arXiv subjects

José Luís da Silva

Publications and source records attributed to José Luís da Silva.

At least 19 recordsLinked to original sources

arXiv2026

Nonlocal, nonlinear Fokker-Planck equations and nonlinear martingale problems

This work is concerned with the existence of mild solutions and the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with nonlocal operators $Ψ(-Δ)$, where $Ψ$ 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.

math.AP↗

arXiv2026

Identification of the residual term in multiplicative self-decomposition using Fox $H$-functions

Multiplicative self-decomposable laws describe random variables that can be decomposed into a product of a scaled-down version of themselves and an independent residual term. Shanbhag et al.~(1977) have shown that the gamma distribution is multiplicative self-decomposable, in particular, the exponential distribution. As a result, they established the multiplicative self-decomposability of the absolute value of a centered normal random variable. A limitation of Shanbhag's result is that the distribution of the residual component is not explicitly identified. In this paper, we aim to fill this gap by providing an explicit distribution of the residual term using a Fox $H$-function. More precisely, the residual term follows an $M$-Wright distribution for the exponential distribution, whereas for the generalized gamma distribution and the absolute value of a centered normal random variable, an $H_{1,1}^{1,0}$ distribution with different parameters.

math.PR↗

arXiv2024

Green Measures for a Class of non-Markov Processes

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 $B_{β,α}$, $0<β\le1$, $0<α\le2$. 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 $dα>2$ and $1<α\le2$. The Green measure then generalizes those measures of all these classes.

math.PR↗

arXiv2023

Cameron--Martin Type Theorem for a Class of non-Gaussian Measures

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.

math.PR↗

arXiv2023

A biorthogonal approach to the infinite dimensional fractional Poisson measure

In this paper we use a biorthogonal approach to the analysis of the infinite dimensional fractional Poisson measure $π_σ^β$, $0<β\leq1$, on the dual of Schwartz test function space $\mathcal{D}'$. The Hilbert space $L^{2}(π_σ^β)$ of complex-valued functions is described in terms of a system of generalized Appell polynomials $\mathbb{P}^{σ,β,α}$ associated to the measure $π_σ^β$. The kernels $C_{n}^{σ,β}(\cdot)$, $n\in\mathbb{N}_{0}$, 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 $\mathbb{P}^{σ,β,α}$, there is a generalized dual Appell system $\mathbb{Q}^{σ,β,α}$ that is biorthogonal to $\mathbb{P}^{σ,β,α}$. The test and generalized function spaces associated to the measure $π_σ^β$ are completely characterized using an integral transform as entire functions.

math.FA↗

arXiv2023

Hydrodynamic limits for the free Kawasaki dynamics of continuous particle systems

An infinite particle system of independent jumping particles in infinite volume is considered. Their construction is recalled,further properties are derived, the relation with hierarchical equations, Poissonian analysis, and second quantization are discussed. The hydrodynamic limit for a general initial distribution satisfying a mixing condition is derived. The long time asymptotic is computed under an extra assumption. The relation with constructions based on infinite volume limits is discussed.

math.PR↗

arXiv2022

Compound Poisson Processes: Potentials, Green Measures and Random Times

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.

math.PR↗

arXiv2022

Fractional Poisson Analysis in Dimension one

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 $π_{λ,β}$, that is, a probability measure in the set of natural (or real) numbers. The Hilbert space $L^{2}(π_{λ,β})$ of complex-valued functions plays a central role in the construction, namely, the test function spaces $(N)_{π_{λ,β}}^κ$, $κ\in[0,1]$ is densely embedded in $L^{2}(π_{λ,β})$. Moreover, $L^{2}(π_{λ,β})$ is also dense in the dual $((N)_{π_{λ,β}}^κ)'=(N)_{π_{λ,β}}^{-κ}$. Hence, we obtain a chain of densely embeddings $(N)_{π_{λ,β}}^κ\subset L^{2}(π_{λ,β})\subset(N)_{π_{λ,β}}^{-κ}$. 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.

math.FA↗

arXiv2021

A White Noise Approach to Stochastic Currents of Brownian Motion

In this paper we study stochastic currents of Brownian motion $ξ(x)$, $x\in\mathbb{R}^{d}$, 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 $ξ(x)$ 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.

math.PR↗

arXiv2021

Asymptotic Behavior of the Subordinated Traveling Waves

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.

math.PR↗

arXiv2021

Strong dissipativity of generalized time-fractional derivatives and quasi-linear (stochastic) partial differential equations

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<t<T, \end{equation*} with (in general nonlinear) operators $A(t,\cdot)$ satisfying general weak monotonicity conditions. Here $k$ is a non-increasing locally Lebesgue-integrable nonnegative function on $[0, \infty)$ with $\underset{s\rightarrow\infty}{\lim}k(s)=0$. Analogous results for the case, where $f$ is replaced by a time-fractional additive noise, are obtained as well. Applications include generalized time-fractional quasi-linear (stochastic) partial differential equations. In particular, time-fractional (stochastic) porous medium and fast diffusion equations with ordinary or fractional Laplace operators or the time-fractional (stochastic) $p$-Laplace equation are covered.

math.AP↗

arXiv2020

Random Time Dynamical Systems I: General Structures

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.

math.DS↗

arXiv2019

Integral Representation of Generalized Grey Brownian Motion

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.

math.PR↗

arXiv2018

Singularity of Generalized Grey Brownian Motion and Time-Changed Brownian Motion

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.

math.PR↗

arXiv2018

From Random Times to Fractional Kinetics

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.

math.PR↗

arXiv2018

Quasi-Linear (Stochastic) Partial Differential Equations with Time-Fractional Derivatives

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.

math.AP↗

arXiv2017

The Stein Characterization of $M$-Wright Distributions

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.

math.PR↗

arXiv2017

Form Factors for Generalized Grey Brownian Motion

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.

math.PR↗