Related papers: The Mather measure and a Large Deviation Principle…
We formulate large deviations principle (LDP) for diffusion pair $(X^\epsilon,\xi^\epsilon)=(X_t^\epsilon,\xi_t^\epsilon)$, where first component has a small diffusion parameter while the second is ergodic Markovian process with fast time.…
The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place…
We introduce a novel generalization of entropy and conditional entropy from which most definitions from the literature can be derived as particular cases. Within this general framework, we investigate the problem of designing…
Entropy, its production, and its change in a dynamical system can be understood from either a fully stochastic dynamic description or from a deterministic dynamics exhibiting chaotic behavior. By taking the former approach based on the…
We extend Hochman's work on exponentially separated self-similar measures on $\mathbb{R}$ to the real analytic setting. More precisely, let $\Phi=\left\{ \varphi_{i}\right\} _{i\in\Lambda}$ be an iterated function system on $I:=[0,1]$…
We revisit entropic formulations of the uncertainty principle for an arbitrary pair of positive operator-valued measures (POVM) $A$ and $B$, acting on finite dimensional Hilbert space. Salicr\'u generalized $(h,\phi)$-entropies, including…
We consider the case of hyperbolic basic sets $\Lambda$ of saddle type for holomorphic maps $f: \mathbb P^2\mathbb C \to \mathbb P^2\mathbb C$. We study equilibrium measures $\mu_\phi$ associated to a class of H\"older potentials $\phi$ on…
We show that for any set $A\subseteq [0,1]^n$ with $\text{Vol}(A)\ge 1/2$ there exists a line $\ell $ such that the one-dimensional Lebesgue measure of $\ell \cap A$ is at least $\Omega ( n^{1/4} )$. The exponent $1/4$ is tight. More…
A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [E. Feireisl et al., Dissipative…
In this paper, we prove the moderate deviations principle (MDP) for a general system of slow-fast dynamics. We provide a unified approach, based on weak convergence ideas and stochastic control arguments, that cover both the averaging and…
In this paper we present an upper bound for the decay of correlation for the stationary stochastic process associated with the Entropy Penalized Method. Let $L(x, v):\Tt^n\times\Rr^n\to \Rr$ be a Lagrangian of the form L(x,v) = {1/2}|v|^2 -…
We investigate the large deviation principle (LDP) of the stationary solutions of stochastic functional differential equations (SFDEs) with infinite delay under small random perturbation. First, we demonstrate the existence and uniqueness…
We consider $n\times n$ Hermitian matrices with i.i.d. entries $X_{ij}$ whose tail probabilities $\mathbb {P}(|X_{ij}|\geq t)$ behave like $e^{-at^{\alpha}}$ for some $a>0$ and $\alpha \in(0,2)$. We establish a large deviation principle for…
One says that the local large deviation principle (LLDP) is satisfied for a family of random vectors $\{\zeta_T\}_{T\ge 0}$ in $\mathbb R^d,$ $d\ge 1,$ if there exists a function $D:\mathbb R^d\to [0,\infty],$ $D\not \equiv \infty,$ such…
In this paper we establish the large deviation principle for the the two-dimensional stochastic Navier-Stokes equations with anisotropic viscosity both for small noise and for short time. The proof for large deviation principle is based on…
We introduce generalized $(\alpha,\beta)$-transformations, which include all $(\alpha,\beta)$ and generalized $\beta$-transformations, and prove that all transitive generalized $(\alpha,\beta)$-transformations satisfy the level-2 large…
We derive the spatially homogeneous Landau equation for Maxwellian molecules from a natural stochastic interacting particle system. More precisely, we control the relative entropy between the joint law of the particle system and the…
Borrowing the idea of topological pressure determining measure-theoretical entropy in topological dynamical systems, we establish a variational principle for upper metric mean dimension with potential in terms of upper measure-theoretical…
In this paper, we establish a moderate deviation principle for an abstract nonlinear equation forced by random noise of L\'evy type. This type of equation covers many hydrodynamical models, including stochastic 2D Navier-Stokes equations,…
We establish multi-scale convergence theory for a class of Hamilton-Jacobi PDEs in space of probability measures. They arise from context of hydrodynamic limit of N-particle deterministic action minimizing (global) Lagrangian dynamics. From…