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We review theoretical models of individual motility as well as collective dynamics and pattern formation of active particles. We focus on simple models of active dynamics with a particular emphasis on nonlinear and stochastic dynamics of…
In this work, we present a mathematical model to describe the adsorption-diffusion process on fractal porous materials. This model is based on the fractal continuum approach and considers the scale-invariant properties of the surface and…
This article is concerned with stochastic differential equations driven by a $d$ dimensional fractional Brownian motion with Hurst parameter $H>1/4$, understood in the rough paths sense. Whenever the coefficients of the equation satisfy a…
We review statistical properties of models generated by the application of a (positive and negative order) fractional derivative operator to a standard random walk and show that the resulting stochastic walks display slowly-decaying…
A functional (lagged) time series regression model involves the regression of scalar response time series on a time series of regressors that consists of a sequence of random functions. In practice, the underlying regressor curve time…
We study a functional linear regression model that deals with functional responses and allows for both functional covariates and high-dimensional vector covariates. The proposed model is flexible and nests several functional regression…
A reasonable description of the degradation process is essential for credible reliability assessment in accelerated degradation testing. Existing methods usually use Markovian stochastic processes to describe the degradation process.…
The process of diffusion is the most elementary stochastic transport process. Brownian motion, the representative model of diffusion, played a important role in the advancement of scientific fields such as physics, chemistry, biology and…
In this article, we present the least squares estimator for the drift parameter in a linear regression model driven by the increment of a fractional Brownian motion sampled at random times. For two different random times, Jittered and…
We present Fractional Diffusion Bridge Models (FDBM), a novel generative diffusion bridge framework driven by an approximation of the rich and non-Markovian fractional Brownian motion (fBM). Real stochastic processes exhibit a degree of…
Concerning bivariate least squares linear regression, the classical approach pursued for functional models in earlier attempts is reviewed using a new formalism in terms of deviation (matrix) traces. Within the framework of classical error…
We address the challenge of estimation in the context of constant linear effect models with dense functional responses. In this framework, the conditional expectation of the response curve is represented by a linear combination of…
This paper is concerned with the stochastic thermodynamics of non-equilibrium Gaussian processes that can exhibit anomalous diffusion. In the systems considered, the noise correlation function is not necessarily related to friction. Thus,…
The characteristic feature of the discrete scale invariant (DSI) processes is the invariance of their finite dimensional distributions by dilation for certain scaling factor. DSI process with piecewise linear drift and stationary increments…
In modern experimental science, there is a common problem of estimating the coefficients of a linear regression in a context where the variables of interest cannot be observed simultaneously. When there is a categorical variable that is…
Let $B=\{(B_{t}^{1},..., B_{t}^{d}), t\geq 0\}$ be a $d$-dimensional fractional Brownian motion with Hurst parameter $H$ and let $R_{t}=% \sqrt{(B_{t}^{1})^{2}+... +(B_{t}^{d})^{2}}$ be the fractional Bessel process. It\^{o}'s formula for…
Fractional Brownian motion (fBm) extends classical Brownian motion by introducing dependence between increments, governed by the Hurst parameter $H\in (0,1)$. Unlike traditional Brownian motion, the increments of an fBm are not independent.…
We introduce the novel concept of a non-stationary iterated function system by considering a countable sequence of distinct set-valued maps $\{\mathcal{F}_k\}_{k\in \mathbb{N}}$ where each $\mathcal{F}_k$ maps $\mathcal{H}(X)\to…
Sensitivity analysis w.r.t. the long-range/memory noise parameter for probability distributions of functionals of solutions to stochastic differential equations is an important stochastic modeling issue in many applications. In this paper…
In attempting to quantify statistically the density structure of the interstellar medium, astronomers have considered a variety of fractal models. Here we argue that, to properly characterise a fractal model, one needs to define precisely…