442 research outputs found
Fokker--Planck and Kolmogorov Backward Equations for Continuous Time Random Walk scaling limits
It is proved that the distributions of scaling limits of Continuous Time
Random Walks (CTRWs) solve integro-differential equations akin to Fokker-Planck
Equations for diffusion processes. In contrast to previous such results, it is
not assumed that the underlying process has absolutely continuous laws.
Moreover, governing equations in the backward variables are derived. Three
examples of anomalous diffusion processes illustrate the theory.Comment: in Proceedings of the American Mathematical Society, Published
electronically July 12, 201
Competitive interference of plant species in monocultures and mixed stands
A dynamic model is presented for calculating the dry matter yields of individual plant species in a mixed stand by means of parameters derived from a spacing experiment with species grown at 2 densities and harvested at regular intervals. Results are given of trials with different crops, including tall and dwarf peas. The model is intended for use under conditions of near-optimum supply of water and nutrients, where the principal competition is for light. A simple method for measuring relative light interception by species in mixed stands is also described
Brownian subordinators and fractional Cauchy problems
A Brownian time process is a Markov process subordinated to the absolute
value of an independent one-dimensional Brownian motion. Its transition
densities solve an initial value problem involving the square of the generator
of the original Markov process. An apparently unrelated class of processes,
emerging as the scaling limits of continuous time random walks, involve
subordination to the inverse or hitting time process of a classical stable
subordinator. The resulting densities solve fractional Cauchy problems, an
extension that involves fractional derivatives in time. In this paper, we will
show a close and unexpected connection between these two classes of processes,
and consequently, an equivalence between these two families of partial
differential equations.Comment: 18 pages, minor spelling correction
Space-time duality for fractional diffusion
Zolotarev proved a duality result that relates stable densities with
different indices. In this paper, we show how Zolotarev duality leads to some
interesting results on fractional diffusion. Fractional diffusion equations
employ fractional derivatives in place of the usual integer order derivatives.
They govern scaling limits of random walk models, with power law jumps leading
to fractional derivatives in space, and power law waiting times between the
jumps leading to fractional derivatives in time. The limit process is a stable
L\'evy motion that models the jumps, subordinated to an inverse stable process
that models the waiting times. Using duality, we relate the density of a
spectrally negative stable process with index to the density of
the hitting time of a stable subordinator with index , and thereby
unify some recent results in the literature. These results also provide a
concrete interpretation of Zolotarev duality in terms of the fractional
diffusion model.Comment: 16 page
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