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Monotone processes, just like martingales, can often be recovered from their final values. Examples include running maxima of supermartingales, as well as running maxima, local times, and various integral functionals of sticky processes…

Renewal processes are broadly used to model stochastic behavior consisting of isolated events separated by periods of quiescence, whose durations are specified by a given probability law. Here, we identify the minimal sufficient statistic…

We consider a particle undergoing Brownian motion in Euclidean space of any dimension, forced by a Gaussian random velocity field that is white in time and smooth in space. We show that conditional on the velocity field, the quenched…

We discuss the identification of a time-dependent potential in a time-fractional diffusion model from a boundary measurement taken at a single point. Theoretically, we establish a conditional Lipschitz stability for this inverse problem.…

We establish a central limit theorem and an invariance principle for stationary random fields, with projective-type conditions. Our result is obtained via an m-dependent approximation method. As applications, we establish invariance…

We define the local empirical process, based on $n$ i.i.d. random vectors in dimension $d$, in the neighborhood of the boundary of a fixed set. Under natural conditions on the shrinking neighborhood, we show that, for these local empirical…

This paper establishes a central limit theorem and an invariance principle for a wide class of stationary random fields under natural and easily verifiable conditions. More precisely, we deal with random fields of the form $X_k =…

In modeling multivariate time series, it is important to allow time-varying smoothness in the mean and covariance process. In particular, there may be certain time intervals exhibiting rapid changes and others in which changes are slow. If…

Extending the ideas of [7], this paper aims at providing a kernel based non-parametric estimation of a new class of time varying AR(1) processes (Xt), with local stationarity and periodic features (with a known period T), inducing the…

Stochastic resetting can be naturally understood as a renewal process governing the evolution of an underlying stochastic process. In this work, we formally derive well-known results of diffusion with resets from a renewal theory…

Let A(n) be a sequence of i.i.d. topical (i.e. isotone and additively homogeneous) operators. Let $x(n,x_0)$ be defined by $x(0,x_0)=x_0$ and $x(n,x_0)=A(n)x(n-1,x_0)$. This can modelize a wide range of systems including, task graphs, train…

This paper studies finite-time stability and instability theorems in probability sense for stochastic nonlinear systems. Firstly, a new sufficient condition is proposed to guarantee that the considered system has a global solution.…

We develop a versatile framework for statistical learning in non-stationary environments. In each time period, our approach applies a stability principle to select a look-back window that maximizes the utilization of historical data while…

We introduce a class of non-local Lagrangians which allow for the variational derivation of non-local conser- vation laws in a self-consistent manner. The formalism developed here generalizes previous approaches, used in the context of…

Introducing the primed inertial coordinate system, for each inertial frame of reference, in addition to the usual inertial coordinate system, we assume that gravity-free space and time possess the Euclidean structures in the primed inertial…

The discussion is limited to first-class parametrized systems, where the definition of time evolution and observables is not trivial, and to finite dimensional systems in order that technicalities do not obscure the conceptual framework.…

We introduce a local multifractal formalism adapted to functions, measures or distributions which display multifractal characteristics that can change with time, or location. We develop this formalism in a general framework and we work out…

In this article, we consider fractional derivatives of local time for $d-$dimensional centered Gaussian processes satisfying certain strong local nondeterminism property. We first give a condition for existence of fractional derivatives of…

We investigate extreme value theory for physical systems with a global conservation law which describe renewal processes, mass transport models and long-range interacting spin models. As shown previously, a special feature is that the…

We prove that any given function can be smoothly approximated by functions lying in the kernel of a linear operator involving at least one fractional component. The setting in which we work is very general, since it takes into account…