Astrid Hilbert

Hamiltonian systems with a stochastic force:nonlinear versus linear, and a girsanov formula

We discuss stochastic perturbations of classical Hamiltonian systems by a white noise force. We prove existence and uniqueness results for the solutions of the equation of motion under general conditions on the classical system, as well as their continuous dependence on the initial conditions. We also prove that the process in phase space is a diffusion with transition probability densities, and Lebesgue measure as c-finite invariant measure. We prove a Girsanov formula relating the solution for a nonlinear force with the one for a linear force, and give asymptotic estimates on functions of the phase space process

Single, Complete, Probability Spaces Consistent With EPR‐Bohm‐Bell Experimental Data

We show that paradoxical consequences of violations of Bell’s inequality are induced by the use of an unsuitable probabilistic description for the EPR‐Bohm‐Bell experiment. The conventional description (due to Bell) is based on a combination of statistical data collected for different settings of polarization beam splitters (PBSs). In fact, such data consists of some conditional probabilities which only partially define a probability space. Ignoring this conditioning leads to apparent contradictions in the classical probabilistic model (due to Kolmogorov). We show how to make a completely consistent probabilistic model by taking into account the probabilities of selecting the settings of the PBSs. Our model matches both the experimental data and is consistent with classical probability theory.

An Approximate Nash Equilibrium for Pure Jump Markov Games of Mean-field-type on Continuous State Space

We investigate mean-field games from the point of view of a large number of indistinguishable players, which eventually converges to infinity. The players are weakly coupled via their empirical measure. The dynamics of the states of the individual players is governed by a non-autonomous pure jump type semi group in a Euclidean space, which is not necessarily smoothing. Investigations are conducted in the framework of non-linear Markovian semi groups. We show that the individual optimal strategy results from a consistent coupling of an optimal control problem with a forward non-autonomous dynamics. In the limit as the number N of players goes to infinity this leads to a jump-type analog of the well-known non-linear McKean–Vlasov dynamics. The case where one player has an individual preference different from the ones of the remaining players is also covered. The two results combined reveal an epsilon-Nash Equilibrium for the N-player games.

Transcience of stochastically perturbed classical Hamiltonian systems and random wave operators

Transcience of stochastically perturbed classical Hamiltonian systems and random wave operators

Estimates Uniform in Time for the Transition Probability of Diffusions with Small Drift and for Stochastically Perturbed Newton Equations

An estimate uniform in time for the transition probability of diffusion processes with small drift is given. This also covers the case of a degenerate diffusion describing a stochastic perturbation of a particle moving according to the Newtons law. Moreover the random wave operator for such a particle is described and the analogue of asymptotic completeness is proven, the latter in the case of a sufficiently small drift.

Nelson‐type Limit for a Particular Class of Lévy Processes

Brownian motion has been constructed in different ways. Einstein was the most outstanding physicists involved in its construction. From a physical point of view a dynamical theory of Brownian motion was favorable. The Ornstein‐Uhlenbeck process models such a dynamical theory and E. Nelson amongst others derived Brownian motion from Ornstein‐Uhlenbeck theory via a scaling limit. In this paper we extend the scaling result to α‐stable Levy processes.

On the functional Hodrick-Prescott Filter with non-compact operators

Abstract We study a version of the functional Hodrick–Prescott filter in the case when the associated operator is not necessarily compact but merely closed and densely defined with closed range. We show that the associated optimal smoothing operator preserves the structure obtained in the compact case when the underlying distribution of the data is Gaussian.

Uniform asymptotic bounds for the heat kernel and the trace of a stochastic geodesic flow

We analyse the asymptotic behaviour of the heat kernel defined by a stochastically perturbed geodesic flow on the cotangent bundle of a Riemannian manifold for small time and small diffusion parameter. This extends Wentzel–Kramers–Brillouin-type methods to a particular case of a degenerate Hamiltonian. We give uniform bounds for the solution of the degenerate Hamiltonian boundary value problem for small time. The results are exploited to derive two-sided estimates and multiplicative asymptotics for the heat kernel semigroup and its trace.

Some remarks on classical, quantum and stochastic dynamical systems

We give a survey of some recent results on stochastic perturbation of classical dynamical systems of Hamiltonian type respectively of gradient type. We also discuss the latters as quantization of classical dynamical systems of the former type. Moreover we examine some relations between classical and quantum systems on manifolds, as well as infinite dimensional versions of these topics.

Evolutionary algorithms for the detection of structural breaks in time series: extended abstract

Detecting structural breaks is an essential task for the statistical analysis of time series, for example, for fitting parametric models to it. In short, structural breaks are points in time at which the behavior of the time series changes. Typically, no solid background knowledge of the time series under consideration is available. Therefore, a black-box optimization approach is our method of choice for detecting structural breaks. We describe a evolutionary algorithm framework which easily adapts to a large number of statistical settings. The experiments on artificial and real-world time series show that the algorithm detects break points with high precision and is computationally very efficient. A reference implementation is availble at the following address: http://www2.imm.dtu.dk/~pafi/SBX/launch.html}