99 research outputs found
Hopf algebras and Markov chains: Two examples and a theory
The operation of squaring (coproduct followed by product) in a combinatorial
Hopf algebra is shown to induce a Markov chain in natural bases. Chains
constructed in this way include widely studied methods of card shuffling, a
natural "rock-breaking" process, and Markov chains on simplicial complexes.
Many of these chains can be explictly diagonalized using the primitive elements
of the algebra and the combinatorics of the free Lie algebra. For card
shuffling, this gives an explicit description of the eigenvectors. For
rock-breaking, an explicit description of the quasi-stationary distribution and
sharp rates to absorption follow.Comment: 51 pages, 17 figures. (Typographical errors corrected. Further fixes
will only appear on the version on Amy Pang's website, the arXiv version will
not be updated.
Moments, cumulants and diagram formulae for non-linear functionals of random measures
This survey provides a unified discussion of multiple integrals, moments,
cumulants and diagram formulae associated with functionals of completely random
measures. Our approach is combinatorial, as it is based on the algebraic
formalism of partition lattices and M\"obius functions. Gaussian and Poisson
measures are treated in great detail. We also present several combinatorial
interpretations of some recent CLTs involving sequences of random variables
belonging to a fixed Wiener chaos.Comment: Survey, preliminary draft. 104 pages. 30 Figure
Mixed Eulerian numbers and Peterson Schubert calculus
In this paper we derive a combinatorial formula for mixed Eulerian numbers in
type from Peterson Schubert calculus. We also provide a simple computation
for mixed -Eulerian numbers in arbitrary Lie types.Comment: 37 page
Coalescent tree based functional representations for some Feynman-Kac particle models
We design a theoretic tree-based functional representation of a class of
Feynman-Kac particle distributions, including an extension of the Wick product
formula to interacting particle systems. These weak expansions rely on an
original combinatorial, and permutation group analysis of a special class of
forests. They provide refined non asymptotic propagation of chaos type
properties, as well as sharp \LL\_p-mean error bounds, and laws of large
numbers for -statistics. Applications to particle interpretations of the top
eigenvalues, and the ground states of Schr\"{o}dinger semigroups are also
discussed
Invariance principles for homogeneous sums: Universality of Gaussian Wiener chaos
We compute explicit bounds in the normal and chi-square approximations of
multilinear homogenous sums (of arbitrary order) of general centered
independent random variables with unit variance. In particular, we show that
chaotic random variables enjoy the following form of universality: (a) the
normal and chi-square approximations of any homogenous sum can be completely
characterized and assessed by first switching to its Wiener chaos counterpart,
and (b) the simple upper bounds and convergence criteria available on the
Wiener chaos extend almost verbatim to the class of homogeneous sums.Comment: Published in at http://dx.doi.org/10.1214/10-AOP531 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Formules d'ItÎ pour l'équation de la chaleur stochastique à travers les théories des chemins rugueux et des structures de régularité
In this thesis, we develop a general theory to prove the existence of several ItÎ formulae on the one-dimensional stochastic heat equation driven by additive space-time white noise. That is denoting by u the solution of this SPDE for any smooth enough function f we define some new notions of stochastic integrals defined upon u, which cannot be defined classically, to deduce new identities involving f(u) and some non-trivial corrections. These new relations are obtained by using the theory of regularity structures and the theory of rough paths. In the first chapter, we obtain a differential and an integral identity involving the reconstruction of some modelled distributions. Then we discuss a general change of variable formula over any Hölder continuous path in the context of rough paths, relating it to the notion of quasi-shuffle algebras and the family of so-called quasi-geometric rough paths. Finally, we apply the general results on quasi-geometric rough paths to the time evolution of u. Using the Gaussian behaviour of the process , most of the terms involved in these equations are also identified with some classical constructions of stochastic calculus.Dans cette thÚse nous développons une théorie générale pour prouver l'existence de plusieurs formules de ItÎ sur l'équation de chaleur stochastique unidimensionnelle dirigée par un bruit blanc en espace-temps. Cela revient a définir de nouvelles notions d'intégrales stochastique sur u, la solution de cette EDPS et à obtenir pour toute fonction assez lisse f des nouvelles identités impliquant f(u) et des termes de correction non triviaux. Ces nouvelles relations sont obtenues en utilisant la théorie des structures de régularité et la théorie des chemins rugueux. Dans le premier chapitre nous obtenons une identité intégrale et une différentielle impliquant la reconstruction de certaines distributions modélisées. Ensuite, nous discutons d'une formule générale de changement de variable pour tout chemins Hölderiens dans le contexte des chemins rugueux en le rapportant à la notion d'algÚbres quasi-shuffle et à la famille des chemins rugueux dits quasi-géométriques. Enfin nous appliquons les résultats généraux sur les chemins rugueux quasi-géométriques à l'évolution temporelle du processus u. En utilisant le comportement gaussien de u, nous identifions la plupart des termes impliqués dans ces équations avec certaines constructions du calcul stochastique
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