Related papers: Fractional integration for irregular martingales
We show that a discrete time martingale with respect to a filtration with atomless innovations is the (infinite) sum of martingales with independent increments. For the continuous time filtration coming from Brownian Motion filtration, we…
This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic…
We determine the sharp constant in the Hardy inequality for fractional Sobolev spaces on half-spaces. Our proof relies on a non-linear and non-local version of the ground state representation.
We introduce three types of partial fractional operators of variable order. An integration by parts formula for partial fractional integrals of variable order and an extension of Green's theorem are proved. These results allow us to obtain…
This paper develops a high-accuracy algorithm for time fractional wave problems, which employs a spectral method in the temporal discretization and a finite element method in the spatial discretization. Moreover, stability and convergence…
In this paper, we introduced the local and global mixed Morrey-type spaces, and some properties of these spaces are also studied. After that, the necessary conditions of the boundedness of fractional integral operators $I_{\alpha}$ are…
In this paper we discuss fractional generalizations of the filtering problem. The "fractional" nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the…
In this paper, we prove capacitary versions of the fractional Sobolev--Poincar\'e inequalities. We characterize localized variant of the boundary fractional Sobolev--Poincar\'e inequalities through uniform fatness condition of the domain in…
Two classes of fractional type variable weights are established in this paper. The first kind of weights ${A_{\vec p( \cdot ),q( \cdot )}}$ are variable multiple weights, which are characterized by the weighted variable boundedness of…
It is an established fact that a finite difference operator approximates a derivative with a fixed algebraic rate of convergence. Nevertheless, we exhibit a new finite difference operator and prove it has spectral accuracy. Its rate of…
We give estimates on discrete fractional integral operators along binary quadratic forms. These operators have been studied for 30 years starting with the investigations of Arkhipov and Oskolkov, but efforts have concentrated on cases where…
In recent years, the theory for Leibniz integral rule in the fractional sense has not been able to get substantial development. As an urgent problem to be solved, we study a Leibniz integral rule for Riemann-Liouville and Caputo type…
In the literature, the left-side of Hermite--Hadamard's inequality is called a midpoint type inequality. In this article, we obtain new integral inequalities of midpoint type for Riemann--Liouville fractional integrals of convex functions…
Recently, fractional differential equations have been investigated via the famous variational iteration method. However, all the previous works avoid the term of fractional derivative and handle them as a restricted variation. In order to…
We discuss here a weak and strong type estimate for fractional integral operators on Morrey spaces, where the underlying measure $\mu$ does not always satisfy the doubling condition.
In this paper, we consider Caputo type fractional stochastic time-delay system with permutable matrices. We derive stochastic analogue of variation of constants formula via a newly defined delayed Mittag-Leffer type matrix function. Thus,…
A recent development in the theory of fractional differential equations with variable coefficients has been a method for obtaining an exact solution in the form of an infinite series involving nested fractional integral operators. This…
This note studies the martingale property of a nonnegative, continuous local martingale Z, given as a nonanticipative functional of a solution to a stochastic differential equation. The condition states that Z is a (uniformly integrable)…
We obtain sharp Hardy inequalities on antisymmetric functions where antisymmetry is understood for multi-dimensional particles. Partially it is an extension of the previously published paper \cite{HL}, where Hardy's inequalities were…
In this paper we consider Sobolev inequalities associated with singular problems for the fractional $p$-Laplacian operator in a bounded domain of $\mathbb{R}^{N}$, $N\geq 2$.