Excursions of diffusion processes and continued fractions

It is well-known that the excursions of a one-dimensional diffusion process can be studied by considering a certain Riccati equation associated with the process. We show that, in many cases of interest, the Riccati equation can be solved in terms of an infinite continued fraction. We examine the probabilistic significance of the expansion. To illustrate our results, we discuss some examples of diffusions in deterministic and in random environments.Comment: 28 pages. Minor changes to Section

Pade approximants of random Stieltjes series

We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function. We study the convergence of the finite truncations of the continued fraction or, equivalently, of the diagonal Pade approximants of the function S(t). By using the Dyson--Schmidt method for an equivalent one-dimensional disordered system, and the results of Marklof et al. (2005), we obtain explicit formulae (in terms of modified Bessel functions) for the almost-sure rate of convergence of these approximants, and for the almost-sure distribution of their poles.Comment: To appear in Proc Roy So

Duration of Non-standard Employment

Non-standard employment is fairly common in Canada, accounting for almost two in five workers aged 16 to 69. Concerns about nonstandard work arise because workers in these jobs tend to have low earnings and are more likely to live in low-income families. They also face greater risk of unemployment and enjoy fewer employer- or government-sponsored benefits. Adding fuel to these concerns is the persistence of nonstandard employment among the people who hold these jobs. For example, of the five million Canadians in non-standard jobs in 1999; half remained in such jobs throughout the following two years. Older workers (45 to 69) were particularly susceptible. The potentially negative aspects of non-standard work are mitigated by many individuals choosing selfemployment, or temporary or part-time jobs. Moreover, non-standard work often serves as a gateway to standard employment. For example, some 60% of individuals without jobs in 1999 who were subsequently employed in 2000 or 2001 initially found nonstandard work. And the temporary nature of non-standard work among youth indicates that for this group non-standard work is typically a stepping stone to permanent full-time employment.Types of employment; non-standard employment

One-dimensional disordered quantum mechanics and Sinai diffusion with random absorbers

We study the one-dimensional Schr\"odinger equation with a disordered potential of the form $V (x) = \phi(x)^2+\phi'(x) + \kappa(x)$ where $\phi(x)$ is a Gaussian white noise with mean $\mu g$ and variance $g$, and $\kappa(x)$ is a random superposition of delta functions distributed uniformly on the real line with mean density $\rho$ and mean strength $v$. Our study is motivated by the close connection between this problem and classical diffusion in a random environment (the Sinai problem) in the presence of random absorbers~: $\phi(x)$ models the force field acting on the diffusing particle and $\kappa(x)$ models the absorption properties of the medium in which the diffusion takes place. The focus is on the calculation of the complex Lyapunov exponent $\Omega(E) = \gamma(E) - \mathrm{i} \pi N(E)$, where $N$ is the integrated density of states per unit length and $\gamma$ the reciprocal of the localisation length. By using the continuous version of the Dyson-Schmidt method, we find an exact formula, in terms of a Hankel function, in the particular case where the strength of the delta functions is exponentially-distributed with mean $v=2g$. Building on this result, we then solve the general case -- in the low-energy limit -- in terms of an infinite sum of Hankel functions. Our main result, valid without restrictions on the parameters of the model, is that the integrated density of states exhibits the power law behaviour N(E) \underset{E\to0+}{\sim} E^\nu \hspace{0.5cm} \mbox{where } \nu=\sqrt{\mu^2+2\rho/g}\:. This confirms and extends several results obtained previously by approximate methods.Comment: LaTeX, 44 pages, 17 pdf figure

Profiles and Transitions of Groups at Risk of Social Exclusion: Lone Parents

This study attempts to answer the following basic question: why do some lone parents escape low income or never enter spells of low income or social assistance (SA), while others remain in low income or on SA for many years? The analysis relies on the 1993-98 longitudinal panel of the Survey of Labour and Income Dynamics (SLID). The main focus is lone mothers, since they account for 93% of low income lone parents. The results make somewhat of a case for investing more in education. However, this is not conclusive. Many lone mothers who are in low income or SA recipients have a post-secondary certification. Also, a higher level of education does not seem to have any benefits in terms of shortening SA spells. The fact that half of new SA recipients exit within the first two years suggest that policies should be well targeted. However, waiting for several years to ascertain who are long term recipients is not the best targeting strategy. Not only is valuable time wasted, but there is evidence that the longer individuals stay on SA, the more difficult it is to exit. A better strategy is to keep probing the characteristics of SA recipients that are associated with long spells and develop programs that are targeted to those characteristics. And since lack of paid work or limited attachment to paid work are common factors among the low income and SA recipients, the main focus should be on providing employment services (such as referrals and employment counseling), coupled with a more generous treatment of earnings under SA and wage subsidies to those able to work a significant number of paid hours.lone mothers; welfare; social assistance

Products of random matrices and generalised quantum point scatterers

To every product of $2\times2$ matrices, there corresponds a one-dimensional Schr\"{o}dinger equation whose potential consists of generalised point scatterers. Products of {\em random} matrices are obtained by making these interactions and their positions random. We exhibit a simple one-dimensional quantum model corresponding to the most general product of matrices in $\text{SL}(2, {\mathbb R})$. We use this correspondence to find new examples of products of random matrices for which the invariant measure can be expressed in simple analytical terms.Comment: 38 pages, 13 pdf figures. V2 : conclusion added ; Definition of function $\Omega$ change