3,647 research outputs found
Stochastic processes that generate polygonal and related random fields
Caption title.Includes bibliographical references (p. 17).Supported by the US Army Research Office. ARO DAAL03-92-G-0115V. Borkar and S.K. Mitter
An Unstructured Mesh Convergent Reaction-Diffusion Master Equation for Reversible Reactions
The convergent reaction-diffusion master equation (CRDME) was recently
developed to provide a lattice particle-based stochastic reaction-diffusion
model that is a convergent approximation in the lattice spacing to an
underlying spatially-continuous particle dynamics model. The CRDME was designed
to be identical to the popular lattice reaction-diffusion master equation
(RDME) model for systems with only linear reactions, while overcoming the
RDME's loss of bimolecular reaction effects as the lattice spacing is taken to
zero. In our original work we developed the CRDME to handle bimolecular
association reactions on Cartesian grids. In this work we develop several
extensions to the CRDME to facilitate the modeling of cellular processes within
realistic biological domains. Foremost, we extend the CRDME to handle
reversible bimolecular reactions on unstructured grids. Here we develop a
generalized CRDME through discretization of the spatially continuous volume
reactivity model, extending the CRDME to encompass a larger variety of
particle-particle interactions. Finally, we conclude by examining several
numerical examples to demonstrate the convergence and accuracy of the CRDME in
approximating the volume reactivity model.Comment: 35 pages, 9 figures. Accepted, J. Comp. Phys. (2018
A Multiscale Guide to Brownian Motion
We revise the Levy's construction of Brownian motion as a simple though still
rigorous approach to operate with various Gaussian processes. A Brownian path
is explicitly constructed as a linear combination of wavelet-based "geometrical
features" at multiple length scales with random weights. Such a wavelet
representation gives a closed formula mapping of the unit interval onto the
functional space of Brownian paths. This formula elucidates many classical
results about Brownian motion (e.g., non-differentiability of its path),
providing intuitive feeling for non-mathematicians. The illustrative character
of the wavelet representation, along with the simple structure of the
underlying probability space, is different from the usual presentation of most
classical textbooks. Similar concepts are discussed for fractional Brownian
motion, Ornstein-Uhlenbeck process, Gaussian free field, and fractional
Gaussian fields. Wavelet representations and dyadic decompositions form the
basis of many highly efficient numerical methods to simulate Gaussian processes
and fields, including Brownian motion and other diffusive processes in
confining domains
Spontaneous magnetisation in the plane
The Arak process is a solvable stochastic process which generates coloured
patterns in the plane. Patterns are made up of a variable number of random
non-intersecting polygons. We show that the distribution of Arak process states
is the Gibbs distribution of its states in thermodynamic equilibrium in the
grand canonical ensemble. The sequence of Gibbs distributions form a new model
parameterised by temperature. We prove that there is a phase transition in this
model, for some non-zero temperature. We illustrate this conclusion with
simulation results. We measure the critical exponents of this off-lattice model
and find they are consistent with those of the Ising model in two dimensions.Comment: 23 pages numbered -1,0...21, 8 figure
The convex minorant of a L\'{e}vy process
We offer a unified approach to the theory of convex minorants of L\'{e}vy
processes with continuous distributions. New results include simple explicit
constructions of the convex minorant of a L\'{e}vy process on both finite and
infinite time intervals, and of a Poisson point process of excursions above the
convex minorant up to an independent exponential time. The Poisson-Dirichlet
distribution of parameter 1 is shown to be the universal law of ranked lengths
of excursions of a L\'{e}vy process with continuous distributions above its
convex minorant on the interval .Comment: Published in at http://dx.doi.org/10.1214/11-AOP658 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
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