a distribution and this fact is effectively used to involve a specific fractional Fokker-Planck equation and its solution.
This well written paper shows that the subdiffusion processes are not only interesting from theoretical point of view, but they are closely related to important applied areas. The reference list represent well both theoretical studies and applications.
This well written paper shows that the subdiffusion processes are not only interesting from theoretical point of view, but they are closely related to important applied areas. The reference list represent well both theoretical studies and applications.
Reviewer: Jordan M. Stoyanov (Newcastle upon Tyne)
MSC:
| 60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |
| 60G99 | Stochastic processes |
| 60G52 | Stable stochastic processes |
Keywords:
subdiffusion; inverse subordinator; first-passage time; \(\alpha \)-stable distribution; fractional Fokker-Planck equationReferences:
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