Time Averages of Markov Processes and Applications to Two-Timescale Problems

We show a decomposition into the sum of a martingale and a deterministic quantity for time averages of the solutions to non-autonomous SDEs and for discrete-time Markov processes. In the SDE case the martingale has an explicit representation in terms of the gradient of the associated semigroup or transition operator. We show how the results can be used to obtain quenched Gaussian concentration inequalities for time averages and to provide insights into the Averaging principle for two-timescale processes.Comment: 18 page

Towards a Quantitative Averaging Principle for Stochastic Differential Equations

This work explores the use of a forward-backward martingale method together with a decoupling argument and entropic estimates between the conditional and averaged measures to prove a strong averaging principle for stochastic differential equations with order of convergence 1/2. We obtain explicit expressions for all the constants involved. At the price of some extra assumptions on the time marginals and an exponential bound in time, we loosen the usual boundedness and Lipschitz assumptions. We conclude with an application of our result to Temperature-Accelerated Molecular Dynamics

Finishing the euchromatic sequence of the human genome

The sequence of the human genome encodes the genetic instructions for human physiology, as well as rich information about human evolution. In 2001, the International Human Genome Sequencing Consortium reported a draft sequence of the euchromatic portion of the human genome. Since then, the international collaboration has worked to convert this draft into a genome sequence with high accuracy and nearly complete coverage. Here, we report the result of this finishing process. The current genome sequence (Build 35) contains 2.85 billion nucleotides interrupted by only 341 gaps. It covers ∼99% of the euchromatic genome and is accurate to an error rate of ∼1 event per 100,000 bases. Many of the remaining euchromatic gaps are associated with segmental duplications and will require focused work with new methods. The near-complete sequence, the first for a vertebrate, greatly improves the precision of biological analyses of the human genome including studies of gene number, birth and death. Notably, the human enome seems to encode only 20,000-25,000 protein-coding genes. The genome sequence reported here should serve as a firm foundation for biomedical research in the decades ahead

Isotope ratios of H, C, and O in CO2 and H2O of the Martian atmosphere

Stable isotope ratios of H, C, and O are powerful indicators of a wide variety of planetary geophysical processes, and for Mars they reveal the record of loss of its atmosphere and subsequent interactions with its surface such as carbonate formation. We report in situ measurements of the isotopic ratios of D/H and O-18/O-16 in water and C-13/C-12, O-18/O-16, O-17/O-16, and (CO)-C-13-O-18/(CO)-C-12-O-16 in carbon dioxide, made in the martian atmosphere at Gale Crater from the Curiosity rover using the Sample Analysis at Mars (SAM)'s tunable laser spectrometer (TLS). Comparison between our measurements in the modern atmosphere and those of martian meteorites such as ALH 84001 implies that the martian reservoirs of CO2 and H2O were largely established similar to 4 billion years ago, but that atmospheric loss or surface interaction may be still ongoing