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Let $\Phi:\mathcal{H}\longrightarrow\mathbb{R\cup}\left\{ +\infty\right\} $ be a closed convex proper function on a real Hilbert space $\mathcal{H}$, and $\partial\Phi:\mathcal{H}\rightrightarrows\mathcal{H}$ its subdifferential. For any…

It has been observed that many spacetimes which feature a near-extremal horizon exhibit the phenomenon of zero-damped modes. This is characterised by the existence of a sequence of quasinormal frequencies which all converge to some purely…

We investigate the time average mean square displacement $\overline{\delta^2}(x(t))=\int_0^{t-\Delta}[x(t^\prime+\Delta)-x(t^\prime)]^2 dt^\prime/(t-\Delta)$ for fractional Brownian and Langevin motion. Unlike the previously investigated…

For an ergodic action of the group $Z^n$ on a probability space and a given arbitrarily slowly decreasing to zero sequence, there exists an integrable function such that the standard ergodic time averages for it converge almost everywhere…

The paper studies asymptotic properties of estimators of multidimensional stochastic differential equations driven by Brownian motions from high-frequency discrete data. Consistency and central limit properties of a class of estimators of…

We obtain exact asymptotic results for the disorder averaged persistence of a Brownian particle moving in a biased Sinai landscape. We employ a new method that maps the problem of computing the persistence to the problem of finding the…

In the context of statistical supervised learning, the noiseless linear model assumes that there exists a deterministic linear relation $Y = \langle \theta_*, X \rangle$ between the random output $Y$ and the random feature vector $\Phi(U)$,…

In this paper, we prove that an Adam-type algorithm with smooth clipping approaches the global minimizer of the regularized non-convex loss function. Adding smooth clipping and taking the state space as the set of all trajectories, we can…

A new class of exclusion type processes acting in continuum with synchronous updating is introduced and studied. Ergodic averages of particle velocities are obtained and their connections to other statistical quantities, in particular to…

We investigate the assymptotic behaviour of the modeling error in approximate deconvolution model in the 3D periodic case, when the order $N$ of deconvolution goes to $\infty$. We consider successively the generalised Helmholz filters of…

Being concerned with ergodicity of McKean--Vlasov SDEs, we establish a general result on exponential ergodicity in the $L^1$-Wasserstein distance. The result is successfully applied to non-degenerate and multiplicative Brownian motion…

We propose a novel framework for approximating the statistical properties of turbulent flows by combining variational methods for the search of unstable periodic orbits with resolvent analysis for dimensionality reduction. Traditional…

In this paper, we consider an inference problem for an Ornstein-Uhlenbeck process driven by a general one-dimensional centered Gaussian process $(G_t)_{t\ge 0}$. The second order mixed partial derivative of the covariance function $ R(t,\,…

We obtain Fisher-Hartwig asymptotics with root and jump type singularities in space-time under the law of the stationary Hermitian Ornstein-Uhlenbeck process, which serve as a dynamical generalization of earlier static results obtained by…

We study a model of the motion by mean curvature of an (1+1) dimensional interface in a 2D Brownian velocity field. For the well-posedness of the model we prove existence and uniqueness for certain degenerate nonlinear stochastic evolution…

We consider the symmetric exclusion process on the $d$-dimensional lattice with translational invariant and ergodic initial data. It is then known that as $t$ diverges the distribution of the process at time $t$ converges to a Bernoulli…

Let the Ornstein-Uhlenbeck process $\{X_t,\,t\geq 0\}$ driven by a fractional Brownian motion $B^H$ described by $d X_t=-\theta X_t dt+ d B_t^H,\, X_0=0$ with known parameter $H\in (0,\frac34)$ be observed at discrete time instants $t_k=kh,…

For a class of idealized chaotic systems (hyperbolic systems) correlations decay exponentially in time. This result is asymptotic and rigorous. The decay rate is related to the Ruelle-Pollicott resonances. Nearly all chaotic model systems,…

The Pearson family of ergodic diffusions with a quadratic diffusion coefficient and a linear force are characterized by explicit dynamics of their integer moments and by explicit relaxation spectral properties towards their steady state.…

We characterize throughout the spectral range of an optical trap the nature of the noise at play and the ergodic properties of the corresponding Brownian motion of an overdamped trapped single microsphere, comparing experimental, analytical…