Regularity of invariant densities for 1D systems with random switching

© 2015 IOP Publishing Ltd & London Mathematical Society.This is a detailed analysis of invariant measures for one-dimensional dynamical systems with random switching. In particular, we prove the smoothness of the invariant densities away from critical points and describe the asymptotics of the invariant densities at critical points.

Stochastic switching in infinite dimensions with applications to random parabolic PDE

© 2015 Society for Industrial and Applied Mathematics.We consider parabolic PDEs with randomly switching boundary conditions. In order to analyze these random PDEs, we consider more general stochastic hybrid systems and prove convergence to, and properties of, a stationary distribution. Applying these general results to the heat equation with randomly switching boundary conditions, we find explicit formulae for various statistics of the solution and obtain almost sure results about its regularity and structure. These results are of particular interest for biological applications as well as for their significant departure from behavior seen in PDEs forced by disparate Gaussian noise. Our general results also have applications to other types of stochastic hybrid systems, such as ODEs with randomly switching right-hand sides.

A theory of hypoellipticity and unique ergodicity for semilinear stochastic PDEs

We present a theory of hypoellipticity and unique ergodicity for semilinear parabolic stochastic PDEs with "polynomial" nonlinearities and additive noise, considered as abstract evolution equations in some Hilbert space. It is shown that if Hörmander's bracket condition holds at every point of this Hilbert space, then a lower bound on the Malliavin covariance operatorμt can be obtained. Informally, this bound can be read as "Fix any finite-dimensional projection on a subspace of sufficiently regular functions. Then the eigenfunctions of μt with small eigenvalues have only a very small component in the image of Π." We also show how to use a priori bounds on the solutions to the equation to obtain good control on the dependency of the bounds on the Malliavin matrix on the initial condition. These bounds are sufficient in many cases to obtain the asymptotic strong Feller property introduced in [HM06]. One of the main novel technical tools is an almost sure bound from below on the size of "Wiener polynomials," where the coefficients are possibly non-adapted stochastic processes satisfying a Lips chitz condition. By exploiting the polynomial structure of the equations, this result can be used to replace Norris' lemma, which is unavailable in the present context. We conclude by showing that the two-dimensional stochastic Navier-Stokes equations and a large class of reaction-diffusion equations fit the framework of our theory.

Mainland size variation informs predictive models of exceptional insular body size change in rodents.

The tendency for island populations of mammalian taxa to diverge in body size from their mainland counterparts consistently in particular directions is both impressive for its regularity and, especially among rodents, troublesome for its exceptions. However, previous studies have largely ignored mainland body size variation, treating size differences of any magnitude as equally noteworthy. Here, we use distributions of mainland population body sizes to identify island populations as 'extremely' big or small, and we compare traits of extreme populations and their islands with those of island populations more typical in body size. We find that although insular rodents vary in the directions of body size change, 'extreme' populations tend towards gigantism. With classification tree methods, we develop a predictive model, which points to resource limitations as major drivers in the few cases of insular dwarfism. Highly successful in classifying our dataset, our model also successfully predicts change in untested cases.

Three Essays on the Effects of Donor Supplied Contraceptives on Fertility, Usage, and Attitudes

After the 2012 London Summit on Family Planning, there have been major strides in advancing the family planning agenda for low and middle-income countries worldwide. Much of the existing infrastructure and funding for family planning access is in the form of supplying free contraceptives to countries. While the average yearly value of donations since 2000 was over 170 million dollars for contraceptives procured for developing countries, an ongoing debate in the empirical literature is whether increases in contraceptive access and supply drive declines in fertility (UNFPA 2014).

This dissertation explores the fertility and behavioral effects of an increase in contraceptive supply donated to Zambia. Zambia, a high-fertility developing country, receives over 80 percent of its contraceptives from multilateral donors and aid agencies. Most contraceptives are donated and provided to women for free at government clinics (DELIVER 2015). I chose Zambia as a case study to measure the relationship between contraceptive supply and fertility because of two donor-driven events that led to an increase in both the quantity and frequency of contraceptives starting in 2008 (UNFPA 2014). Donations increased because donors and the Zambian government started a systematic method of forecasting contraceptive need on December 2007, and the Mexico City Policy was lifted in January 2009.

In Chapter 1, I investigate whether a large change in quantity and frequency of donated contraceptives affected fertility, using available data on contraceptive donations to Zambia, and birth records from the 2007 and 2013 Demographic and Health Surveys. I use a difference-in-difference framework to estimate the fertility effects of a supply chain improvement program that started in 2011, and was designed to ensure more regularity of contraceptive supply. The increase in total contraceptive supply after the Mexico City Policy was rescinded is associated with a 12 percent reduction in fertility relative to the before period, after controlling for demographic characteristics and time controls. There is evidence that a supply chain improvement program led to significant fertility declines for regions that received the program after the Mexico City Policy was rescinded.

In Chapter 2, I explore the effects of the large increase in donated contraceptives on modern contraceptive uptake. According to the 2007 and 2013 Demographic and Health Surveys, there was a dramatic increase in current use of injectables, implants, and IUDs. Simultaneously, declines occurred in usage of condoms, lactational amenorrhea method (LAM), and traditional methods. In this chapter, I estimate the effect of the increase in donations on uptake, composition of contraceptive usage, and usage of methods based on distance to contraceptive access points. The results show the post-2007 period is associated with an increase in usage of injectables and the pill among women living further away from access points.

In Chapter 3, I explore attitudes towards the contraceptive supply system, and identify areas for improvement, based on qualitative interviews with 14 experts and 61 Zambian users and non-users of contraceptives. The interviews uncover systemic barriers that prevent women from consistently accessing methods, and individual barriers that exacerbate the deficiencies in supply chain procedures. I find that 39 out of 61 women interviewed, both users and non-users, had personal experiences with stock out. The qualitative results suggest that the increase in contraceptives brought to the country after 2007 may have not contributed to as large of a decline in fertility because of bottlenecks in the supply chain, and problems in maintaining stock levels at clinics. I end the chapter with a series of four recommendations for improvements in the supply chain going forward, in light of recent commitments by the Zambian government during the 2012 London Summit on Family Planning.