505 research outputs found
On dynamical systems perturbed by a null-recurrent motion: the general case
We consider a perturbed ordinary differential equation where the perturbation is only significant when a one-dimensional null recurrent diffusion is close to zero. We investigate the first order correction to the unperturbed system and prove a central limit theorem type result, i.e., that the normalized deviation process converges in law in the space of continuous functions to a limit process which we identify. We show that this limit process has a component which only moves when the limit of the null-recurrent fast motion equals zero. The set of these times forms a zero-measure Cantor set and therefore the limiting process cannot be described by a standard SDE. We characterize this process by its infinitesimal generator (with appropriate boundary conditions) and we also characterize the process as the weak solution of an SDE that depends on the local time of the fast motion process. We also investigate the long time behavior of such a system when the unperturbed motion is trivial. In this case, we show that the long-time limit is constant on a set of full Lebesgue measure with probability 1, but it has nontrivial drift and diffusion components that move only when the fast motion equals zero.The authors are grateful to D. Dolgopyat for introducing them to the problem and to L. Koralov and D. Dolgopyat for their helpful suggestions during invaluable discussions and for reading the manuscript. We also thank P.E. Jabin for a discussion on Section 7. While working on the paper, Z. Pajor-Gyulai was partially supported by the NSF Grant Numbers 1309084 and DMS1101635. M. Salins was partially supported by the NSF Grant Number 1407615. The authors are also grateful for the anonymous referee for pointing out numerous typos and giving many suggestions, in particularly pointing us to several relevant papers that greatly improved the quality of the paper. (1309084 - NSF; DMS1101635 - NSF; 1407615 - NSF)Accepted manuscrip
On-line nonparametric estimation
A survey of some recent results on nonparametric on-line estimation is presented. The first result deals with an on-line estimation for a smooth signal S(t) in the classic 'signal plus Gaussian white noise' model. Then an analogous on-line estimator for the regression estimation problem with equidistant design is described and justified. Finally some preliminary results related to the on-line estimation for the diffusion observed process are described
Small Noise Asymptotics for Invariant Densities for a Class of Diffusions: A Control Theoretic View (with Erratum)
The uniqueness argument in the proof of Theorem 5, p. 483, of "Small noise
asymptotics for invariant densities for a class of diffusions: a control
theoretic view, J. Math. Anal. and Appl. (2009) " is flawed. We give here a
corrected proof.Comment: 23 pages; Journal of Mathematical Analysis and Applications, 200
Statistical approach to inverse boundary problems for partial differential equations
Inverse problems for elliptic and parabolic partial differential equations are considered. It is assumed that a solution of the equation is observed in white Gaussian noise with a small spectral density. The goal is to recover smooth but unknown boundary or initial conditions based on the noisy data. It is shown that the second order minimax estimators are linear as the spectral density of the noise goes to zero
On estimation of the linearized drift for nonlinear stochastic differential equations
The estimation of linearized drift for stochastic differential equations with equilibrium points is considered. It is proved that the linearized drift matrix can be estimated efficiently if the initial condition for the system is chosen close enough to the equilibrium point. Some bounds for initial conditions providing the asymptotical efficiency of estimators are found
Stability of gyroscopic systems under small random excitations
Gyroscopic systems with two degrees of freedom under small random perturbations are investigated by use of the stochastic averaging principle. It is proved that the principal term of the Lyapunov exponent for the original system coincides with the Lyapunov exponent for the averaged system. An explicit formula for the averaged Lyapunov exponent is derived. The averaged moment Lyapunov exponent is considered as well. An example is given in which an unstable gyroscopical system is stabilized by noise of the Sratonovich type
Moment instabilities in multidimensional systems with noise
We present a systematic study of moment evolution in multidimensional
stochastic difference systems, focusing on characterizing systems whose
low-order moments diverge in the neighborhood of a stable fixed point. We
consider systems with a simple, dominant eigenvalue and stationary, white
noise. When the noise is small, we obtain general expressions for the
approximate asymptotic distribution and moment Lyapunov exponents. In the case
of larger noise, the second moment is calculated using a different approach,
which gives an exact result for some types of noise. We analyze the dependence
of the moments on the system's dimension, relevant system properties, the form
of the noise, and the magnitude of the noise. We determine a critical value for
noise strength, as a function of the unperturbed system's convergence rate,
above which the second moment diverges and large fluctuations are likely.
Analytical results are validated by numerical simulations. We show that our
results cannot be extended to the continuous time limit except in certain
special cases.Comment: 21 pages, 15 figure
A homogenization theorem for Langevin systems with an application to Hamiltonian dynamics
This paper studies homogenization of stochastic differential systems. The
standard example of this phenomenon is the small mass limit of Hamiltonian
systems. We consider this case first from the heuristic point of view,
stressing the role of detailed balance and presenting the heuristics based on a
multiscale expansion. This is used to propose a physical interpretation of
recent results by the authors, as well as to motivate a new theorem proven
here. Its main content is a sufficient condition, expressed in terms of
solvability of an associated partial differential equation ("the cell
problem"), under which the homogenization limit of an SDE is calculated
explicitly. The general theorem is applied to a class of systems, satisfying a
generalized detailed balance condition with a position-dependent temperature.Comment: 32 page
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