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In this paper we introduce the notion of fractional martingale as the fractional derivative of order $\alpha$ of a continuous local martingale, where $\alpha\in(-{1/2},{1/2})$, and we show that it has a nonzero finite variation of order…
We derive fractional Brownian motion and stochastic processes with multifractal properties using a framework of network of Gaussian conditional probabilities. This leads to the derivation of new representations of fractional Brownian…
In this paper, we apply high level versions of Jacobi's derivative formula to number theory such as quarternary quadratic forms and convolution sums of some arithmetical functions.
This paper deals with the well posedness of an integrodifferential equation that describes a vortex filament associated to a 3D turbulent fluid flow. This equation is driven by a fractional Brownian motion of Hurst parameter H>1/2. We prove…
We consider a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion. One does not have uniqueness for the solutions of the corresponding stochastic…
In this paper we consider the asymptotic behaviour of all moments of the interparticle distance and of all mixed moments of an isotropic Brownian stochastic flow which serves as a smooth approximation of the Arratia flow.
We prove existence and uniqueness of the solution of a stochastic shell--model. The equation is driven by an infinite dimensional fractional Brownian--motion with Hurst--parameter $H\in (1/2,1)$, and contains a non--trivial coefficient in…
We construct an iterated stochastic integral with fractional Brownian motion with H > 1/2. The first integrand is a deterministic function, and each successive integral is with respect to an independent fBm. We show that this symmetric…
This is a review of statistical inference methodology for stochastic differential equations driven by fractional Brownian motion, otherwise called fractional diffusions. The first section reviews the theory needed to rigorously define them.…
We provide a new, concise proof of weak existence and uniqueness of solutions to the stochastic differential equation for the multidimensional skew Brownian motion. We also present an application to Brownian particles with skew-elastic…
While the theory of diffusion of a single Brownian particle in confined geometries is well-established by now, we discuss here the theoretical framework necessary to generalize the theory of diffusion to dense suspensions of strongly…
When a probe particle immersed in a fluid with nonlinear interactions is subject to strong driving, the cumulants of the stochastic force acting on the probe are nonlinear functionals of the driving protocol. We present a Volterra series…
We study asymptotic properties of the system of interacting diffusion particles on the real line which transfer a mass [arXiv:1408.0628]. The system is a natural generalization of the coalescing Brownian motions. The main difference is that…
We present a fundamental classification of forces relevant in nonequilibrium structure formation under collective flow in Brownian many-body systems. The internal one-body force field is systematically split into contributions relevant for…
The paper is devoted to the existence of integral functionals $\int_0^\infty f(X(t))\,{\mathrm{d}t}$ for several classes of processes in $\mathbb{R}$ with $d\ge 3$. Some examples such as Brownian motion, fractional Brownian motion, compound…
A stochastic sewing lemma which is applicable for processes taking values in Banach spaces is introduced. Applications to additive functionals of fractional Brownian motion of distributional type are discussed.
We introduce a fundamental theory for the kinetics of systems of classical particles. The theory represents a unification of kinetic theory, Brownian motion and field theory. It is self-consistent and is the dynamic generalization of the…
We introduce a one-dimensional stochastic system where particles perform independent diffusions and interact through pairwise coagulation events, which occur at a nontrivial rate upon collision. Under appropriate conditions on the diffusion…
We propose and investigate a simple model which describes the kinetics of aggregation of Brownian particles with stochastic self-replication. An exact solution and the scaling theory are presented alongside numerical simulation which fully…
This study makes the first attempt to use the 2/3-order fractional Laplacian modeling of enhanced diffusing movements of random turbulent particle resulting from nonlinear inertial interactions. A combined effect of the inertial…