Related papers: Functional limit theorems for the fractional Ornst…

In this article we derive a self-normalized functional limit theorem for strictly stationary linear processes with i.i.d. heavy-tailed innovations and random coefficients under the condition that all partial sums of the series of…

This paper establishes limit theorems for a class of stochastic hybrid systems (continuous deterministic dynamic coupled with jump Markov processes) in the fluid limit (small jumps at high frequency), thus extending known results for jump…

We consider Langevin equation involving fractional Brownian motion with Hurst index $H\in(0,\frac12)$. Its solution is the fractional Ornstein-Uhlenbeck process and with unknown drift parameter $\theta$. We construct the estimator that is…

This article establishes cutoff thermalization (also known as the cutoff phenomenon) for a class of generalized Ornstein-Uhlenbeck systems $(X^\varepsilon_t(x))_{t\geqslant 0}$ with $\varepsilon$-small additive L\'evy noise and initial…

This article studies typical dynamics and fluctuations for a slow-fast dynamical system perturbed by a small fractional Brownian noise. Based on an ergodic theorem with explicit rates of convergence, which may be of independent interest, we…

Fluctuations and noise may alter the behavior of dynamical systems considerably. For example, oscillations may be sustained by demographic fluctuations in biological systems where a stable fixed point is found in the absence of noise. We…

The fractional Ornstein-Uhleneck (fOU) process is described by the overdamped Langevin equation $\dot{x}(t)+\gamma x=\sqrt{2 D}\xi(t)$, where $\xi(t)$ is the fractional Gaussian noise with the Hurst exponent $0<H<1$. For $H\neq 1/2$ the fOU…

An important result in classical stochastic thermodynamics is the work fluctuation--dissipation relation (FDR), which states that the dissipated work done along a slow process is proportional to the resulting work fluctuations. Here we show…

We consider a perturbation of a Hilbert space-valued Ornstein--Uhlenbeck process by a class of singular nonlinear non-autonomous maximal monotone time-dependent drifts. The only further assumption on the drift is that it is bounded on balls…

We consider the asymptotic behaviour of the fluctuation process for large stochastic systems of interacting particles driven by both idiosyncratic and common noise with an interaction kernel \(k \in L^2(\R^d) \cap L^\infty(\R^d)\). Our…

For an arbitrary Hilbert space-valued Ornstein-Uhlenbeck process we construct the Ornstein-Uhlenbeck Bridge connecting a starting point $x$ and an endpoint $y$ that belongs to a certain linear subspace of full measure. We derive also a…

Consider the random variable $\mathrm{Tr}( f_1(W)A_1\dots f_k(W)A_k)$ where $W$ is an $N\times N$ Hermitian Wigner matrix, $k\in\mathbb{N}$, and choose (possibly $N$-dependent) regular functions $f_1,\dots, f_k$ as well as bounded…

We prove sharp bounds for the growth rate of eigenfunctions of the Ornstein-Uhlenbeck operator and its natural generalizations. The bounds are sharp even up to lower order terms and have important applications to geometric flows.

We prove a sequence of limiting results about weakly dependent stationary and regularly varying stochastic processes in discrete time. After deducing the limiting distribution for individual clusters of extremes, we present a new type of…

The generalization of fractional Brownian motion in infinite-dimensional white and grey noise spaces has been recently carried over, following the Mandelbrot-Van Ness representation, through Riemann-Liouville type fractional operators. Our…

In this paper we consider a triangular array of branching processes with non-stationary immigration. We prove a weak convergence of properly normalized branching processes with immigration to deterministic function under assumption that…

In this paper, we study the cut-off phenomenon under the total variation distance of $d$-dimensional Ornstein-Uhlenbeck processes which are driven by L\'evy processes. That is to say, under the total variation distance, there is an abrupt…

We revisit the Ornstein-Uhlenbeck (OU) process as the fundamental mathematical description of linear irreversible phenomena, with fluctuations, near an equilibrium. By identifying the underlying circulating dynamics in a stationary process…

For a class of stochastic models with Gaussian and rough mean-reverting volatility that embeds the genuine rough Stein-Stein model, we study the weak approximation rate when using a Euler type scheme with integrated kernels. Our first…

For a strictly stationary sequence of random variables we derive functional convergence of the joint partial sum and partial maxima process under joint regular variation with index $\alpha \in (0,2)$ and weak dependence conditions. The…