461 research outputs found
Measured quantum probability distribution functions for Brownian motion
The quantum analog of the joint probability distributions describing a
classical stochastic process is introduced. A prescription is given for
constructing the quantum distribution associated with a sequence of
measurements. For the case of quantum Brownian motion this prescription is
illustrated with a number of explicit examples. In particular it is shown how
the prescription can be extended in the form of a general formula for the
Wigner function of a Brownian particle entangled with a heat bath.Comment: Phys. Rev. A, in pres
Foundations for Relativistic Quantum Theory I: Feynman's Operator Calculus and the Dyson Conjectures
In this paper, we provide a representation theory for the Feynman operator
calculus. This allows us to solve the general initial-value problem and
construct the Dyson series. We show that the series is asymptotic, thus proving
Dyson's second conjecture for QED. In addition, we show that the expansion may
be considered exact to any finite order by producing the remainder term. This
implies that every nonperturbative solution has a perturbative expansion. Using
a physical analysis of information from experiment versus that implied by our
models, we reformulate our theory as a sum over paths. This allows us to relate
our theory to Feynman's path integral, and to prove Dyson's first conjecture
that the divergences are in part due to a violation of Heisenberg's uncertainly
relations
Exponential Mixing for a Stochastic PDE Driven by Degenerate Noise
We study stochastic partial differential equations of the reaction-diffusion
type. We show that, even if the forcing is very degenerate (i.e. has not full
rank), one has exponential convergence towards the invariant measure. The
convergence takes place in the topology induced by a weighted variation norm
and uses a kind of (uniform) Doeblin condition.Comment: 10 pages, 1 figur
Stochastic Dynamical Structure (SDS) of Nonequilibrium Processes in the Absence of Detailed Balance. III: potential function in local stochastic dynamics and in steady state of Boltzmann-Gibbs type distribution function
From a logic point of view this is the third in the series to solve the
problem of absence of detailed balance. This paper will be denoted as SDS III.
The existence of a dynamical potential with both local and global meanings in
general nonequilibrium processes has been controversial. Following an earlier
explicit construction by one of us (Ao, J. Phys. {\bf A37}, L25 '04,
arXiv:0803.4356, referred to as SDS II), in the present paper we show
rigorously its existence for a generic class of situations in physical and
biological sciences. The local dynamical meaning of this potential function is
demonstrated via a special stochastic differential equation and its global
steady-state meaning via a novel and explicit form of Fokker-Planck equation,
the zero mass limit. We also give a procedure to obtain the special stochastic
differential equation for any given Fokker-Planck equation. No detailed balance
condition is required in our demonstration. For the first time we obtain here a
formula to describe the noise induced shift in drift force comparing to the
steady state distribution, a phenomenon extensively observed in numerical
studies. The comparison to two well known stochastic integration methods, Ito
and Stratonovich, are made ready. Such comparison was made elsewhere (Ao, Phys.
Life Rev. {\bf 2} (2005) 117. q-bio/0605020).Comment: latex. 13 page
A Pathwise Ergodic Theorem for Quantum Trajectories
If the time evolution of an open quantum system approaches equilibrium in the
time mean, then on any single trajectory of any of its unravelings the time
averaged state approaches the same equilibrium state with probability 1. In the
case of multiple equilibrium states the quantum trajectory converges in the
mean to a random choice from these states.Comment: 8 page
An open Problem on Strongly Consistent Learning of the Best Prediction for Gaussian Processes
Brownian markets
Financial market dynamics is rigorously studied via the exact generalized
Langevin equation. Assuming market Brownian self-similarity, the market return
rate memory and autocorrelation functions are derived, which exhibit an
oscillatory-decaying behavior with a long-time tail, similar to empirical
observations. Individual stocks are also described via the generalized Langevin
equation. They are classified by their relation to the market memory as heavy,
neutral and light stocks, possessing different kinds of autocorrelation
functions
The interplay of intrinsic and extrinsic bounded noises in genetic networks
After being considered as a nuisance to be filtered out, it became recently
clear that biochemical noise plays a complex role, often fully functional, for
a genetic network. The influence of intrinsic and extrinsic noises on genetic
networks has intensively been investigated in last ten years, though
contributions on the co-presence of both are sparse. Extrinsic noise is usually
modeled as an unbounded white or colored gaussian stochastic process, even
though realistic stochastic perturbations are clearly bounded. In this paper we
consider Gillespie-like stochastic models of nonlinear networks, i.e. the
intrinsic noise, where the model jump rates are affected by colored bounded
extrinsic noises synthesized by a suitable biochemical state-dependent Langevin
system. These systems are described by a master equation, and a simulation
algorithm to analyze them is derived. This new modeling paradigm should enlarge
the class of systems amenable at modeling.
We investigated the influence of both amplitude and autocorrelation time of a
extrinsic Sine-Wiener noise on: the Michaelis-Menten approximation of
noisy enzymatic reactions, which we show to be applicable also in co-presence
of both intrinsic and extrinsic noise, a model of enzymatic futile cycle
and a genetic toggle switch. In and we show that the
presence of a bounded extrinsic noise induces qualitative modifications in the
probability densities of the involved chemicals, where new modes emerge, thus
suggesting the possibile functional role of bounded noises
Robustness and Generalization
We derive generalization bounds for learning algorithms based on their
robustness: the property that if a testing sample is "similar" to a training
sample, then the testing error is close to the training error. This provides a
novel approach, different from the complexity or stability arguments, to study
generalization of learning algorithms. We further show that a weak notion of
robustness is both sufficient and necessary for generalizability, which implies
that robustness is a fundamental property for learning algorithms to work
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