Convergence of the stochastic Euler scheme for locally Lipschitz coefficients

Stochastic differential equations are often simulated with the Monte Carlo Euler method. Convergence of this method is well understood in the case of globally Lipschitz continuous coefficients of the stochastic differential equation. The important case of superlinearly growing coefficients, however, has remained an open question. The main difficulty is that numerically weak convergence fails to hold in many cases of superlinearly growing coefficients. In this paper we overcome this difficulty and establish convergence of the Monte Carlo Euler method for a large class of one-dimensional stochastic differential equations whose drift functions have at most polynomial growth.Comment: Published at http://www.springerlink.com/content/g076w80730811vv3 in the Foundations of Computational Mathematics 201

Algebraic structure of stochastic expansions and efficient simulation

We investigate the algebraic structure underlying the stochastic Taylor solution expansion for stochastic differential systems.Our motivation is to construct efficient integrators. These are approximations that generate strong numerical integration schemes that are more accurate than the corresponding stochastic Taylor approximation, independent of the governing vector fields and to all orders. The sinhlog integrator introduced by Malham & Wiese (2009) is one example. Herein we: show that the natural context to study stochastic integrators and their properties is the convolution shuffle algebra of endomorphisms; establish a new whole class of efficient integrators; and then prove that, within this class, the sinhlog integrator generates the optimal efficient stochastic integrator at all orders.Comment: 19 page

Force generation in small ensembles of Brownian motors

The motility of certain gram-negative bacteria is mediated by retraction of type IV pili surface filaments, which are essential for infectivity. The retraction is powered by a strong molecular motor protein, PilT, producing very high forces that can exceed 150 pN. The molecular details of the motor mechanism are still largely unknown, while other features have been identified, such as the ring-shaped protein structure of the PilT motor. The surprisingly high forces generated by the PilT system motivate a model investigation of the generation of large forces in molecular motors. We propose a simple model, involving a small ensemble of motor subunits interacting through the deformations on a circular backbone with finite stiffness. The model describes the motor subunits in terms of diffusing particles in an asymmetric, time-dependent binding potential (flashing ratchet potential), roughly corresponding to the ATP hydrolysis cycle. We compute force-velocity relations in a subset of the parameter space and explore how the maximum force (stall force) is determined by stiffness, binding strength, ensemble size, and degree of asymmetry. We identify two qualitatively different regimes of operation depending on the relation between ensemble size and asymmetry. In the transition between these two regimes, the stall force depends nonlinearly on the number of motor subunits. Compared to its constituents without interactions, we find higher efficiency and qualitatively different force-velocity relations. The model captures several of the qualitative features obtained in experiments on pilus retraction forces, such as roughly constant velocity at low applied forces and insensitivity in the stall force to changes in the ATP concentration.Comment: RevTex 9 pages, 4 figures. Revised version, new subsections in Sec. III, removed typo

Large current noise in nanoelectromechanical systems close to continuous mechanical instabilities

We investigate the current noise of nanoelectromechanical systems close to a continuous mechanical instability. In the vicinity of the latter, the vibrational frequency of the nanomechanical system vanishes, rendering the system very sensitive to charge fluctuations and, hence, resulting in very large (super-Poissonian) current noise. Specifically, we consider a suspended single-electron transistor close to the Euler buckling instability. We show that such a system exhibits an exponential enhancement of the current noise when approaching the Euler instability which we explain in terms of telegraph noise.Comment: 11 pages, 12 figures; v2: minor changes, published versio

Magnetization precession due to a spin polarized current in a thin nanoelement: numerical simulation study

In this paper a detailed numerical study (in frames of the Slonczewski formalism) of magnetization oscillations driven by a spin-polarized current through a thin elliptical nanoelement is presented. We show that a sophisticated micromagnetic model, where a polycrystalline structure of a nanoelement is taken into account, can explain qualitatively all most important features of the magnetization oscillation spectra recently observed experimentally (S.I. Kiselev et al., Nature, vol. 425, p. 380 (2003), namely: existence of several equidistant spectral bands, sharp onset and abrupt disappearance of magnetization oscillations with increasing current, absence of the out-of-plane regime predicted by a macrospin model and the relation between frequencies of so called small-angle and quasichaotic oscillations. However, a quantitative agreement with experimental results (especially concerning the frequency of quasichaotic oscillations) could not be achieved in the region of reasonable parameter values, indicating that further model refinement is necessary for a complete understanding of the spin-driven magnetization precession even in this relatively simple experimental situation.Comment: Submitted to Phys. Rev. B; In this revised version figure positions on the page have been changed to ensure correct placements of the figure caption

The governing bodies of Lutheran secondary schools in Australia : an exploratory study

The governing bodies of Lutheran secondary colleges and schools in Australia are comprised of a cross-section of community-minded people, who have accepted the invitation to serve as college council members. The council members bring a range of experiences and expertise to the task they have accepted. This paper profiles the membership of college councils. It reports, to a limited degree, the alignment or otherwise of the way councils operate against the expectation of the By-laws of the Lutheran Church of Australia in regard to the operation of college councils. The paper also explores the views of both council members and principals on some issues, and uncovers the degree of difference or similarity in these views. The research method used for the study is the survey technique. Questionnaires were sent to the principals, council members and council chairpersons of all the Lutheran secondary colleges in Australia, except one. The research reveals that there are a significant number of issues of organisation and behaviours of Councils on which there are quite differing views, both between Councils, between individual council members, and between councils and principals

Pure multiplicative stochastic resonance of anti-tumor model with seasonal modulability

The effects of pure multiplicative noise on stochastic resonance in an anti-tumor system modulated by a seasonal external field are investigated by using theoretical analyses of the generalized potential and numerical simulations. For optimally selected values of the multiplicative noise intensity quasi-symmetry of two potential minima and stochastic resonance are observed. Theoretical results and numerical simulations are in good quantitative agreement.Comment: 5 pages, 5 figure

Synchronization of a Stochastic Reaction-Diffusion System On a Thin Two-Layer Domain

A system of semilinear parabolic stochastic partial differential equations with additive space-time noise is considered on the union of thin bounded tubular domains D1,ε := Γ × (0, ε) and D2,ε := Γ × (−ε, 0) joined at the common base Γ ⊂ Rd, where d ≥ 1. The equations are coupled by an interface condition on Γ which involves a reaction intensity k(x , ε), where x = (x , xd+1) ∈ Rd+1 with x ∈ Γ and |xd+1| < ε. Random influences are included through additive space-time Brownian motion, which depend only on the base spatial variable x ∈ Γ and not on the spatial variable xd+1 in the thin direction. Moreover, the noise is the same in both layers D1,ε and D2,ε. Limiting properties of the global random attractor are established as the thinness parameter of the domain ε → 0, i.e., as the initial domain becomes thinner, when the intensity function possesses the property limε→0 ε−1k(x , ε) = +∞. In particular, the limiting dynamics is described by a single stochastic parabolic equation with the averaged diffusion coefficient and a nonlinearity term, which essentially indicates synchronization of the dynamics on both sides of the common base Γ. Moreover, in the case of nondegenerate noise we obtain stronger synchronization phenomena in comparison with analogous results in the deterministic case previously investigated by Chueshov and Rekalo [EQUADIFF-2003, F. Dumortier et al., eds., World Scientific, Hackensack, NJ, 2005, pp. 645–650; Sb. Math., 195 (2004), pp. 103–128]

Dynamic model of gene regulation for the lac operon

Gene regulatory network is a collection of DNA which interact with each other and with other matter in the cell. The lac operon is an example of a relatively simple genetic network and is one of the best-studied structures in the Escherichia coli bacteria. In this work we consider a deterministic model of the lac operon with a noise term, representing the stochastic nature of the regulation. The model is written in terms of a system of simultaneous first order differential equations with delays. We investigate an analytical and numerical solution and analyse the range of values for the parameters corresponding to a stable solution

Dimer diffusion in a washboard potential

The transport of a dimer, consisting of two Brownian particles bounded by a harmonic potential, moving on a periodic substrate is investigated both numerically and analytically. The mobility and diffusion of the dimer center of mass present distinct properties when compared with those of a monomer under the same transport conditions. Both the average current and the diffusion coefficient are found to be complicated non-monotonic functions of the driving force. The influence of dimer equilibrium length, coupling strength and damping constant on the dimer transport properties are also examined in detail.Comment: Final revised version. 7 pages, 6 figure