124 research outputs found
Random environment on coloured trees
In this paper, we study a regular rooted coloured tree with random labels
assigned to its edges, where the distribution of the label assigned to an edge
depends on the colours of its endpoints. We obtain some new results relevant to
this model and also show how our model generalizes many other probabilistic
models, including random walk in random environment on trees, recursive
distributional equations and multi-type branching random walk on .Comment: Published in at http://dx.doi.org/10.3150/07-BEJ101 the Bernoulli
(http://isi.cbs.nl/bernoulli/) by the International Statistical
Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm
Bindweeds or random walks in random environments on multiplexed trees and their asympotics
We report on the asymptotic behaviour of a new model of random walk, we term
the bindweed model, evolving in a random environment on an infinite multiplexed
tree. The term \textit{multiplexed} means that the model can be viewed as a
nearest neighbours random walk on a tree whose vertices carry an internal
degree of freedom from the finite set , for some integer . The
consequence of the internal degree of freedom is an enhancement of the tree
graph structure induced by the replacement of ordinary edges by multi-edges,
indexed by the set . This indexing conveys the
information on the internal degree of freedom of the vertices contiguous to
each edge. The term \textit{random environment} means that the jumping rates
for the random walk are a family of edge-indexed random variables, independent
of the natural filtration generated by the random variables entering in the
definition of the random walk; their joint distribution depends on the index of
each component of the multi-edges. We study the large time asymptotic behaviour
of this random walk and classify it with respect to positive recurrence or
transience in terms of a specific parameter of the probability distribution of
the jump rates. This classifying parameter is shown to coincide with the
critical value of a matrix-valued multiplicative cascade on the ordinary tree
(\textit{i.e.} the one without internal degrees of freedom attached to the
vertices) having the same vertex set as the state space of the random walk.
Only results are presented here since the detailed proofs will appear
elsewhere
Dynamical systems with heavy-tailed random parameters
Motivated by the study of the time evolution of random dynamical systems
arising in a vast variety of domains --- ranging from physics to ecology ---,
we establish conditions for the occurrence of a non-trivial asymptotic
behaviour for these systems in the absence of an ellipticity condition. More
precisely, we classify these systems according to their type and --- in the
recurrent case --- provide with sharp conditions quantifying the nature of
recurrence by establishing which moments of passage times exist and which do
not exist. The problem is tackled by mapping the random dynamical systems into
Markov chains on with heavy-tailed innovation and then using
powerful methods stemming from Lyapunov functions to map the resulting Markov
chains into positive semi-martingales.Comment: 24 page
Generalizations of forest fires with ignition at origin
We study generalizations of the Forest Fire model introduced in~\cite{BTRB}
and~\cite{Volk} by allowing the rates at which the tree grow to depend on their
location, introducing long-range burning, as well as continuous-space
generalization of the model. We establish that in all the models in
consideration the time required to reach site at distance from the origin
is of order at most for any
Angular asymptotics for multi-dimensional non-homogeneous random walks with asymptotically zero drift
We study the first exit time from an arbitrary cone with apex at the origin by a non-homogeneous random walk (Markov chain) on () with mean drift that is asymptotically zero. Specifically, if the mean drift at \bx \in \Z^d is of magnitude O(\| \bx\|^{-1}), we show that a.s. for any cone. On the other hand, for an appropriate drift field with mean drifts of magnitude \| \bx\|^{-\beta}, , we prove that our random walk has a limiting (random) direction and so eventually remains in an arbitrarily narrow cone. The conditions imposed on the random walk are minimal: we assume only a uniform bound on nd moments for the increments and a form of weak isotropy. We give several illustrative examples, including a random walk in random environment model
Superdiffusive planar random walks with polynomial space-time drifts
We quantify superdiffusive transience for a two-dimensional random walk in which the vertical coordinate is a martingale and the horizontal coordinate has a positive drift that is a polynomial function of the individual coordinates and of the present time. We describe how the model was motivated through an heuristic connection to a self-interacting, planar random walk which interacts with its own centre of mass via an excluded-volume mechanism, and is conjectured to be superdiffusive with a scale exponent 3 / 4 . The self-interacting process originated in discussions with Francis Comets
Strong transience for one-dimensional Markov chains with asymptotically zero drifts
For near-critical, transient Markov chains on the non-negative integers in
the Lamperti regime, where the mean drift at decays as as , we quantify degree of transience via existence of moments for
conditional return times and for last exit times, assuming increments are
uniformly bounded. Our proof uses a Doob -transform, for the transient
process conditioned to return, and we show that the conditioned process is also
of Lamperti type with appropriately transformed parameters. To do so, we obtain
an asymptotic expansion for the ratio of two return probabilities, evaluated at
two nearby starting points; a consequence of this is that the return
probability for the transient Lamperti process is a regularly-varying function
of the starting point.Comment: 26 pages; v2: minor revisions, expanded discussio
Non-homogeneous random walks with non-integrable increments and heavy-tailed random walks on strips
We study asymptotic properties of spatially non-homogeneous random walks with non-integrable increments, including transience, almost-sure bounds, and existence and non existence of moments for first-passage and last-exit times. In our proofs we also make use of estimates for hitting probabilities and large deviations bounds. Our results are more general than existing results in the literature, which consider only the case of sums of independent (typically, identically distributed) random variables. We do not assume the Markov property. Existing results that we generalize include a circle of ideas related to the Marcinkiewicz-Zygmund strong law of large numbers, as well as more recent work of Kesten and Maller. Our proofs are robust and use martingale methods. We demonstrate the benefit of the generality of our results by applications to some non-classical models, including random walks with heavy-tailed increments on two-dimensional strips, which include, for instance, certain generalized risk processes
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