We consider a gradient interface model on the lattice with interaction potential which is a nonconvex perturbation of a convex potential. Using a technique which decouples the neighboring vertices sites into even and odd vertices, we show for a class of non-convex potentials: the uniqueness of ergodic component for \nablaϕ-Gibbs measures, the decay of covariances, the scaling limit and the strict convexity of the surface tension.
This paper is concerned with the approximation of the effective conductivity $σ(A,μ)$ associated to an elliptic operator $\nabla_x A(x,η) \nabla_x$ where for $x\in \R^d$, $d\geq 1$, $A(x,η)$ is a bounded elliptic random symmetric $d\times d$ matrix and $η$ takes value in an ergodic probability space $(X,μ)$. Writing $A^N(x,η)$ the periodization of $A(x,η)$ on the torus $T^d_N$ of dimension $d$ and side $N$ we prove that for $μ$-almost all $η$ $$ \lim_{N\to +\infty}σ(A^N,η)=σ(A,μ) $$ We extend this result to non-symmetric operators $\nabla_x (a+E(x,η)) \nabla_x$ corresponding to diffusions in e…
We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients $a^ω$. The diffusion is formally associated with $L^ωu = \nabla\cdot(a^ω\nabla u)$, and we make sense of it through Dirichlet forms theory. We prove for $X$ a quenched invariance principle, under some moment conditions on the environment; the key tool is the sublinearity of the corrector obtained by Moser's iteration scheme.
We study a continuous-time random walk on $\mathbb{Z}^d$ in an environment of random conductances taking values in $(0,\infty)$. For a static environment, we extend the quenched local limit theorem to the case of a general speed measure, given suitable ergodicity and moment conditions on the conductances and on the speed measure. Under stronger moment conditions, an annealed local limit theorem is also derived. Furthermore, an annealed local limit theorem is exhibited in the case of time-dependent conductances, under analogous moment and ergodicity assumptions. This dynamic local limit theorem…
We consider the $\nabla ϕ$ interface model with a uniformly convex interaction potential possessing Hölder continuous second derivatives. Combining ideas of Naddaf and Spencer with methods from quantitative homogenization, we show that the surface tension (or free energy) associated to the model is at least $C^{2,β}$ for some $β>0$. We also prove a fluctuation-dissipation relation by identifying its Hessian with the covariance matrix characterizing the scaling limit of the model. Finally, we obtain a quantitative rate of convergence for the Hessian of the finite-volume surface tension to that …
In this note we prove some sufficient conditions for ergodicity of a Levy-type process, such that on the test functions the generator of the respective semigroup is of the form $$ Lf(x) = a(x)f'(x) + \int_{\mathbb{R}}{ \left( f(x+u)-f(x)- \nabla f(x)\cdot u \mathbb{I}_{|u|\leq 1} \right) ν(x,du)}, \quad f\in C_{\infty}^{2}(\mathbb{R}). $$ Here $ν(x,du)$ is a Levy-type kernel and $a(\cdot): \mathbb{R}\to \mathbb{R}$. We consider the case when the tails are of polynomial decay as well as the case when the decay is (sub)-exponential. For the proof the Foster-Lyapunov approach is used.
We study the surface quasi-geostrophic equation with an irregular spatial perturbation $$ \partial_{t }θ+ u\cdot\nablaθ= -ν(-Δ)^{γ/2}θ+ ζ,\qquad u=\nabla^{\perp}(-Δ)^{-1}θ, $$ on $[0,\infty)\times\mathbb{T}^{2}$, with $ν\geq 0$, $γ\in [0,3/2)$ and $ζ\in B^{-2+κ}_{\infty,\infty}(\mathbb{T}^{2})$ for some $κ>0$. This covers the case of $ζ= (-Δ)^{α/2}ξ$ for $α<1$ and $ξ$ a spatial white noise on $\mathbb{T}^{2}$. Depending on the relation between $γ$ and $α$, our setting is subcritical, critical or supercritical in the language of Hairer's regularity structures \cite{Hai14}. Based on purely analy…
A goodness of fit test for the drift coefficient of an ergodic diffusion process is presented. The test is based on the score marked empirical process. The weak convergence of the proposed test statistic is studied under the null hypotheses and it is proved that the limit process is a continuous Gaussian process. The structure of its covariance function allows to calculate the limit distribution and it turns out that it is a function of a standard Brownian motion and so exact reject regions can be constructed. The proposed test is asymptotically distribution free and it is consistent under any…
We establish the existence, uniqueness and attraction properties of an ergodic invariant measure for the Boussinesq Equations in the presence of a degenerate stochastic forcing acting only in the temperature equation and only at the largest spatial scales. The central challenge is to establish time asymptotic smoothing properties of the Markovian dynamics corresponding to this system. Towards this aim we encounter a Lie bracket structure in the associated vector fields with a complicated dependence on solutions. This leads us to develop a novel Hörmander-type condition for infinite-dimensional…
This brief pedagogical note re-proves a simple theorem on the convergence, in $L_2$ and in probability, of time averages of non-stationary time series to the mean of expectation values. The basic condition is that the sum of covariances grows sub-quadratically with the length of the time series. I make no claim to originality; the result is widely, but unevenly, spread bit of folklore among users of applied probability. The goal of this note is merely to even out that distribution.
We consider models of gradient type, which are the densities of a collection of real-valued random variables $ϕ:=\{ϕ_x: x \in Λ\}$ given by $Z^{-1}\exp({-\sum\nolimits_{j \sim k}V(ϕ_j-ϕ_k)})$. We focus our study on the case that $V(\nablaϕ) = [1+(\nablaϕ)^2]^α$ with $0 < α< 1/2$, which is a non-convex potential. We introduce an auxiliary field $t_{jk}$ for each edge and represent the model as the marginal of a model with log-concave density. Based on this method, we prove that finite moments of the fields $\left<[v \cdot ϕ]^p \right>$ are bounded uniformly in the volume. This leads to the exis…
We study the ergodic properties of a class of multidimensional piecewise Ornstein-Uhlenbeck processes with jumps, which contains the limit of the queueing processes arising in multiclass many-server queues with heavy-tailed arrivals and/or asymptotically negligible service interruptions in the Halfin-Whitt regime as special cases. In these queueing models, the Itô equations have a piecewise linear drift, and are driven by either (1) a Brownian motion and a pure-jump Lévy process, or (2) an anisotropic Lévy process with independent one-dimensional symmetric $α$-stable components, or (3) an anis…
We introduce a class of $M \times M$ sample covariance matrices $\mathcal Q$ which subsumes and generalizes several previous models. The associated population covariance matrix $Σ= \mathbb E \cal Q$ is assumed to differ from the identity by a matrix of bounded rank. All quantities except the rank of $Σ- I_M$ may depend on $M$ in an arbitrary fashion. We investigate the principal components, i.e.\ the top eigenvalues and eigenvectors, of $\mathcal Q$. We derive precise large deviation estimates on the generalized components $\langle \mathbf w, \boldsymbol ξ_i \rangle$ of the outlier and non-out…
We consider a discrete-time Markov chain $\boldsymbolΦ$ on a general state-space ${\sf X}$, whose transition probabilities are parameterized by a real-valued vector $\boldsymbolθ$. Under the assumption that $\boldsymbolΦ$ is geometrically ergodic with corresponding stationary distribution $π(\boldsymbolθ)$, we are interested in estimating the gradient $\nabla α(\boldsymbolθ)$ of the steady-state expectation $$α(\boldsymbolθ) = π( \boldsymbolθ) f.$$ To this end, we first give sufficient conditions for the differentiability of $α(\boldsymbolθ)$ and for the calculation of its gradient via a seque…
The Ensemble Kalman filter and Ensemble square root filters are data assimilation methods used to combine high dimensional nonlinear models with observed data. These methods have proved to be indispensable tools in science and engineering as they allow computationally cheap, low dimensional ensemble state approximation for extremely high dimensional turbulent forecast models. From a theoretical perspective, these methods are poorly understood, with the exception of a recently established but still incomplete nonlinear stability theory. Moreover, recent numerical and theoretical studies of cata…
We study the joint limit distribution of the $k$ largest eigenvalues of a $p\times p$ sample covariance matrix $XX^\T$ based on a large $p\times n$ matrix $X$. The rows of $X$ are given by independent copies of a linear process, $X_{it}=\sum_j c_j Z_{i,t-j}$, with regularly varying noise $(Z_{it})$ with tail index $α\in(0,4)$. It is shown that a point process based on the eigenvalues of $XX^\T$ converges, as $n\to\infty$ and $p\to\infty$ at a suitable rate, in distribution to a Poisson point process with an intensity measure depending on $α$ and $\sum c_j^2$. This result is extended to random …
Consider an ergodic stationary random field $A$ on the ambient space $\mathbb R^d$. In order to establish concentration properties for nonlinear functions $Z(A)$, it is standard to appeal to functional inequalities like Poincaré or logarithmic Sobolev inequalities in the probability space. These inequalities are however only known to hold for a restricted class of laws (product measures, Gaussian measures with integrable covariance, or more general Gibbs measures with nicely behaved Hamiltonians). In this contribution, we introduce variants of these inequalities, which we refer to as multiscal…
