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By using white noise analysis, we study the integral kernel $\xi(x)$, $x\in\mathbb{R}^{d}$, of stochastic currents corresponding to fractional Brownian motion with Hurst parameter $H\in(0,1)$. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and $d\ge1$ we show that the kernel $\xi(x)$ is well-defined as a Hida distribution for all $H\in(0,1/2]$. For $x=0$ and $d=1$, $\xi(0)$ is a Hida distribution for all $H\in(0,1)$. For $d\ge2$, then $\xi(0)$ is a Hida distribution only for $H\in(0,1/d)$. To cover the case $H\in[1/d,1)$ we have to truncate the delta function so that $\xi^{(N)}(0)$ is a Hida distribution whenever $2N(H-1)+Hd>1$., Comment: 17 pages

Starting with the overall wave function expression for the electron in an Aharonov-Bohm potential, we derive a version of the Kuramoto Model describing the phase dynamics of the bound state of the quantum mechanical system., Comment: 14 pages, 6 figures, 1 table

We construct the Feynman integral for the Schr\"odinger propagator in the polar conjugate momentum space, which describes the bound state Aharonov-Bohm effect, as a well-defined white noise functional., Comment: 28 pages, 1 figure

Pseudo-haptic techniques are becoming increasingly popular in human-computer interaction. They replicate haptic sensations by leveraging primarily visual feedback rather than mechanical actuators. These techniques bridge the gap between the real and virtual worlds by exploring the brain's ability to integrate visual and haptic information. One of the many advantages of pseudo-haptic techniques is that they are cost-effective, portable, and flexible. They eliminate the need for direct attachment of haptic devices to the body, which can be heavy and large and require a lot of power and maintenance. Recent research has focused on applying these techniques to extended reality and mid-air interactions. To better understand the potential of pseudo-haptic techniques, the authors developed a novel taxonomy encompassing tactile feedback, kinesthetic feedback, and combined categories in multimodal approaches, ground not covered by previous surveys. This survey highlights multimodal strategies and potential avenues for future studies, particularly regarding integrating these techniques into extended reality and collaborative virtual environments., Comment: 26 pages, 6 figures, accepted for publication in IEEE Access

In this paper, we investigate the Green measure for a class of non-Gaussian processes in $\mathbb{R}^{d}$. These measures are associated with the family of generalized grey Brownian motions $B_{\beta,\alpha}$, $0<\beta\le1$, $0<\alpha\le2$. This family includes both fractional Brownian motion, Brownian motion, and other non-Gaussian processes. We show that the perpetual integral exists with probability $1$ for $d\alpha>2$ and $1<\alpha\le2$. The Green measure then generalizes those measures of all these classes., Comment: 13 pages

In this paper, we study the quasi-invariant property of a class of non-Gaussian measures. These measures are associated with the family of generalized grey Brownian motions. We identify the Cameron--Martin space and derive the explicit Radon-Nikodym density in terms of the Wiener integral with respect to the fractional Brownian motion. Moreover, we show an integration by parts formula for the derivative operator in the directions of the Cameron--Martin space. As a consequence, we derive the closability of both the derivative and the corresponding gradient operators., Comment: 36 pages, 1 figure

This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $\Psi(-\Delta)$, where $\Psi$ is a Bernstein function. As applications, the existence and uniqueness of solutions to the corresponding nonlinear martingale problems are proved. Furthermore, it is shown that these solutions form a nonlinear Markov process in the sense of McKean such that their one-dimensional time marginal law densities are the solutions to the nonlocal nonlinear Fokker-Planck equation. Hence, McKean's program envisioned in his PNAS paper from 1966 is realized for these nonlocal PDEs., Comment: arXiv admin note: text overlap with arXiv:2210.05612

In this paper we study the existence of Green measures for Markov processes with a nonlocal jump generator. The jump generator has no second moment and satisfies a suitable condition on its Fourier transform. We also study the same problem for certain classes of random time changes Markov processes with jump generator., Comment: 20 pages

In this paper we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure $\pi_{\sigma}^{\beta}$, $0<\beta\leq1$, on the dual of Schwartz test function space $\mathcal{D}'$. The Hilbert space $L^{2}(\pi_{\sigma}^{\beta})$ of complex-valued functions is described in terms of a system of generalized Appell polynomials $\mathbb{P}^{\sigma,\beta,\alpha}$ associated to the measure $\pi_{\sigma}^{\beta}$. The kernels $C_{n}^{\sigma,\beta}(\cdot)$, $n\in\mathbb{N}_{0}$, of the monomials may be expressed in terms of the Stirling operators of the first and second kind as well as the falling factorials in infinite dimensions. Associated to the system $\mathbb{P}^{\sigma,\beta,\alpha}$, there is a generalized dual Appell system $\mathbb{Q}^{\sigma,\beta,\alpha}$ that is biorthogonal to $\mathbb{P}^{\sigma,\beta,\alpha}$. The test and generalized function spaces associated to the measure $\pi_{\sigma}^{\beta}$ are completely characterized using an integral transform as entire functions., Comment: 39 pages, 1 figure

In this paper, we use a biorthogonal approach (Appell system) to construct and characterize the spaces of test and generalized functions associated to the fractional Poisson measure $\pi_{\lambda,\beta}$, that is, a probability measure in the set of natural (or real) numbers. The Hilbert space $L^{2}(\pi_{\lambda,\beta})$ of complex-valued functions plays a central role in the construction, namely, the test function spaces $(N)_{\pi_{\lambda,\beta}}^{\kappa}$, $\kappa\in[0,1]$ is densely embedded in $L^{2}(\pi_{\lambda,\beta})$. Moreover, $L^{2}(\pi_{\lambda,\beta})$ is also dense in the dual $((N)_{\pi_{\lambda,\beta}}^{\kappa})'=(N)_{\pi_{\lambda,\beta}}^{-\kappa}$. Hence, we obtain a chain of densely embeddings $(N)_{\pi_{\lambda,\beta}}^{\kappa}\subset L^{2}(\pi_{\lambda,\beta})\subset(N)_{\pi_{\lambda,\beta}}^{-\kappa}$. The characterization of these spaces is realized via integral transforms and chain of spaces of entire functions of different types and order of growth. Wick calculus extends in a straightforward manner from Gaussian analysis to the present non-Gaussian framework. Finally, in Appendix B we give an explicit relation between (generalized) Appell polynomials and Bell polynomials., Comment: 32 pages, 0 figures

In this paper we study stochastic currents of Brownian motion $\xi(x)$, $x\in\mathbb{R}^{d}$, by using white noise analysis. For $x\in\mathbb{R}^{d}\backslash\{0\}$ and for $x=0\in\mathbb{R}$ we prove that the stochastic current $\xi(x)$ is a Hida distribution. Moreover for $x=0\in\mathbb{R}^{d}$ with $d>1$ we show that the stochastic current is not a Hida distribution., Comment: 10 pages

In this paper we introduce the concept of random time changes in dynamical systems. The subordination principle may be applied to study the long time behavior of the random time systems. We show, under certain assumptions on the class of random time, that the subordinated system exhibits a slower time decay which is determined by the random time characteristics. In the path asymptotic a random time change is reflected in the new velocity of the resulting dynamics., Comment: 17 pages. arXiv admin note: text overlap with arXiv:2008.03390

In this paper we investigate the long-time behavior of the subordination of the constant speed traveling waves by a general class of kernels. We use the Feller--Karamata Tauberian theorem in order to study the long-time behavior of the upper and lower wave. As a result we obtain the long-time behavior for the propagation of the front of the wave., Comment: 24 pages, 1 figure

In this paper strong dissipativity of generalized time-fractional derivatives on Gelfand triples of properly in time weighted $L^p$-path spaces is proved. In particular, the classical Caputo derivative is included as a special case. As a consequence one obtains the existence and uniqueness of solutions to evolution equations on Gelfand triples with generalized time-fractional derivatives. These equations are of type \begin{equation*} \frac{d}{dt} (k * u)(t) + A(t, u(t)) = f(t), \quad 0

In this paper we introduce and study three classes of fractional periodic processes. An application to ring polymers is investigated. We obtain a closed analytic expressions for the form factors, the Debye functions and their asymptotic decay. The relation between the end-to-halftime and radius of gyration is computed for these classes of periodic processes.

In this paper we investigate the representation of a class of non Gaussian processes, namely generalized grey Brownian motion, in terms of a weighted integral of a stochastic process which is a solution of a certain stochastic differential equation. In particular the underlying process can be seen as a non Gaussian extension of the Ornstein-Uhlenbeck process, hence generalizing the representation results of Muravlev as well as Harms and Stefanovits to the non Gaussian case., Comment: arXiv admin note: text overlap with arXiv:1708.06784, arXiv:1807.07867

The generalized grey Brownian motion is a time continuous self-similar with stationary increments stochastic process whose one dimensional distributions are the fundamental solutions of a stretched time fractional differential equation. Moreover, the distribution of the time-changed Brownian motion by an inverse stable process solves the same equation, hence both processes have the same one dimensional distribution. In this paper we show the mutual singularity of the probability measures on the path space which are induced by generalized grey Brownian motion and the time-changed Brownian motion though they have the same one dimensional distribution. This singularity property propagates to the probability measures of the processes which are solutions to the stochastic differential equations driven by these processes., Comment: 18 pages

In this paper we study the effect of the subordination by a general random time-change to the solution of a model on spatial ecology in terms of its evolution density. In particular on traveling waves for a non-local spatial logistic equation. We study the Cesaro limit of the subordinated dynamics in a number of particular cases related to the considered fractional derivative making use of the Karamata-Tauberian theorem., Comment: 39 pages

In this paper we investigate the energy functions for a class of non Gaussian processes. These processes are characterized in terms of the Mittag-Leffler function. We obtain closed analytic form for the energy function, in particular we recover the Brownian and fractional Brownian motion cases., Comment: arXiv admin note: text overlap with arXiv:1708.06784

In this paper we construct a Markov process which has as invariant measure the fractional Edwards measure based on a $d$-dimensional fractional Brownian motion, with Hurst index $H$ in the case of $Hd=1$. We use the theory of classical Dirichlet forms. However since the corresponding self-intersection local time of fractional Brownian motion is not Meyer-Watanabe differentiable in this case, we show the closability of the form via quasi translation invariance of the fractional Edwards measure along shifts in the corresponding fractional Cameron-Martin space.

In this paper use the Stein method to characterize the $M$-Wright distribution $M_{\frac{1}{3}}$ and its symmetrization. The Stein operator is associated with the general Airy equation and the corresponding Stein equation is nothing but a general inhomogeneous Airy equation., Comment: 10, pages, 1 fugure

In this paper we investigate the form factors of paths for a class of non Gaussian processes. These processes are characterized in terms of the Mittag-Leffler function. In particular, we obtain a closed analytic form for the form factors, the Debye function, and can study their asymptotic decay., Comment: 18 pages, 3 figures

In this paper we develop a method to solve evolution equations on Gelfand triples with time-fractional derivative based on monotonicity techniques. Applications include deterministic and stochastic quasi-linear partial differential equations with time-fractional derivatives, including time-fractional (stochastic) porous media equations (including the case where the Laplace operator is also fractional) and $p$-Laplace equations as special cases., Comment: published version in SIAM Journal on Mathematical Analysis

We prove that the self-intersection local times for generalized grey Brownian motion $B^{\beta,\alpha}$ in arbitrary dimension $d$ is a well defined object in a suitable distribution space for $d\alpha<2$.

In this note we prove that the probability measures generated by two generalized grey Brownian motions with different parameters are singular with respect to each other. This result can be interpreted as an extension of the Feldman-H\'ajek dichotomy of Gaussian measures to a family of non-Gaussian measures, Comment: 8 pages

We apply the subordination principle to construct kinetic fractional statistical dynamics in the continuum in terms of solutions to Vlasov-type hierarchies. As a by-product we obtain the evolution of the density of particles in the fractional kinetics in terms of a non-linear Vlasov-type kinetic equation. As an application we study the intermittency of the fractional mesoscopic dynamics., Comment: Published in Methods of Functional Analysis and Topology (MFAT), available at http://mfat.imath.kiev.ua/article/?id=890. arXiv admin note: text overlap with arXiv:1604.03807

Let $\sigma$ be a non-atomic, infinite Radon measure on $\mathbb R^d$, for example, $d\sigma(x)=z\,dx$ where $z>0$. We consider a system of freely independent particles $x_1,\dots,x_N$ in a bounded set $\Lambda\subset\mathbb R^d$, where each particle $x_i$ has distribution $\frac1{\sigma(\Lambda)}\,\sigma$ on $\Lambda$ and the number of particles, $N$, is random and has Poisson distribution with parameter $\sigma(\Lambda)$. If the particles were classically independent rather than freely independent, this particle system would be the restriction to $\Lambda$ of the Poisson point process on $\mathbb R^d$ with intensity measure $\sigma$. In the case of free independence, this particle system is not the restriction of the free Poisson process on $\mathbb R^d$ with intensity measure $\sigma$. Nevertheless, we prove that this is true in an approximative sense: if bounded sets $\Lambda^{(n)}$ ($n\in\mathbb N$) are such that $\Lambda^{(1)}\subset\Lambda^{(2)}\subset\Lambda^{(3)}\subset\dotsm$ and $\bigcup_{n=1}^\infty \Lambda^{(n)}=\mathbb R^d$, then the corresponding particle system in $\Lambda^{(n)}$ converges (as $n\to\infty$) to the free Poisson process on $\mathbb R^d$ with intensity measure $\sigma$. We also prove the following $N/V$-limit: Let $N^{(n)}$ be a determinstic sequence of natural numbers such that $\lim_{n\to\infty}N^{(n)}/\sigma(\Lambda^{(n)})=1$. Then the system of $N^{(n)}$ freely independent particles in $\Lambda^{(n)}$ converges (as $n\to\infty$) to the free Poisson process. We finally extend these results to the case of a free L\'evy white noise (in particular, a free L\'evy process) without free Gaussian part.

In this paper we shall establish an existence and uniqueness result for solutions of multidimensional, time dependent, stochastic differential equations driven simultaneously by a multidimensional fractional Brownian motion with Hurst parameter $H > \frac{1}{2} and a multidimensional standard Brownian motion under a weaker condition than the Lipschitz one., Comment: 21 pages

In this paper we establish a substitution formula for stochastic differential equation driven by generalized grey noise. We then apply this formula to investigate the absolute continuity of the solution with respect to the Lebesgue measure and the positivity of the density. Finally, we derive an upper bound and show the smoothness of the density., Comment: 17 pages

We present the expansion of the multifractional Brownian (mBm) local time in higher dimensions, in terms of Wick powers of white noises (or multiple Wiener integrals). If a suitable number of kernels is subtracted, they exist in the sense of generalized white noise functionals. Moreover we show the convergence of the regularized truncated local times for mBm in the sense of Hida distributions.

We construct an infinite dimensional analysis with respect to non-Gaussian measures of Mittag-Leffler type which we call Mittag-Leffler measures. It turns out that the well-known Wick ordered polynomials in Gaussian analysis cannot be generalized to this non-Gaussian case. Instead of using Wick ordered polynomials we prove that a system of biorthogonal polynomials, called Appell system, is applicable to the Mittag-Leffler measures. Therefore we are able to introduce a test function and a distribution space. As an application we construct Donsker's delta in a non-Gaussian setting as a weak integral in the distribution space.

Through chaos decomposition we improve the Varadhan estimate for the rate of convergence of the centered approximate self-intersection local time of planar Brownian motion., Comment: 5 pages

In this paper we investigate the class of grey Brownian motions $B_{\alpha,\beta}$ ($0<\alpha<2$, $0<\beta\leq1$). We show that grey Brownian motion admits different representations in terms of certain known processes, such as fractional Brownian motion, multivariate elliptical distribution or as a subordination. The weak convergence of the increments of $B_{\alpha,\beta}$ in $t$, $w$-variables are studied. Using the Berman criterium we show that $B_{\alpha,\beta}$ admits a $\lambda$-square integrable local time $L^{B_{\alpha,\beta}}(\cdot,I)$ almost surely ($\lambda$ Lebesgue measure). Moreover, we prove that this local time can be weak-approximated by the number of crossings $C^{B_{\alpha,\beta}^{\varepsilon}}(x,I)$, of level $x$, of the convolution approximation $B_{\alpha,\beta}^{\varepsilon}$ of grey Brownian motion., Comment: 20 pages

In this paper we investigate two variants of $\alpha$-stable processes, namely tempered stable subordinators and modified tempered stable process as well as their renormalization. We study the weak convergence in the Skorohod space and prove that they satisfy the uniform tightness condition. Finally, applications to the $\alpha$-dependence of the solutions of SDEs driven by these processes are discussed., Comment: 20 pages

In this work we extend Varadhan's construction of the Edwards polymer model to the case of fractional Brownian motions in $\R^d$, for any dimension $d\geq 2$, with arbitrary Hurst parameters $H\leq 1/d$., Comment: 14 pages

The Mittag-Leffler function $E_{\alpha}$ being a natural generalization of the exponential function, an infinite-dimensional version of the fractional Poisson measure would have a characteristic functional \[ C_{\alpha}(\phi) :=E_{\alpha}(\int (e^{i\phi(x)}-1)d\mu (x)) \] which we prove to fulfill all requirements of the Bochner-Minlos theorem. The identity of the support of this new measure with the support of the infinite-dimensional Poisson measure ($\alpha =1$) allows the development of a fractional infinite-dimensional analysis modeled on Poisson analysis through the combinatorial harmonic analysis on configuration spaces. This setting provides, in particular, explicit formulas for annihilation, creation, and second quantization operators. In spite of the identity of the supports, the fractional Poisson measure displays some noticeable differences in relation to the Poisson measure, which may be physically quite significant., Comment: 16 pages

In this work we present expansions of intersection local times of fractional Brownian motions in $\R^d$, for any dimension $d\geq 1$, with arbitrary Hurst coefficients in $(0,1)^d$. The expansions are in terms of Wick powers of white noises (corresponding to multiple Wiener integrals), being well-defined in the sense of generalized white noise functionals. As an application of our approach, a sufficient condition on $d$ for the existence of intersection local times in $L^2$ is derived, extending the results of D. Nualart and S. Ortiz-Latorre in "Intersection Local Time for Two Independent Fractional Brownian Motions" (J. Theoret. Probab.,20(4)(2007), 759-767) to different and more general Hurst coefficients., Comment: 28 pages