7,206 research outputs found
Stochastic Domination in Space-Time for the Contact Process
Liggett and Steif (2006) proved that, for the supercritical contact process
\non certain graphs, the upper invariant measure stochastically dominates an
\ni.i.d.\\ Bernoulli product measure. In particular, they proved this for
\n and (for infection rate sufficiently large) -ary homogeneous
\ntrees .
\n In this paper we prove some space-time versions of their results. We do this
\nby combining their methods with specific properties of the contact process and
\ngeneral correlation inequalities.
\n One of our main results concerns the contact process on with .
\nWe show that, for large infection rate, there exists a subset of the
\nvertices of , containing a "positive fraction" of all the vertices of
\n, such that the following holds: The contact process on observed on
\n stochastically dominates an independent spin-flip process. (This is
\nknown to be false for the contact process on graphs having subexponential
\ngrowth.)
\n We further prove that the supercritical contact process on
\nobserved on certain -dimensional space-time slabs stochastically dominates
\nan i.i.d.\\ Bernoulli product measure, from which we conclude strong mixing
\nproperties important in the study of certain random walks in random
\nenvironment
Stochastic domination: the contact process, Ising models and FKG measures
We prove for the contact process on , and many other graphs, that the
upper invariant measure dominates a homogeneous product measure with large
density if the infection rate is sufficiently large. As a
consequence, this measure percolates if the corresponding product measure
percolates. We raise the question of whether domination holds in the symmetric
case for all infinite graphs of bounded degree. We study some asymmetric
examples which we feel shed some light on this question. We next obtain
necessary and sufficient conditions for domination of a product measure for
``downward'' FKG measures. As a consequence of this general result, we show
that the plus and minus states for the Ising model on dominate the same
set of product measures. We show that this latter fact fails completely on the
homogenous 3-ary tree. We also provide a different distinction between
and the homogenous 3-ary tree concerning stochastic domination and Ising
models; while it is known that the plus states for different temperatures on
are never stochastically ordered, on the homogenous 3-ary tree, almost
the complete opposite is the case. Next, we show that on , the set of
product measures which the plus state for the Ising model dominates is strictly
increasing in the temperature. Finally, we obtain a necessary and sufficient
condition for a finite number of variables, which are both FKG and
exchangeable, to dominate a given product measure.Comment: 27 page
A theoretical investigation of the effect of proliferation & adhesion on monoclonal conversion in the colonic crypt
The surface epithelium lining the intestinal tract renews itself rapidly by a coordinated programme of cell proliferation, migration and differentiation events that is initiated in the crypts of Lieberkühn. It is generally believed that colorectal cancer arises due to mutations that disrupt the normal cellular dynamics of the crypts. Using a spatially structured cell-based model of a colonic crypt, we investigate the likelihood that the progeny of a mutated cell will dominate, or be sloughed out of, a crypt. Our approach is to perform multiple simulations, varying the spatial location of the initial mutation, and the proliferative and adhesive properties of the mutant cells, to obtain statistical distributions for the probability of their domination. Our simulations lead us to make a number of predictions. The process of monoclonal conversion always occurs, and does not require that the cell which initially gave rise to the population remains in the crypt. Mutations occurring more than one to two cells from the base of the crypt are unlikely to become the dominant clone. The probability of a mutant clone persisting in the crypt is sensitive to dysregulation of adhesion. By comparing simulation results with those from a simple one-dimensional stochastic model of population dynamics at the base of the crypt, we infer that this sensitivity is due to direct competition between wild-type and mutant cells at the base of the crypt. We also predict that increases in the extent of the spatial domain in which the mutant cells proliferate can give rise to counter-intuitive, non-linear changes to the probability of their fixation, due to effects that cannot be captured in simpler models
Travelling wave solutions to the KPP equation with branching noise arising from initial conditions with compact support
We consider the one-dimensional KPP-equation driven by space-time white noise
and extend the construction of travelling wave solutions arising from Heavyside
initial data from [Tribe, 1996, MR1396765] to non-negative continuous functions
with compact support. As an application the existence of travelling wave
solutions is used to prove that the support of any solution is recurrent. As a
by-product, several upper measures are introduced that allow for a stochastic
domination of any solution to the SPDE at a fixed point in time.Comment: 28 page
A theoretical investigation of the effect of proliferation and\ud adhesion on monoclonal conversion in the colonic crypt
Colorectal cancers are initiated by the accumulation of mutations in the colonic epithelium. Using a spatially structured cell-based model of a colonic crypt, we investigate the likelihood that the progeny of a mutated cell will dominate, or be sloughed out of, a crypt. Our approach is to perform multiple simulations, varying the spatial location of the initial mutation, and its proliferative and adhesive properties, to obtain statistical distributions for the probability of domination. Our simulations lead us to make a number of predictions. The process of monoclonal conversion always occurs, and does not require that the cell which initially gave rise to the population remains in the crypt. Mutations occurring more than one to two cells from the base of the crypt are unlikely to become the dominant clone. The probability of a mutant clone persisting in the crypt is sensitive to dysregulation of adhesion, and comparison with a one-dimensional model suggests that this is caused by competition directly at the base of the crypt.\ud
We also predict that increases in the extent of the spatial domain in which the mutant cells proliferate cause counter-intuitive non-linear changes to the probability of its fixation, due to effects that cannot be captured in simpler models
- …