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Non-ergodicity impacts statistical inference in a diverse range of disciplines inside and outside of physics. However the concept of ergodicity is used inconsistently, and may refer to several nonequivalent notions. To help address this, we…
We adapt and study a variance reduction approach for the homogenization of elliptic equations in divergence form. The approach, borrowed from atomistic simulations and solid-state science [von Pezold et al, Physical Review B 2010; Wei et…
We introduce a variant of the asymmetric random average process with continuous state variables where the maximal transport is restricted by a cutoff. For periodic boundary conditions, we show the existence of a phase transition between a…
The nature of diffusion is usually studied for particles or time-evolving systems. Similar in principle, such studies can be conducted by tracking how a given function of observable properties evolves over time-akin to the evolution of…
Stochastic Volterra equations (SVEs) serve as mathematical models for the time evolutions of random systems with memory effects and irregular behaviour. We introduce neural stochastic Volterra equations as a physics-inspired architecture,…
The general framework of Entropic Dynamics (ED) is used to construct non-relativistic models of relational quantum mechanics from well known inference principles -- probability, entropy and information geometry. Although only partially…
Variational inequalities have recently attracted considerable interest in machine learning as a flexible paradigm for models that go beyond ordinary loss function minimization (such as generative adversarial networks and related deep…
We describe classes of ergodic dynamical systems for which some statistical properties are known exactly. These systems have integer dimension, are not globally dissipative, and are defined by a probability density and a two-form. This…
In this paper we introduce and exploit the real replica approach for a minimal generalization of the Hopfield model, by assuming the learned patterns to be distributed accordingly to a standard unit Gaussian. We consider the high storage…
This work is concerned with the stability properties of linear stochastic differential equations with random (drift and diffusion) coefficient matrices, and the stability of a corresponding random transition matrix (or exponential…
We study the consistency of stochastic dynamic programs under converging probability distributions and other approximations. Utilizing results on the epi-convergence of expectation functions with varying measures and integrands, and the…
We develop and implement new probabilistic strategy for proving exponential ergodicity for interacting diffusion processes on unbounded lattice. The concept of the solution used is rather weak as we construct the process in infinite…
This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third grade fluid equations in bounded domain $D\subset\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First, we…
This paper is concerned with ergodic properties of inhomogeneous Markov processes. Since the transition probabilities depend on initial times, the existing methods to obtain invariant measures for homogeneous Markov processes are not…
Explicit conditions are presented for the existence, uniqueness and ergodicity of the strong solution to a class of generalized stochastic porous media equations. Our estimate of the convergence rate is sharp according to the known optimal…
The study of epidemic models plays an important role in mathematical epidemiology. There are many researches on epidemic models using ordinary differential equations, partial differential equations or stochastic differential equations. In…
The aim of this work is to present, in self-contained form, results concerning fundamental and the most important questions related to linear stochastic Volterra equations of convolution type. The paper is devoted to study the existence and…
In this paper, we investigate the exponential ergodicity in a Wasserstein-type distance for a damping Hamiltonian dynamics with state-dependent and non-local collisions, which indeed is a special case of piecewise deterministic Markov…
Classical cellular automata represent a class of explicit discrete spacetime lattice models in which complex large-scale phenomena emerge from simple deterministic rules. With the goal to uncover different physically distinct classes of…
In uncertainty quantification, a stochastic modelling is often applied, where parameters are substituted by random variables. We investigate linear dynamical systems of ordinary differential equations with a quantity of interest as output.…