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To characterize local finite-time properties associated with transient chaos in open dynamical systems, we introduce an escape rate and fractal dimensions suitable for this purpose in a coarse-grained description. We numerically illustrate…
We consider certain one dimensional ordinary stochastic differential equations driven by additive Brownian motion of variance $\varepsilon ^2$. When $\varepsilon =0$ such equations have an unstable non-hyperbolic fixed point and the drift…
A bifurcating system subject to multiplicative noise can display on-off intermittency. Using a canonical example, we investigate the extreme sensitivity of the intermittent behavior to the nature of the noise. Through a perturbative…
We consider a finite impulse response system with centered independent sub-Gaussian design covariates and noise components that are not necessarily identically distributed. We derive non-asymptotic near-optimal estimation and prediction…
We consider the problem of the construction of the Goodness-of-Fit test in the case of continuous time observations of a diffusion process with small noise. The null hypothesis is parametric and we use a minimum distance estimator of the…
We consider the problem of stochastic exit from a planar domain, whose boundary is an unstable periodic orbit, and which contains a stable periodic orbit. This problem arises when investigating the distribution of noise-induced phase slips…
In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in…
The dynamics of escape from an attractive state due to random perturbations is of central interest to many areas in science. Previous studies of escape in chaotic systems have rather focused on the case of unbounded noise, usually assumed…
In this paper we derive lower bounds in minimax sense for estimation of the instantaneous volatility if the diffusion type part cannot be observed directly but under some additional Gaussian noise. Three different models are considered. Our…
Recent studies have shown that in the presence of noise both fronts propagating into a metastable state and so-called pushed fronts propagating into an unstable state, exhibit diffusive wandering about the average position. In this paper we…
The current series of papers is concerned with stochastic stability of monotone dynamical systems by identifying the basic dynamical units that can survive in the presence of noise interference. In the first of the series, for the…
We consider solutions to nonlinear hyperbolic systems of balance laws with stiff relaxation and formally derive a parabolic-type effective system describing the late-time asymptotics of these solutions. We show that many examples from…
This paper addresses the estimation of uncertain distributed diffusion coefficients in elliptic systems based on noisy measurements of the model output. We formulate the parameter identification problem as an infinite dimensional…
We consider the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. We assume that the initial velocity is a finite-energy and L^q-summable perturbation of the Oseen vortex with…
We study statistical inference for small-noise-perturbed multiscale dynamical systems under the assumption that we observe a single time series from the slow process only. We construct estimators for both averaging and homogenization…
The minimum achievable statistical uncertainty in the estimation of physical parameters is determined by the quantum Fisher information. Its computation for noisy systems is still a challenging problem. Using a variational approach, we…
A mathematical model of autoresonance in nonlinear systems with combined parametric and external chirped frequency excitation is considered. Solutions with a growing amplitude and a bounded phase mismatch are associated with the…
The enigmatic stability of population oscillations within ecological systems is analyzed. The underlying mechanism is presented in the framework of two interacting species free to migrate between two spatial patches. It is shown that that…
Stochastic resetting and noise-enhanced stability are two phenomena which can affect the lifetime and relaxation of nonequilibrium states. They can be considered as measures of controlling the efficiency of the completion process when a…
We study the dynamics of fronts when both inertial effects and external fluctuations are taken into account. Stochastic fluctuations are introduced as multiplicative noise arising from a control parameter of the system. Contrary to the…