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SS10 - Analyse stochastique / SS10 - Stochastic Analysis
Org: M. Barlow (UBC) et/and D. Bakry (Toulouse)
- MICHEL BENAIM, Neuchatel
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- NINA GANTERT, Inst. Math. Stochastik, Universitaet Karlsruhe
Deviations of random walk in random scenery
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Let (Zn)n Î N0 be a d-dimensional random walk in random
scenery, i.e., Zn = åk=0n-1 Y(Sk) with (Sk)k Î N0 a random walk in Zd and ( Y(z) )z Î Zd an
i.i.d. scenery, independent of the walk. The walker's steps have mean
zero and finite variance.
We identify the speed and the rate of the logarithmic decay of
P([ 1/(n)] Zn > bn) for various choices of sequences (bn)n in
[1,¥). Depending on (bn)n and the upper tails of the
scenery, we identify different regimes for the speed of decay and
different variational formulas for the rate functions. In contrast to
recent work by A. Asselah and F. Castell, we consider sceneries
unbounded to infinity. It turns out that there are interesting
connections to large deviation properties of self-intersections of the
walk, which have been studied recently by X. Chen.
- RICHARD KENYON, CNRS / Paris Sud
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- VLADA LIMIC, University of British Columbia
Attracting edge in some strongly reinforced walks
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Reinforcement is observed frequently in nature and society, where
beneficial interactions tend to be repeated. Edge reinforced random
walker on a graph remembers the number of times each edge was
traversed in the past, and decides to make the next random step with
probabilities favouring places visited before. Using martingale
techniques and comparison with the generalized Urn scheme, it is shown
in [1] that the edge reinforced random walker on a graph of bounded
degree, with the reinforcement weight function W(k) = kr, r > 1,
traverses a random attracting edge at all large times, with
probability 1. A remarkably short argument of Sellke [2] shows that
attracting edge exists if and only if
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å
k
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1
W(k)
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< ¥, \leqno(1) |
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whenever the underlying graph has no odd cycle. The conjecture that
condition (1) implies existence of attracting edge when the underlying
graph is a triangle is still open.
Progress has been made recently [3] towards better understanding of
attracting edge property for convex and increasing weights W with
property (1).
References
- [1]
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V. Limic,
Attracting edge property for a class of reinforced random walks.
Ann. Probab. 31(2003), 1615-1654.
- [2]
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T. Sellke,
Reinforced random walks on the d-dimensional integer lattice.
Preprint, 1994.
- [3]
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V. Limic and P. Tarrès, in progress.
- PIERRE MATHIEU, CMI Université de Provence
Centered Markov chains
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A reversible measure turns a Markov operator into a symmetric operator
and spectral theory can then be used to study the transition kernel.
We introduce the more general notion of `centered measure' and prove
an upper estimate on the decay of the transition probabilities of
Carne-Varopoulos type. The connection with the rate of escape in the
case of random walks on discrete groups will be discussed.
Ref: http://www.cmi.univ-mrs.fr/~pmathieu/Papiers/CarneVaro.ps
- PIERRE TARRES, Université Paul Sabatier, Toulouse, and Université de
Neuchâtel
Self-Interacting Random Walks
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A self-interacting random walk is a random process evolving in an
environment depending on its past behavior.
The notion of Edge-Reinforced Random Walk (ERRW) was introduced in
1986 by Coppersmith and Diaconis [2] on a discrete graph, with the
probability of a move along an edge being proportional to the number
of visits to this edge. In the same spirit, Pemantle introduced in
1988 [5] the Vertex-Reinforced Random Walk (VRRW), the probability of
a move to an adjacent vertex being then proportional to the number of
visits to this vertex (and not to the edge leading to the vertex). The
Self-Interacting Diffusion (SID) is a continuous counterpart to these
notions.
Although introduced by similar definitions, these processes show some
significantly different behaviors, leading in their understanding to
various methods. While the study of ERRW essentially requires some
probabilistic tools, corresponding to some local properties, the
comprehension of VRRW and SID needs a joint understanding of on one
hand a dynamical system governing the general evolution, and on the
other hand some probabilistic phenomena, acting as a perturbation,
and sometimes changing the nature of this dynamical system.
The purpose of our talk is to present our recent results on the
subject [1], [3], [4], [6].
References
- [1]
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M. Benaïm and P. Tarrès,
Dynamics of vertex-reinforced random walks.
In progress, 2004.
- [2]
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D. Coppersmith and P. Diaconis,
Random walks with reinforcement.
Unpublished manuscript, 1986.
- [3]
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V. Limic and P. Tarrès.
In progress, 2004.
- [4]
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T. Mountford and P. Tarrès,
An asymptotic result for brownian polymers.
Preprint, 2004.
- [5]
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R. Pemantle,
Random processes with reinforcement.
MIT Doctoral Dissertation, 1988.
- [6]
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P. Tarrès,
VRRW on Z eventually gets stuck on five points.
Annals of Probability, 2004, to appear.
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