96 research outputs found
Penalizations of the Brownian motion by a functional of its local times
In this article, we study the family of probability measures (indexed by a
positive real number t), obtained by penalization of the Brownian motion by a
given functional of its local times at time t. We prove that this family tends
to a limit measure when t goes to infinity if the functional satisfies some
conditions of domination, and we check these conditions in several particular
cases
The distribution of eigenvalues of randomized permutation matrices
In this article we study in detail a family of random matrix ensembles which
are obtained from random permutations matrices (chosen at random according to
the Ewens measure of parameter ) by replacing the entries equal to
one by more general non-vanishing complex random variables. For these
ensembles, in contrast with more classical models as the Gaussian Unitary
Ensemble, or the Circular Unitary Ensemble, the eigenvalues can be very
explicitly computed by using the cycle structure of the permutations. Moreover,
by using the so-called virtual permutations, first introduced by Kerov,
Olshanski and Vershik, and studied with a probabilistic point of view by
Tsilevich, we are able to define, on the same probability space, a model for
each dimension greater than or equal to one, which gives a meaning to the
notion of almost sure convergence when the dimension tends to infinity. In the
present paper, depending on the precise model which is considered, we obtain a
number of different results of convergence for the point measure of the
eigenvalues, some of these results giving a strong convergence, which is not
common in random matrix theory
A new kind of augmentation of filtrations
Let be a
filtered probability space satisfying the usual assumptions: it is usually not
possible to extend to (the -algebra generated by
) a coherent family of probability measures
, each of them being defined on . It is known
that for instance, on the Wiener space, this extension problem has a positive
answer if one takes the filtration generated by the coordinate process, but can
have a negative answer if one takes its usual augmentation. On the other hand,
the usual assumptions are crucial in order to obtain the existence of regular
versions of paths for most stochastic processes of interest, such as the local
time of the standard Brownian motion, stochastic integrals, etc. In order to
fix this problem, we introduce a new property for filtrations, intermediate
between the right continuity and the usual conditions. We show that most of the
important results of the theory of stochastic processes which are generally
proved under the usual augmentation, such as the existence of regular version
of trajectories or the d\'ebut theorem, still hold under the N-augmentation;
moreover this new augmentation allows the extension of a coherent family of
probability measures whenever this is possible with the original filtration.Comment: A new reference to earlier work by K. Bichteler is adde
The bead process for beta ensembles
The bead process introduced by Boutillier is a countable interlacing of the
determinantal sine-kernel point processes. We construct the bead process for
general sine beta processes as an infinite dimensional Markov chain whose
transition mechanism is explicitly described. We show that this process is the
microscopic scaling limit in the bulk of the Hermite beta corner process
introduced by Gorin and Shkolnikov, generalizing the process of the minors of
the Gaussian unitary and orthogonal ensembles. In order to prove our results,
we use bounds on the variance of the point counting of the circular and the
Gaussian beta ensembles, proven in a companion paper
A remarkable -finite measure associated with last passage times and penalisation problems
In this paper, we give a global view of the results we have obtained in
relation with a remarkable class of submartingales, called , and its
links with a universal sigma-finite measure and penalization problems on the
space of continuous and cadlag paths.Comment: This is a contribution for the special volume for the 60th birthday
of Eckhard Plate
On sigma-finite measures related to the Martin boundary of recurrent Markov chains
In our monograph with B. Roynette and M. Yor, we construct a sigma-finite
measure related to penalisations of different stochastic processes, including
the Brownian motion in dimension 1 or 2, and a large class of linear
diffusions. In the last chapter of the monograph, we define similar measures
from recurrent Markov chains satisfying some technical conditions. In the
present paper, we give a classification of these measures, in function of the
minimal Martin boundary of the Markov chain considered at the beginning. We
apply this classification to the examples considered at the end of our
monograph
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