Paul Raynaud de Fitte

Existence of Weak Solutions to Stochastic Evolution Inclusions

Abstract We prove the existence of a weak mild solution to the Cauchy problem for the semilinear stochastic differential inclusion in a Hilbert space where W is a Wiener process, A is a linear operator that generates a C 0-semigroup, F and G are multifunctions with convex compact values satisfying some growth condition, and with respect to the second variable, a condition weaker than the Lipschitz condition. The weak solution is constructed in the sense of Young measures.

Some variational convergence results with applications to evolution inclusions

We study variational convergence for integral functionals defined on L H ∞ ([0, 1];dt) × y([0,1]; \( \mathbb{Y} \) ) where ℍ is a separable Hilbert space, \( \mathbb{D} \) is a Polish space and y[0,1]; \( \mathbb{D} \) ) is the space of Young measures on [0,1] × \( \mathbb{D} \) , and we investigate its applications to evolution inclusions. We prove the dependence of solutions with respect to the control Young measures and apply it to the study of the value function associated with these control problems. In this framework we then prove that the value function is a viscosity subsolution of the associated HJB equation. Some limiting properties for nonconvex integral functionals in proximal analysis are also investigated.

Radonification of cylindrical semimartingales by a single Hilbert-Schmidt operator

It is proved that in Hilbert spaces a single Hilbert–Schmidt operator radonifies cylindrical semimartingales to strong semimartingales. This improves a result due to Badrikian and Ustunel (also L. Schwartz), who needed composition of three Hilbert–Schmidt operators.

Parametrized Kantorovich-Rubinštein theorem and application to the coupling of random variables

We prove a version for random measures of the celebrated Kantorovich-Rubinstein duality theorem and we give an application to the coupling of random variables which extends and unifies known results.

On the fiber product of Young measures with application to a control problem with measures

This paper studies, in the context of separable metric spaces, the stable convergence of the fiber product for Young measures with applications to a control problem governed by an ordinary differential equations where the controls are Young measures. Essentially we study some variational properties of the value functions and the existence of quasi-saddle points of these functions which occurs in this dynamic control problem, and also their link with the viscosity solution of the associated Hamilton-Jacobi-Bellman equation.

Bochner-almost periodicity for stochastic processes

We compare several notions of almost periodicity for continuous processes defined on the time interval I = ℝ or I = [0, + ∞) with values in a separable Banach space 𝔼 (or more generally a separable completely regular topological space): almost periodicity in distribution, in probability, in quadratic mean, almost sure almost periodicity, almost equi-almost periodicity. In the deterministic case, all these notions reduce to Bochner-almost periodicity, which is equivalent to Bohr-almost periodicity when I = ℝ, and to asymptotic Bohr-almost periodicity when I = [0, + ∞).

Some variational convergence results for a class of evolution inclusions of second order using Young measures

This paper has two main parts. In the first part, we discuss the existence and uniqueness of the W E 2,1 -solution u μ,ν of a second order differential equation with two boundary points conditions in a finite dimensional space, governed by controls μ, ν which are measures on a compact metric space. We also discuss the dependence on the controls and the variational properties of the value function V h(t, μ) := supν∈ℜ h(u μ, ν(t)), associated with a bounded lower semicontinuous function h. In the second main part, we discuss the limiting behaviour of a sequence of dynamics governed by second order evolution inclusions with two boundary points conditions. We prove that (up to extracted sequences) the solutions stably converge to a Young measure ν and we show that the limit measure ν satisfies a Fatou-type lemma in Mathematical Economics with variational-type inclusion property.

Some Problems in Second Order Evolution Inclusions with Boundary Condition: A Variational Approach

We prove, under appropriate assumptions, the existence of solutions for a second order evolution inclusion with boundary conditions via a variational approach.

\(\mathfrak{S}\)-Uniform Scalar Integrability and Strong Laws of Large Numbers for Pettis Integrable Functions with Values in a Separable Locally Convex Space

Generalizing techniques developed by Cuesta and Matrán for Bochner integrable random vectors of a separable Banach space, we prove a strong law of large numbers for Pettis integrable random elements of a separable locally convex space E. This result may be seen as a compactness result in a suitable topology on the set of Pettis integrable probabilities on E.