On the Fractional Wave Equation

In this paper we study the time-fractional wave equation of order 1 < ν < 2 and give a probabilistic interpretation of its solution. In the case 0 < ν < 1 , d = 1 , the solution can be interpreted as a time-changed Brownian motion, while for 1 < ν < 2 it coincides with the density of a symmetric stable process of order 2 / ν . We give here an interpretation of the fractional wave equation for d > 1 in terms of laws of stable d−dimensional processes. We give a hint at the case of a fractional wave equation for ν > 2 and also at space-time fractional wave equations.

Extension of Mikhlin multiplier theorem to fractional derivatives and stable processes

In this paper, we prove a new generalized Mikhlin multiplier theorem whose conditions are given with respect to fractional derivatives in integral forms with two different integration intervals. We also discuss the connection between fractional derivatives and stable processes and prove a version of Mikhlin theorem under a condition given in terms of the infinitesimal generator of symmetric stable process. The classical Mikhlin theorem is shown to be a corollary of this new generalized version in this paper.

The synchronization of coupled stochastic systems driven by symmetric α-stable process and Brownian motion

The synchronization of stochastic differential equations (SDEs) driven by symmetric ?-stable process and Brownian Motion is investigated in pathwise sense. This coupled dynamical system is a new mathematical model, where one of the systems is driven by Gaussian noise, another one is driven by non- Gaussian noise. In this paper, we prove that the synchronization still persists for this coupled dynamical system. Examples and simulations are given.

Harnack inequalities for SDEs with multiplicative noise and non-regular drift

The log-Harnack inequality and Harnack inequality with powers for semigroups associated to SDEs with non-degenerate diffusion coefficient and non-regular time-dependent drift coefficient are established, based on the recent papers [7, 21]. We consider two cases in this work: (1) the drift fulfills the LPS-type integrability, and (2) the drift is uniformly Hölder continuous with respect to the spatial variable. Finally, by using explicit heat kernel estimates for the stable process with drift, the Harnack inequality for the stochastic differential equation driven by symmetric stable process is also proved.

On fractional derivatives of the local time of a symmetric stable process as a doubly indexed process

In this paper we prove two main results. The first one is to prove the regularity of fractional derivatives of local time of symmetric stable process with index

On Stochastic Equations with Measurable Coefficients Driven by Symmetric Stable Processes

We consider a one-dimensional stochastic equation , , with respect to a symmetric stable process of index . It is shown that solving this equation is equivalent to solving of a 2-dimensional stochastic equation with respect to the semimartingale and corresponding matrix . In the case of we provide new sufficient conditions for the existence of solutions of both equations with measurable coefficients. The existence proofs are established using the method of Krylov's estimates for processes satisfying the 2-dimensional equation. On another hand, the Krylov's estimates are based on some analytical facts of independent interest that are also proved in the paper.