Random time change and related evolution equations. Time asymptotic behavior

Abstract

In this paper we investigate the long time behavior of solutions to fractional in time evolution equations which appear as results of random time changes in Markov processes. We consider inverse subordinators as random times and use the subordination principle for the solutions to forward Kolmogorov equations. The class of subordinators for which asymptotic analysis may be realized is described. ∞ 0 u 0 (x, τ )G t (τ )dτ,

Green measures for Markov processes

“…It is known that the marginal distribution \nu t of Y (t) holds a subordination formula, see [7] \nu t (\mathrm{ x) = \int \infty 0 D t (\tau )\mu \tau (\mathrm{ x) \mathrm{ \tau ,…”

Subordination Approach to Space-Time Fractional Diffusion

“…The technique used in the present work for deriving the integral representation for the subordination kernel does not rely on the scaling property and can be extended to equations with more general nonlocal operators in space, such as those considered in [36], as well as operators with a general memory kernel in time, as in [24,37].…”

“…In [14,15], subordination principles for the multi-dimensional space-time fractional diffusion equation are deduced. In the case of time-fractional evolution equations with general time-fractional operators, subordination principles have been studied and employed in [23][24][25][26][27][28].…”

From time to times

“…On the other hand, we find technical difficulties in handling general inverse subordinators. Such limitations can be overcome for certain sub-classes of inverse subordinators, see, e.g., [19,20]. Let us underline that the random time change approach turns to be a very effective tools in modeling several physical systems, spanning from ecological to biological ones, see, e.g., [23] and references therein, also in view of additional applications.…”