On the spectral gap of Brownian motion with jump boundary

In this paper we consider the Brownian motion with jump boundary and present a new proof of a recent result of Li, Leung and Rakesh concerning the exact convergence rate in the one-dimensional case. Our methods are dierent and mainly probabilistic relying on coupling methods adapted to the special situation under investigation. Moreover we answer a question raised by Ben-Ari and Pinsky concerning the dependence of the spectral gap from the jump distribution in a multi-dimensional setting.

The thermohydrodynamical picture of Brownian motion via a generalized Smoluchowsky equation

Via an operator continued fraction scheme, we expand Kramers equation in the high friction limit. Then all distribution moments are expressed in terms of the first momemt (particle density). The latter satisfies a generalized Smoluchowsky equation. As an application, we present the nonequilibrium thermodynamics and hydrodynamical picture for the one-dimensional Brownian motion. (C) 2000 Elsevier B.V. B.V. All rights reserved.

Non-Markovian expansion in quantum Brownian motion

We consider the non-Markovian Langevin evolution of a dissipative dynamical system in quantum mechanics in the path integral formalism. After discussing the role of the frequency cutoff for the interaction of the system with the heat bath and the kernel and noise correlator that follow from the most common choices, we derive an analytic expansion for the exact non-Markovian dissipation kernel and the corresponding colored noise in the general case that is consistent with the fluctuation-dissipation theorem and incorporates systematically non-local corrections. We illustrate the modifications to results obtained using the traditional (Markovian) Langevin approach in the case of the exponential kernel and analyze the case of the non-Markovian Brownian motion. We present detailed results for the free and the quadratic cases, which can be compared to exact solutions to test the convergence of the method, and discuss potentials of a general nonlinear form. © 2013 Elsevier B.V. All rights reserved.

Anisotropic Brownian motion in ordered phases of DNA fragments

Using Fluorescence Recovery After Photobleaching, we investigate the Brownian motion of DNA rod-like fragments in two distinct anisotropic phases with a local nematic symmetry. The height of the measurement volume ensures the averaging of the anisotropy of the in-plane diffusive motion parallel or perpendicular to the local nematic director in aligned domains. Still, as shown in using a model specifically designed to handle such a situation and predicting a non-Gaussian shape for the bleached spot as fluorescence recovery proceeds, the two distinct diffusion coefficients of the DNA particles can be retrieved from data analysis. In the first system investigated (a ternary DNA-lipid lamellar complex), the magnitude and anisotropy of the diffusion coefficient of the DNA fragments confined by the lipid bilayers are obtained for the first time. In the second, binary DNA-solvent system, the magnitude of the diffusion coefficient is found to decrease markedly as DNA concentration is increased from isotropic to cholesteric phase. In addition, the diffusion coefficient anisotropy measured within cholesteric domains in the phase coexistence region increases with concentration, and eventually reaches a high value in the cholesteric phase.

The von Mises-Fisher distribution of the first exit point from the hypersphere of the drifted Brownian motion and the density of the first exit time

A characterization is provided for the von Mises–Fisher random variable, in terms of first exit point from the unit hypersphere of the drifted Wiener process. Laplace transform formulae for the first exit time from the unit hypersphere of the drifted Wiener process are provided. Post representations in terms of Bell polynomials are provided for the densities of the first exit times from the circle and from the sphere.

Interim analysis of clinical trials: Simulation studies of fractional Brownian motion

Interim clinical trial monitoring procedures were motivated by ethical and economic considerations. Classical Brownian motion (Bm) techniques for statistical monitoring of clinical trials were widely used. Conditional power argument and α-spending function based boundary crossing probabilities are popular statistical hypothesis testing procedures under the assumption of Brownian motion. However, it is not rare that the assumptions of Brownian motion are only partially met for trial data. Therefore, I used a more generalized form of stochastic process, called fractional Brownian motion (fBm), to model the test statistics. Fractional Brownian motion does not hold Markov property and future observations depend not only on the present observations but also on the past ones. In this dissertation, we simulated a wide range of fBm data, e.g., H = 0.5 (that is, classical Bm) vs. 0.5< H <1, with treatment effects vs. without treatment effects. Then the performance of conditional power and boundary-crossing based interim analyses were compared by assuming that the data follow Bm or fBm. Our simulation study suggested that the conditional power or boundaries under fBm assumptions are generally higher than those under Bm assumptions when H > 0.5 and also matches better with the empirical results. ^

On ergodic least-squares estimators of the generalized diffusion coefficient for fractional Brownian motion

We analyse a class of estimators of the generalized diffusion coefficient for fractional Brownian motion Bt of known Hurst index H, based on weighted functionals of the single time square displacement. We show that for a certain choice of the weight function these functionals possess an ergodic property and thus provide the true, ensemble-averaged, generalized diffusion coefficient to any necessary precision from a single trajectory data, but at expense of a progressively higher experimental resolution. Convergence is fastest around H ? 0.30, a value in the subdiffusive regime.

On the Moving Boundary Hitting Probability for the Brownian Motion

2000 Mathematics Subject Classification: 60J65.

A geometriai Brown-mozgás feltevésének elfogadhatósága a reálopciók értékelésében (The acceptability of the assumption of geometric brownian motion in the valuation of real options)

A reálopciók a döntési rugalmasság megtestesítőiként jelen vannak a vállalatvezetők mindennapjaiban, és cégtől függően jelentős értéket képviselhetnek. Értékelésük a hagyományos diszkontált pénzáramlás módszerekkel csak korlátozottan lehetséges, ezért alternatívaként felmerül a pénzügyi opcióárazás módszertana, amelynek hagyományos változatai az alaptermék alakulásáról geometriai Brown-mozgást feltételeznek. A cikk ezt a feltevést veszi górcső alá a reálopciókra történő alkalmazás szempontjából, és megmutatja, hogy habár önkényesnek tűnhet, valójában nem pusztán egy matematikai szempontból kényelmes megoldás, hanem pénzügyileg is elfogadható feltétel. _______ Real options represent the fl exibility of decision-making, and are thus part of the everyday work of corporate executives, often having great value. Valuing them with the use of traditional Discounted Cash Flow models has limited relevance, therefore arises the alternative methodology of fi nancial option pricing, the traditional versions of which assume that the price of the underlying asset follows Geometric Brownian Motion. The paper examines this assumption from the aspect of real option valuation and shows that although it might seem arbitrary, it is not only a mathematically convenient choice, but also a fi nancially acceptable one.

Comments on ‘‘Modeling fractional stochastic systems as non-random fractional dynamics driven Brownian motions

Applied Mathematical Modelling, Vol.33

Brownian bridge and other path-dependent Gaussian processes vectorial simulation

The iterative simulation of the Brownian bridge is well known. In this article, we present a vectorial simulation alternative based on Gaussian processes for machine learning regression that is suitable for interpreted programming languages implementations. We extend the vectorial simulation of path-dependent trajectories to other Gaussian processes, namely, sequences of Brownian bridges, geometric Brownian motion, fractional Brownian motion, and Ornstein-Ulenbeck mean reversion process.

Symmetric Brownian motor

In this paper, we present a model of a symmetric Brownian motor which changes the sign of its velocity when the temperature gradient is inverted. The velocity, external work, and efficiency are studied as a function of the temperatures of the baths and other relevant parameters. The motor shows a current reversal when another parameter (a phase shift) is varied. Analytical predictions and results from numerical simulations are performed and agree very well. Generic properties of this type of motor are discussed.

Brownian pump powered by a white-noise flasing ratchet

A Brownian pump of particles powered by a stochastic flashing ratchet mechanism is studied. The pumping device is embedded in a finite region and bounded by particle reservoirs. In the steady state, we exactly calculate the spatial density profile, the concentration ratio between both reservoirs and the particle flux. We propose a simulation framework for the consistent evaluation of such observable quantities.

Tight coupling in thermal Brownian motors

We study analytically a thermal Brownian motor model and calculate exactly the Onsager coefficients. We show how the reciprocity relation holds and that the determinant of the Onsager matrix vanishes. Such a condition implies that the device is built with tight coupling. This explains why Carnot¿s efficiency can be achieved in the limit of infinitely slow velocities. We also prove that the efficiency at maximum power has the maximum possible value, which corresponds to the Curzon-Alhborn bound. Finally, we discuss the model acting as a Brownian refrigerator.