32 research outputs found

    Lenses in Skew Brownian Flow

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    We consider a stochastic flow in which individual particles follow skew Brownian motions, with each one of these processes driven by the same Brownian motion. One does not have uniqueness for the solutions of the corresponding stochastic differential equation simultaneously for all real initial conditions. Due to this lack of the simultaneous strong uniqueness for the whole system of stochastic differential equations, the flow contains lenses, that is, pairs of skew Brownian motions which start at the same point, bifurcate, and then coalesce in a finite time. The paper contains qualitative and quantitative (distributional) results on the geometry of the flow and lenses.Comment: Published at http://dx.doi.org/10.1214/009117904000000711 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    On the continuity of local times of Borel right Markov processes

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    The problem of finding a necessary and sufficient condition for the continuity of the local times for a general Markov process is still open. Barlow and Hawkes have completely treated the case of the L\'{e}vy processes, and Marcus and Rosen have solved the case of the strongly symmetric Markov processes. We treat here the continuity of the local times of Borel right processes. Our approach unifies that of Barlow and Hawkes and of Marcus and Rosen, by using an associated Gaussian process, that appears as a limit in a CLT involving the local time process.Comment: Published at http://dx.doi.org/10.1214/009117906000000980 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    A characterization of the infinitely divisible squared Gaussian processes

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    We show that, up to multiplication by constants, a Gaussian process has an infinitely divisible square if and only if its covariance is the Green function of a transient Markov process.Comment: Published at http://dx.doi.org/10.1214/009117905000000684 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Escape from the boundary in Markov population processes

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    Density dependent Markov population processes in large populations of size NN were shown by Kurtz (1970, 1971) to be well approximated over finite time intervals by the solution of the differential equations that describe their average drift, and to exhibit stochastic fluctuations about this deterministic solution on the scale N\sqrt N that can be approximated by a diffusion process. Here, motivated by an example from evolutionary biology, we are concerned with describing how such a process leaves an absorbing boundary. Initially, one or more of the populations is of size much smaller than NN, and the length of time taken until all populations have sizes comparable to NN then becomes infinite as N→∞N \to \infty. Under suitable assumptions, we show that in the early stages of development, up to the time when all populations have sizes at least N1−αN^{1-\alpha}, for 1/3<α<11/3 < \alpha < 1, the process can be accurately approximated in total variation by a Markov branching process. Thereafter, the process is well approximated by the deterministic solution starting from the original initial point, but with a random time delay. Analogous behaviour is also established for a Markov process approaching an equilibrium on a boundary, where one or more of the populations become extinct.Comment: 50 page

    A Skorokhod map on measure valued paths with applications to priority queues

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    The Skorokhod map on the half-line has proved to be a useful tool for studying processes with non-negativity constraints. In this work we introduce a measure-valued analog of this map that transforms each element ? of a certain class of c`adl`ag paths that take values in the space of signed measures on [0, ?) to a c\'ad\'ag path that takes values in the space of non-negative measures on [0,∞)[0, \infty) in such a way that for each x>0x>0, the path t→ζt[0,x]t\to \zeta_t [0, x] is transformed via a Skorokhod map on the half-line, and the regulating functions for different x>0x>0 are coupled. We establish regularity properties of this map and show that the map provides a convenient tool for studying queueing systems in which tasks are prioritized according to a continuous parameter. One such priority assignment rule is the well known earliest-deadline-first priority rule. We study it both for the single and the many server queueing systems. We show how the map provides a framework within which to form fluid model equations, prove uniqueness of solutions to these equations and establish convergence of scaled state processes to the fluid model. In particular, for these models, we obtain new convergence results in time-inhomogeneous settings, which appear to fall outside the purview of existing approaches and is essential when studying the EDF policy for many servers queues. Based on Joint work with Rami Atar, Anup Biswas and Kavita Ramanan.Non UBCUnreviewedAuthor affiliation: TechnionFacult