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

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

Potential theory of infinite dimensional Lévy processes

AbstractWe study the potential theory of a large class of infinite dimensional Lévy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e., excessive functions with compact level sets. Then many techniques from classical potential theory carry over to this infinite dimensional setting. Thus a number of potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as, e.g., formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron–Martin space is polar, in the Lévy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data

Branching processes for the fragmentation equation

International audienceWe investigate branching properties of the solution of a stochastic differential equation of fragmentation (SDEF) and we properly associate a continuous time càdlàg Markov process on the space S# of all fragmentation sizes, introduced by J. Bertoin. A binary fragmentation kernel induces a specific class of integral type branching kernels and taking as base process the solution of the initial (SDEF), we construct a branching process corresponding to a rate of loss of mass greater than a given strictly positive size d. It turns out that this branching process takes values in the set of all finite configurations of sizes greater than d. The process on S# is then obtained by letting d tend to zero. A key argument for the convergence of the branching processes is given by the Bochner-Kolmogorov theorem. The construction and the proof of the path regularity of the Markov processes are based on several newly developed potential theoretical tools, in terms of excessive functions and measures, compact Lyapunov functions, and some appropriate absorbing sets