The impact of daylight-saving time (DST) on patients with delayed sleep-wake phase disorder (DSWPD).

Due to time zones, sun time and local time rarely match. The difference between local and sun time, which we designate by Solar Jet Lag (SoJL), depends on location within a time zone and can range from zero to several hours. Daylight saving time (DST) simply adds 1 h to SoJL, independently of the location. We hypothesised that the impact of DST is particularly problematic in patients with delayed sleep-wake phase disorder (DSWPD), worsening their sleep debt. DSWPD is characterised by a chronic misalignment between the internal and social timing, reflected by an inability to fall asleep and wake-up at conventional or socially acceptable times. We analysed the clinical records of 162 DSWPD patients from a sleep medicine centre in Lisbon, Portugal (GMTzone), and separated them into two groups: the ones diagnosed across DST or across Standard Time (ST). We included 82 patients (54.9% male; age: median [Q1 , Q3 ] 34.5 [25.0, 45.3]; range 16-92; 54 in DST and 28 in ST) who had Dim Light Melatonin Onset (DLMO) measured as a marker for the circadian phase and sleep timing (onset, SO, mid-point, MS and end, SE) self-reported separately for work- and work-free days. Differences between ST and DST were compared using Mann-Whitney or Student's t-tests. On a weekly average, patients in DST slept less (difference between medians of 37 min. p < .01), mainly due to sleep on workdays (SDw, p < .01), which also correlated with SoJL (rsp = .38, p < .01). While the time from DLMO to SO was similar in those in ST or those in DST, the time from DLMO to SE was significantly shorter for those in DST. The average duration between DLMO and sleep end was close to 10.5 h in ST, the biological night length described in the literature. Our results favour perennial ST and suggest assigning time-zones close to sun time to prevent social jetlag and sleep deprivation.

MARKOV PROCESSES CONDITIONED ON THEIR LOCATION AT LARGE EXPONENTIAL TIMES.

Suppose that (Xt ) t ≥0 is a one-dimensional Brownian motion with negative drift -µ. It is possible to make sense of conditioning this process to be in the state 0 at an independent exponential random time and if we kill the conditioned process at the exponential time the resulting process is Markov. If we let the rate parameter of the random time go to 0, then the limit of the killed Markov process evolves like X conditioned to hit 0, after which time it behaves as X killed at the last time X visits 0. Equivalently, the limit process has the dynamics of the killed "bang-bang" Brownian motion that evolves like Brownian motion with positive drift +µ when it is negative, like Brownian motion with negative drift -µ when it is positive, and is killed according to the local time spent at 0. An extension of this result holds in great generality for a Borel right process conditioned to be in some state a at an exponential random time, at which time it is killed. Our proofs involve understanding the Campbell measures associated with local times, the use of excursion theory, and the development of a suitable analogue of the "bang-bang" construction for a general Markov process. As examples, we consider the special case when the transient Borel right process is a one-dimensional diffusion. Characterizing the limiting conditioned and killed process via its infinitesimal generator leads to an investigation of the h-transforms of transient one-dimensional diffusion processes that goes beyond what is known and is of independent interest.

Subdiffusion in the Presence of Reactive Boundaries: A Generalized Feynman-Kac Approach.

We derive, through subordination techniques, a generalized Feynman-Kac equation in the form of a time fractional Schrödinger equation. We relate such equation to a functional which we name the subordinated local time. We demonstrate through a stochastic treatment how this generalized Feynman-Kac equation describes subdiffusive processes with reactions. In this interpretation, the subordinated local time represents the number of times a specific spatial point is reached, with the amount of time spent there being immaterial. This distinction provides a practical advance due to the potential long waiting time nature of subdiffusive processes. The subordinated local time is used to formulate a probabilistic understanding of subdiffusion with reactions, leading to the well known radiation boundary condition. We demonstrate the equivalence between the generalized Feynman-Kac equation with a reflecting boundary and the fractional diffusion equation with a radiation boundary. We solve the former and find the first-reaction probability density in analytic form in the time domain, in terms of the Wright function. We are also able to find the survival probability and subordinated local time density analytically. These results are validated by stochastic simulations that use the subordinated local time description of subdiffusion in the presence of reactions.

GENERALIZED SELF INTERSECTION LOCAL TIME FOR A SUPERPROCESS OVER A STOCHASTIC FLOW.

This paper examines the existence of the self-intersection local time for a superprocess over a stochastic flow in dimensions d ≤ 3, which through constructive methods, results in a Tanaka like representation. The superprocess over a stochastic flow is a superprocess with dependent spatial motion, and thus Dynkin's proof of existence, which requires multiplicity of the log-Laplace functional, no longer applies. Skoulakis and Adler's method of calculating moments is extended to higher moments, from which existence follows.

A financial market with singular drift and no arbitrage.

We study a financial market where the risky asset is modelled by a geometric Itô-Lévy process, with a singular drift term. This can for example model a situation where the asset price is partially controlled by a company which intervenes when the price is reaching a certain lower barrier. See e.g. Jarrow and Protter (J Bank Finan 29:2803-2820, 2005) for an explanation and discussion of this model in the Brownian motion case. As already pointed out by Karatzas and Shreve (Methods of Mathematical Finance, Springer, Berlin, 1998) (in the continuous setting), this allows for arbitrages in the market. However, the situation in the case of jumps is not clear. Moreover, it is not clear what happens if there is a delay in the system. In this paper we consider a jump diffusion market model with a singular drift term modelled as the local time of a given process, and with a delay θ>0 in the information flow available for the trader. We allow the stock price dynamics to depend on both a continuous process (Brownian motion) and a jump process (Poisson random measure). We believe that jumps and delays are essential in order to get more realistic financial market models. Using white noise calculus we compute explicitly the optimal consumption rate and portfolio in this case and we show that the maximal value is finite as long as θ>0. This implies that there is no arbitrage in the market in that case. However, when θ goes to 0, the value goes to infinity. This is in agreement with the above result that is an arbitrage when there is no delay. Our model is also relevant for high frequency trading issues. This is because high frequency trading often leads to intensive trading taking place on close to infinitesimal lengths of time, which in the limit corresponds to trading on time sets of measure 0. This may in turn lead to a singular drift in the pricing dynamics. See e.g. Lachapelle et al. (Math Finan Econom 10(3):223-262, 2016) and the references therein.

Favorite Sites of a Persistent Random Walk.

We consider favorite (i.e., most visited) sites of a symmetric persistent random walk on ℤ , a discrete-time process typified by the correlation of its directional history. We show that the cardinality of the set of favorite sites is eventually at most three. This is a generalization of a result by Tóth for a simple random walk, used to partially prove a longstanding conjecture by Erdos and Róvósz. The original conjecture asserting that for the simple random walk on integers the cardinality of the set of favorite sites is eventually at most two was recently disproved by Ding and Shen.

Structure aware Runge-Kutta time stepping for spacetime tents.

We introduce a new class of Runge-Kutta type methods suitable for time stepping to propagate hyperbolic solutions within tent-shaped spacetime regions. Unlike standard Runge-Kutta methods, the new methods yield expected convergence properties when standard high order spatial (discontinuous Galerkin) discretizations are used. After presenting a derivation of nonstandard order conditions for these methods, we show numerical examples of nonlinear hyperbolic systems to demonstrate the optimal convergence rates. We also report on the discrete stability properties of these methods applied to linear hyperbolic equations.

Renormalized self-intersection local time of bifractional Brownian motion.

Let B H , K = { B H , K ( t ) , t ≥ 0 } be a d-dimensional bifractional Brownian motion with Hurst parameters H ∈ ( 0 , 1 ) and K ∈ ( 0 , 1 ] . Assuming d ≥ 2 , we prove that the renormalized self-intersection local time ∫ 0 T ∫ 0 t δ ( B H , K ( t ) - B H , K ( s ) ) d s d t - E ( ∫ 0 T ∫ 0 t δ ( B H , K ( t ) - B H , K ( s ) ) d s d t ) exists in L 2 if and only if H K d < 3 / 2 , where δ denotes the Dirac delta function. Our work generalizes the result of the renormalized self-intersection local time for fractional Brownian motion.

Dynamical emergence of Markovianity in local time scheme.

Recently we pointed out the so-called local time scheme as a novel approach to quantum foundations that solves the preferred pointer-basis problem. In this paper, we introduce and analyse in depth a rather non-standard dynamical map that is imposed by the scheme. On the one hand, the map does not allow for introducing a properly defined generator of the evolution nor does it represent a quantum channel. On the other hand, the map is linear, positive, trace preserving and unital as well as completely positive, but is not divisible and therefore non-Markovian. Nevertheless, we provide quantitative criteria for dynamical emergence of time-coarse-grained Markovianity, for exact dynamics of an open system, as well as for operationally defined approximation of a closed or open many-particle system. A closed system never reaches a steady state, whereas an open system may reach a unique steady state given by the Lüders-von Neumann formula; where the smaller the open system, the faster a steady state is attained. These generic findings extend the standard open quantum systems theory and substantially tackle certain cosmological issues.

A high-order local time stepping finite volume solver for one-dimensional blood flow simulations: application to the ADAN model.

In recent years, the complexity of vessel networks for one-dimensional blood flow models has significantly increased, because of enhanced anatomical detail or automatic peripheral vasculature generation, for example. This fact, along with the application of these models in uncertainty quantification and parameter estimation poses the need for extremely efficient numerical solvers. The aim of this work is to present a finite volume solver for one-dimensional blood flow simulations in networks of elastic and viscoelastic vessels, featuring high-order space-time accuracy and local time stepping (LTS). The solver is built on (i) a high-order finite volume type numerical scheme, (ii) a high-order treatment of the numerical solution at internal vertexes of the network, often called junctions, and (iii) an accurate LTS strategy. The accuracy of the proposed methodology is verified by empirical convergence tests. Then, the resulting LTS scheme is applied to arterial networks of increasing complexity and spatial scale heterogeneity, with a number of one-dimensional segments ranging from a few tens up to several thousands and vessel lengths ranging from less than a millimeter up to tens of centimeters, in order to evaluate its computational cost efficiency. The proposed methodology can be extended to any other hyperbolic system for which network applications are relevant. Copyright © 2016 John Wiley & Sons, Ltd.