We deal with the general structure of (noncommutative) stochastic processes by using the standard techniques of Operator Algebras. Any stochastic process is associated to a state on a universal object, i.e. the free product $C^*$-algebra in a natural way. In this setting one recovers the classical (i.e. commutative) probability scheme and many others, like those associated to the Monotone, Boolean and the $q$-deformed canonical commutation relations including the Bose/Fermi and Boltzmann cases. Natural symmetries like stationarity and exchangeability, as well as the ergodic properties of the stochastic processes are reviewed in detail for many interesting cases arising from Quantum Physics and Probability.
The objective of this paper is to characterize the structure of the set $Θ$ for a continuous ergodic upper probability $\mathbb{V}=\sup_{P\inΘ}P$ (Theorem \ref {main result}): . $Θ$ contains a finite number of ergodic probabilities; . Any invariant probability in $Θ$ is a convex combination of those ergodic ones in $Θ$; . Any probability in $Θ$ coincides with an invariant one in $Θ$ on the invariant $σ$-algebra. The last property has already been obtained in \textsl{Cerreia-Vioglio, Maccheroni, and Marinacci} \cite{ergodictheorem}, which firstly studied the ergodicity of such capacities. As an…
We formulate an ergodic theory for the (almost sure) limit $\mathcal{P}^\text{co}_{\tilde{\mathcal{E}}}$ of a sequence $(\mathcal{P}^\text{co}_{\mathcal{E}_n})$ of successive dynamic imprecise probability kinematics (DIPK, introduced in Caprio and Gong, 2021) updates of a set $\mathcal{P}^\text{co}_{\mathcal{E}_0}$ representing the initial beliefs of an agent. As a consequence, we formulate a strong law of large numbers.
Let $(X_n)_{n \in\mathbb{N}}$ be a $V$-geometrically ergodic Markov chain on a measurable space $\mathbb{X}$ with invariant probability distribution $π$. In this paper, we propose a discretization scheme providing a computable sequence $(\widehatπ_k)_{k\ge 1}$ of probability measures which approximates $π$ as $k$ growths to infinity. The probability measure $\widehatπ_k$ is computed from the invariant probability distribution of a finite Markov chain. The convergence rate in total variation of $(\widehatπ_k)_{k\ge 1}$ to $π$ is given. As a result, the specific case of first order autoregressiv…
Let $f=(f^x\mid x\in S)$, $S\subset\mathbb{Z}^m$, be a semigroup of ergodic measure-preserving transformations of a probability space $(Ω,\mathsf{P})$ and $h$ a real random function on $S$, such that $h(x+y,ω)\le h(x,ω)+h(y,f^xω)$ for all $x,y\in S$ and $ω\inΩ$. We prove that there exists a sublinear function $q\colon O\to[-\infty;\infty)$ defined on $O=\mathrm{int}(\mathrm{cone}(S))$, and a set $W\subsetΩ$ of full probability, such that $h(x_n,ω)/\lvert x_n\rvert\to q(x)$ for all $ω\in W$ and all sequences $(x_n)\subset S$ with asymptotic direction $x\in O$. The moment condition for this refl…
In many applications, it is often necessary to sample the mean value of certain quantity with respect to a probability measure μ on the level set of a smooth function $ξ: \mathbb{R}^d\rightarrow \mathbb{R}^k$, $1\le k < d$. A specially interesting case is the so-called conditional probability measure, which is useful in the study of free energy calculation and model reduction of diffusion processes. By Birkhoff's ergodic theorem, one approach to estimate the mean value is to compute the time average along an infinitely long trajectory of an ergodic diffusion process on the level set whose inva…
We analyze the distribution of eigenvectors for mesoscopic, mean-field perturbations of diagonal matrices in the bulk of the spectrum. Our results apply to a generalized $N\times N$ Rosenzweig-Porter model. We prove that the eigenvectors entries are asymptotically Gaussian with a specific variance, localizing them onto a small, explicit part of the spectrum. For a well spread initial spectrum, this variance profile universally follows a heavy-tailed Cauchy distribution. In the case of smooth entries, we also obtain a strong form of quantum unique ergodicity as an overwhelming probability bound…
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.
Different versions of percolation games on $\mathbb{Z}^{2}$, with parameters $p$ and $q$ that indicate, respectively, the probability with which a site in $\mathbb{Z}^{2}$ is labeled a trap and the probability with which it is labeled a target, are shown to have probability $0$ of culminating in draws when $p+q > 0$. We show that, for fixed $p$ and $q$, the probability of draw in each of these games is $0$ if and only if a certain $1$-dimensional probabilistic cellular automaton (PCA) $F_{p,q}$ with a size-$3$ neighbourhood is ergodic. This allows us to conclude that $F_{p,q}$ is ergodic whene…
This paper is concerned with an extended Galton-Watson process so as to allow individuals to live and reproduce for more than one unit time. We assume that each individual can live $k$ seasons (time-units) with probability $h_k$, and produce $m$ offspring with probability $p_m$ during each season. These can be seen as Galton-Watson processes with countably infinitely many types in which particles of type $i$ may only have offspring of type $i+1$ and type $1$. Let $\textbf{M}$ be its mean progeny matrix and $γ$ be the convergence radius of the power series $\sum_{k\geq 0}r^k(\textbf{M}^k)_{ij}$…
In the first part of the note we analyze the long time behaviour of a two dimensional stochastic Navier--Stokes equations system on a torus with a degenerate, one dimensional noise. In particular, for some initial data and noises we identify the invariant probability measure for the system and give a sufficient condition under which it is unique and stochastically stable. In the second part of the note, we consider a simple example of a finite-dimensional system of stochastic differential equations driven by a one dimensional Wiener process with a drift, that displays some similarity with the …
Consider $N\times N$ symmetric one-dimensional random band matrices with general distribution of the entries and band width $W \geq N^{3/4+\varepsilon}$ for any $\varepsilon>0$. In the bulk of the spectrum and in the large $N$ limit, we obtain the following results. (i) The semicircle law holds up to the scale $N^{-1+\varepsilon}$ for any $\varepsilon>0$. (ii) The eigenvalues locally converge to the point process given by the Gaussian orthogonal ensemble at any fixed energy. (iii) All eigenvectors are delocalized, meaning their ${\rm L}^\infty$ norms are all simultaneously bounded by $N^{-\fra…
