240 research outputs found
Statistical Analysis of a Semilinear Hyperbolic System Advected by a White in Time Random Velocity Field
We study a system of semilinear hyperbolic equations passively advected by
smooth white noise in time random velocity fields. Such a system arises in
modeling non-premixed isothermal turbulent flames under single-step kinetics of
fuel and oxidizer. We derive closed equations for one-point and multi-point
probability distribution functions (PDFs) and closed form analytical formulas
for the one point PDF function, as well as the two-point PDF function under
homogeneity and isotropy. Exact solution formulas allows us to analyze the
ensemble averaged fuel/oxidizer concentrations and the motion of their level
curves. We recover the empirical formulas of combustion in the thin reaction
zone limit and show that these approximate formulas can either underestimate or
overestimate average concentrations when reaction zone is not tending to zero.
We show that the averaged reaction rate slows down locally in space due to
random advection induced diffusion; and that the level curves of ensemble
averaged concentration undergo diffusion about mean locations.Comment: 18 page
Improved linear response for stochastically driven systems
The recently developed short-time linear response algorithm, which predicts
the average response of a nonlinear chaotic system with forcing and dissipation
to small external perturbation, generally yields high precision of the response
prediction, although suffers from numerical instability for long response times
due to positive Lyapunov exponents. However, in the case of stochastically
driven dynamics, one typically resorts to the classical fluctuation-dissipation
formula, which has the drawback of explicitly requiring the probability density
of the statistical state together with its derivative for computation, which
might not be available with sufficient precision in the case of complex
dynamics (usually a Gaussian approximation is used). Here we adapt the
short-time linear response formula for stochastically driven dynamics, and
observe that, for short and moderate response times before numerical
instability develops, it is generally superior to the classical formula with
Gaussian approximation for both the additive and multiplicative stochastic
forcing. Additionally, a suitable blending with classical formula for longer
response times eliminates numerical instability and provides an improved
response prediction even for long response times
Ergodicity, Decisions, and Partial Information
In the simplest sequential decision problem for an ergodic stochastic process
X, at each time n a decision u_n is made as a function of past observations
X_0,...,X_{n-1}, and a loss l(u_n,X_n) is incurred. In this setting, it is
known that one may choose (under a mild integrability assumption) a decision
strategy whose pathwise time-average loss is asymptotically smaller than that
of any other strategy. The corresponding problem in the case of partial
information proves to be much more delicate, however: if the process X is not
observable, but decisions must be based on the observation of a different
process Y, the existence of pathwise optimal strategies is not guaranteed.
The aim of this paper is to exhibit connections between pathwise optimal
strategies and notions from ergodic theory. The sequential decision problem is
developed in the general setting of an ergodic dynamical system (\Omega,B,P,T)
with partial information Y\subseteq B. The existence of pathwise optimal
strategies grounded in two basic properties: the conditional ergodic theory of
the dynamical system, and the complexity of the loss function. When the loss
function is not too complex, a general sufficient condition for the existence
of pathwise optimal strategies is that the dynamical system is a conditional
K-automorphism relative to the past observations \bigvee_n T^n Y. If the
conditional ergodicity assumption is strengthened, the complexity assumption
can be weakened. Several examples demonstrate the interplay between complexity
and ergodicity, which does not arise in the case of full information. Our
results also yield a decision-theoretic characterization of weak mixing in
ergodic theory, and establish pathwise optimality of ergodic nonlinear filters.Comment: 45 page
Associação entre características de desempenho de tilápia-do-nilo ao longo do período de cultivo.
O objetivo deste trabalho foi estimar as herdabilidades e a estrutura de correlações genéticas entre as características de desempenho de tilápia-do-nilo (Oreochromis niloticus) da linhagem GIFT, em diferentes estágios do ciclo de produção. As tilápias foram cultivadas em tanques - rede. Mediu-se ganho em peso diário total, peso vivo e ganho em peso diário, em quatro períodos, com intervalos de aproximadamente 30 dias. Foram realizadas análises unicaracter para as medidas, em todas as biometrias e, nas análises bicaracter, as medidas de mesma característica foram combinadas duas a duas e com o ganho em peso diário total. As estimações de herdabilidade variaram de 0,15 a 0,11 para peso vivo, 0,16 a 0,09 para ganho em peso diário e 0,17 a 0,12 para ganho em peso diário total, nas análises unicaracter. Os valores estimados de correlação genética para peso vivo e ganho em peso diário, associados ao ganho em peso diário total, variaram entre 0,37 a 0,98 e 0,74 a 0,8 respectivamente. A forte associação genética estimada entre peso vivo em biometrias intermediárias e ganho em peso diário total sugere que a seleção para velocidade de crescimento pode ser realizada de forma precoce
Observability and nonlinear filtering
This paper develops a connection between the asymptotic stability of
nonlinear filters and a notion of observability. We consider a general class of
hidden Markov models in continuous time with compact signal state space, and
call such a model observable if no two initial measures of the signal process
give rise to the same law of the observation process. We demonstrate that
observability implies stability of the filter, i.e., the filtered estimates
become insensitive to the initial measure at large times. For the special case
where the signal is a finite-state Markov process and the observations are of
the white noise type, a complete (necessary and sufficient) characterization of
filter stability is obtained in terms of a slightly weaker detectability
condition. In addition to observability, the role of controllability in filter
stability is explored. Finally, the results are partially extended to
non-compact signal state spaces
On the exchange of intersection and supremum of sigma-fields in filtering theory
We construct a stationary Markov process with trivial tail sigma-field and a
nondegenerate observation process such that the corresponding nonlinear
filtering process is not uniquely ergodic. This settles in the negative a
conjecture of the author in the ergodic theory of nonlinear filters arising
from an erroneous proof in the classic paper of H. Kunita (1971), wherein an
exchange of intersection and supremum of sigma-fields is taken for granted.Comment: 20 page
Upper estimate of martingale dimension for self-similar fractals
We study upper estimates of the martingale dimension of diffusion
processes associated with strong local Dirichlet forms. By applying a general
strategy to self-similar Dirichlet forms on self-similar fractals, we prove
that for natural diffusions on post-critically finite self-similar sets
and that is dominated by the spectral dimension for the Brownian motion
on Sierpinski carpets.Comment: 49 pages, 7 figures; minor revision with adding a referenc
An Optimal Execution Problem with Market Impact
We study an optimal execution problem in a continuous-time market model that
considers market impact. We formulate the problem as a stochastic control
problem and investigate properties of the corresponding value function. We find
that right-continuity at the time origin is associated with the strength of
market impact for large sales, otherwise the value function is continuous.
Moreover, we show the semi-group property (Bellman principle) and characterise
the value function as a viscosity solution of the corresponding
Hamilton-Jacobi-Bellman equation. We introduce some examples where the forms of
the optimal strategies change completely, depending on the amount of the
trader's security holdings and where optimal strategies in the Black-Scholes
type market with nonlinear market impact are not block liquidation but gradual
liquidation, even when the trader is risk-neutral.Comment: 36 pages, 8 figures, a modified version of the article "An optimal
execution problem with market impact" in Finance and Stochastics (2014
Analytic Controllability of Time-Dependent Quantum Control Systems
The question of controllability is investigated for a quantum control system
in which the Hamiltonian operator components carry explicit time dependence
which is not under the control of an external agent. We consider the general
situation in which the state moves in an infinite-dimensional Hilbert space, a
drift term is present, and the operators driving the state evolution may be
unbounded. However, considerations are restricted by the assumption that there
exists an analytic domain, dense in the state space, on which solutions of the
controlled Schrodinger equation may be expressed globally in exponential form.
The issue of controllability then naturally focuses on the ability to steer the
quantum state on a finite-dimensional submanifold of the unit sphere in Hilbert
space -- and thus on analytic controllability. A relatively straightforward
strategy allows the extension of Lie-algebraic conditions for strong analytic
controllability derived earlier for the simpler, time-independent system in
which the drift Hamiltonian and the interaction Hamiltonia have no intrinsic
time dependence. Enlarging the state space by one dimension corresponding to
the time variable, we construct an augmented control system that can be treated
as time-independent. Methods developed by Kunita can then be implemented to
establish controllability conditions for the one-dimension-reduced system
defined by the original time-dependent Schrodinger control problem. The
applicability of the resulting theorem is illustrated with selected examples.Comment: 13 page
Stochastic Line-Motion and Stochastic Conservation Laws for Non-Ideal Hydromagnetic Models. I. Incompressible Fluids and Isotropic Transport Coefficients
We prove that smooth solutions of non-ideal (viscous and resistive)
incompressible magnetohydrodynamic equations satisfy a stochastic law of flux
conservation. This property involves an ensemble of surfaces obtained from a
given, fixed surface by advecting it backward in time under the plasma velocity
perturbed with a random white-noise. It is shown that the magnetic flux through
the fixed surface is equal to the average of the magnetic fluxes through the
ensemble of surfaces at earlier times. This result is an analogue of the
well-known Alfven theorem of ideal MHD and is valid for any value of the
magnetic Prandtl number. A second stochastic conservation law is shown to hold
at unit Prandtl number, a random version of the generalized Kelvin theorem
derived by Bekenstein-Oron for ideal MHD. These stochastic conservation laws
are not only shown to be consequences of the non-ideal MHD equations, but are
proved in fact to be equivalent to those equations. We derive similar results
for two more refined hydromagnetic models, Hall magnetohydrodynamics and the
two-fluid plasma model, still assuming incompressible velocities and isotropic
transport coefficients. Finally, we use these results to discuss briefly the
infinite-Reynolds-number limit of hydromagnetic turbulence and to support the
conjecture that flux-conservation remains stochastic in that limit.Comment: 20 pages, no figures, submitted to J. Math. Phys
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