76 research outputs found
On the quasi-regularity of non-sectorial Dirichlet forms by processes having the same polar sets
We obtain a criterion for the quasi-regularity of generalized (non-sectorial)
Dirichlet forms, which extends the result of P.J. Fitzsimmons on the
quasi-regularity of (sectorial) semi-Dirichlet forms. Given the right (Markov)
process associated to a semi-Dirichlet form, we present sufficient conditions
for a second right process to be a standard one, having the same state space.
The above mentioned quasi-regularity criterion is then an application. The
conditions are expressed in terms of the associated capacities, nests of
compacts, polar sets, and quasi-continuity. A second application is on the
quasi-regularity of the generalized Dirichlet forms obtained by perturbing a
semi-Dirichlet form with kernels .Comment: Correction of typos and other minor change
A topological characterization of complete distributive lattices
AbstractAn ordered compact space is a compact topological space X, endowed with a partially ordered relation, whose graph is a closed set of X × X (cf. [4]). An important subclass of these spaces is that of Priestley spaces, characterized by the following property: for every x, y ϵ X with x ≰ y there is an increasing clopen set A (i.e. A is closed and open and such that a ϵ A, a ⩽ z implies that z ϵ A) which separates x from y, i.e., x ϵ A and y ≱ A. It is known (cf. [5, 6]) that there is a dual equivalence between the category Ld01 of distributive lattices with least and greatest element and the category P of Priestley spaces.In this paper we shall prove that a lattice L ϵ Ld01 is complete if and only if the associated Priestley space X verifies the condition: (E0) D ⊆ X, D is increasing and open implies D∗ is increasing clopen (where A∗ denotes the least increasing set which includes A).This result generalizes a well-known characterization of complete Boolean algebras in terms of associated Stone spaces (see [2, Ch. III, Section 4, Lemma 1], for instance).We shall also prove that an ordered compact space that fulfils (E0) is necessarily a Priestley space
Stochastic equation of fragmentation and branching processes related to avalanches
We give a stochastic model for the fragmentation phase of a snow avalanche.
We construct a fragmentation-branching process related to the avalanches, on
the set of all fragmentation sizes introduced by J. Bertoin. A fractal property
of this process is emphasized. We also establish a specific stochastic equation
of fragmentation. It turns out that specific branching Markov processes on
finite configurations of particles with sizes bigger than a strictly positive
threshold are convenient for describing the continuous time evolution of the
number of the resulting fragments. The results are obtained by combining
analytic and probabilistic potential theoretical tools.Comment: 17 page
Scaling property for fragmentation processes related to avalanches
International audienceWe emphasize a scaling property for the continuous time fragmentation processes related to a stochastic model for the fragmentation phase of an avalanche. We present numerical results that confirm the validity of the scaling property for our model, based on the appropriate stochastic differential equation of fragmentation and on a fractal property of the solution
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