Long- and short-time asymptotics of the first-passage time of the Ornstein-Uhlenbeck and other mean-reverting processes

The first-passage problem of the Ornstein-Uhlenbeck process to a boundary is a long-standing problem with no known closed-form solution except in specific cases. Taking this as a starting-point, and extending to a general mean-reverting process, we investigate the long- and short-time asymptotics using a combination of Hopf-Cole and Laplace transform techniques. As a result we are able to give a single formula that is correct in both limits, as well as being exact in certain special cases. We demonstrate the results using a variety of other models

Analytical approximation to the multidimensional Fokker--Planck equation with steady state

The Fokker--Planck equation is a key ingredient of many models in physics, and related subjects, and arises in a diverse array of settings. Analytical solutions are limited to special cases, and resorting to numerical simulation is often the only route available; in high dimensions, or for parametric studies, this can become unwieldy. Using asymptotic techniques, that draw upon the known Ornstein--Uhlenbeck (OU) case, we consider a mean-reverting system and obtain its representation as a product of terms, representing short-term, long-term, and medium-term behaviour. A further reduction yields a simple explicit formula, both intuitive in terms of its physical origin and fast to evaluate. We illustrate a breadth of cases, some of which are `far' from the OU model, such as double-well potentials, and even then, perhaps surprisingly, the approximation still gives very good results when compared with numerical simulations. Both one- and two-dimensional examples are considered.Comment: Updated version as publishe

An exactly solvable self-convolutive recurrence

We consider a self-convolutive recurrence whose solution is the sequence of coefficients in the asymptotic expansion of the logarithmic derivative of the confluent hypergeometic function $U(a,b,z)$. By application of the Hilbert transform we convert this expression into an explicit, non-recursive solution in which the $n$th coefficient is expressed as the $(n-1)$th moment of a measure, and also as the trace of the $(n-1)$th iterate of a linear operator. Applications of these sequences, and hence of the explicit solution provided, are found in quantum field theory as the number of Feynman diagrams of a certain type and order, in Brownian motion theory, and in combinatorics

Distribution of the time at which the deviation of a Brownian motion is maximum before its first-passage time

We calculate analytically the probability density $P(t_m)$ of the time $t_m$ at which a continuous-time Brownian motion (with and without drift) attains its maximum before passing through the origin for the first time. We also compute the joint probability density $P(M,t_m)$ of the maximum $M$ and $t_m$. In the driftless case, we find that $P(t_m)$ has power-law tails: $P(t_m)\sim t_m^{-3/2}$ for large $t_m$ and $P(t_m)\sim t_m^{-1/2}$ for small $t_m$. In presence of a drift towards the origin, $P(t_m)$ decays exponentially for large $t_m$. The results from numerical simulations are in excellent agreement with our analytical predictions.Comment: 13 pages, 5 figures. Published in Journal of Statistical Mechanics: Theory and Experiment (J. Stat. Mech. (2007) P10008, doi:10.1088/1742-5468/2007/10/P10008

Associations between the K232A polymorphism in the diacylglycerol-O-transferase 1 (DGAT1) gene and performance in Irish Holstein-Friesian dairy cattle

peer-reviewedSelection based on genetic polymorphisms requires accurate quantification of the effect or association of the polymorphisms with all traits of economic importance. The objective of this study was to estimate, using progeny performance data on 848 Holstein-Friesian bulls, the association between a non-conservative alanine to lysine amino acid change (K232A) in exon 8 of the diacylglycerol-O-transferase 1 (DGAT1) gene and milk production and functionality in the Irish Holstein-Friesian population. The DGAT1 gene encodes the diacylglycerol-O-transferase microsomal enzyme necessary to catalyze the final step in triglyceride synthesis. Weighted mixed model methodology, accounting for the additive genetic relationships among animals, was used to evaluate the association between performance and the K232A polymorphism. The minor allele frequency (K allele) was 0.32. One copy of the K allele was associated (P < 0.001) with 77 kg less milk yield, 4.22 kg more fat yield, 0.99 kg less protein yield, and 1.30 and 0.28 g/kg greater milk fat and protein concentration, respectively; all traits were based on predicted 305-day production across the first five lactations. The K232A polymorphism explained 4.8%, 10.3% and 1.0% of the genetic variance in milk yield, fat yield and protein yield, respectively. There was no association between the K232A polymorphism and fertility, functional survival, calving performance, carcass traits, or any conformation trait with the exception of rump width and carcass conformation. Using the current economic values for the milk production traits in the Irish total merit index, one copy of the K allele is worth €5.43 in expected profitability of progeny. Results from this study will be useful in quantifying the cost-benefit of including the K232A polymorphism in the Irish national breeding programme

THE SPANNING SET AS A MEASURE OF MOVEMENT VARIABILITY

The variability of an individual’s movement pattern is an increasingly important focus of research in sport and exercise biomechanics. Inter-trial variability of a single variable is typically assessed using mean deviation or coefficient of variation, however, recent alternatives to these have been proposed such as the spanning set technique. This paper presents an investigation into the validity of the spanning set measure. Variability scores using the spanning set were compared against more traditional measures of variability (mean deviation, coefficient of variation and variance ratio). Results indicate that the spanning set is biased towards early-phase variability and may inaccurately describe the overall level of movement variability

Effective mass and quantum lifetime in a Si/Si0.87Ge0.13/Si two-dimensional hole gas

Measurements of Shubnikov de Haas oscillations in the temperature range 0.3–2 K have been used to determine an effective mass of 0.23 m0 in a Si/Si0.87Ge0.13/Si two-dimensional hole gas. This value is in agreement with theoretical predictions and with that obtained from cyclotron resonance measurements. The ratio of the transport time to the quantum lifetime is found to be 0.8. It is concluded that the 4 K hole mobility of 11 000 cm2 V−1 s−1 at a carrier sheet density of 2.2×1011 cm−2 is limited by interface roughness and short-range interface charge scattering