136 research outputs found
Populations with interaction and environmental dependence: from few, (almost) independent, members into deterministic evolution of high densities
Many populations, e.g. of cells, bacteria, viruses, or replicating DNA
molecules, start small, from a few individuals, and grow large into a
noticeable fraction of the environmental carrying capacity . Typically, the
elements of the initiating, sparse set will not be hampering each other and
their number will grow from in a branching process or Malthusian
like, roughly exponential fashion, , where is the size at
discrete time , is the offspring mean per individual (at the
low starting density of elements, and large ), and a sum of i.i.d.
random variables. It will, thus, become detectable (i.e. of the same order as
) only after around generations, when its density will
tend to be strictly positive. Typically, this entity will be random, even if
the very beginning was not at all stochastic, as indicated by lower case ,
due to variations during the early development. However, from that time
onwards, law of large numbers effects will render the process deterministic,
though initiated by the random density at time log , expressed through the
variable . Thus, acts both as a random veil concealing the start and a
stochastic initial value for later, deterministic population density
development. We make such arguments precise, studying general density and also
system-size dependent, processes, as . As an intrinsic size
parameter, may also be chosen to be the time unit. The fundamental ideas
are to couple the initial system to a branching process and to show that late
densities develop very much like iterates of a conditional expectation
operator.Comment: presented at IV Workshop on Branching Processes and their
Applications at Badajoz, Spain, 10-13 April, 201
A new stochastic differential equation approach for waves in a random medium
We present a mathematical approach that simplifies the theoretical treatment
of electromagnetic localization in random media and leads to closed form
analytical solutions. Starting with the assumption that the dielectric
permittivity of the medium has delta-correlated spatial fluctuations, and using
the Ito lemma, we derive a linear stochastic differential equation for a one
dimensional random medium. The equation leads to localized wave solutions. The
localized wave solutions have a localization length that scales inversely with
the square of the frequency of the wave in the low frequency regime, whereas in
the high frequency regime, this length varies inversely with the frequency to
the power of two thirds
Large Deviations for processes on half-line
We consider a sequence of processes defined on half-line for all non negative
t. We give sufficient conditions for Large Deviation Principle (LDP) to hold in
the space of continuous functions with a new metric that is more sensitive to
behaviour at infinity than the uniform metric. LDP is established for Random
Walks, Diffusions, and CEV model of ruin, all defined on the half-line. LDP in
this space is "more precise" than that with the usual metric of uniform
convergence on compacts.Comment: 23 page
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