arXiv subjects
Publications and source records attributed to Remi Leandre.
We give a review of our recent works related to the Malliavin Calculus of Bismut type for non-Markovian generators. Part IV is new and relates the Malliavin Calculus and the general theory of elliptic pseudo-differential operators.
We generalize for the Bilaplacian the Eells-Elworthy- Malliavin construction of the Brownian motion on a Riemannian manifold.
We give an Ito type formula for a semi group whose generarator is a positive pseudo-differential operator which does not in general satisfies maximumu principle.
We give a translation in semi-group theory of Wentzel-Freidlin estimates for Poisson process. We consider the case of the upper bound.
We translate in semigroup theory Bismut's way of the Malliavin calculus.
We define and present an example of a deformation quantization product on a Hida space of test functions endowed with a Wick product.
We define measures on central extension of current groups in any dimension by using infinite dimensional Brownian motion.
We give two stochastic diffeologies on the free loop space which allow us to define stochastic equivariant cohomology theories in the Chen-Souriau sense and to establish a link with cyclic cohomology. With the second one, we can establish a stochastic fixed point theorem.
We propose a stochastic extension of deformation quantization on a Hilbert space. The Moyal product is defined in this context on the space of functionals belonging to all of the Sobolev spaces of the Malliavin calculus.
A derivation operator and a divergence operator are defined on the algebra of bounded operators on the symmetric Fock space over the complexification of a real Hilbert space $\eufrak{h}$ and it is shown that they satisfy similar properties as the derivation and divergence operator on the Wiener space over $\eufrak{h}$. The derivation operator is then used to give sufficient conditions for the existence of smooth Wigner densities for pairs of operators satisfying the canonical commutation relations. For $\eufrak{h}=L^2(\mathbb{R}_+)$, the divergence operator is shown to coincide with the Hudson-Parthasarathy quantum stochastic integral for adapted integrable processes and with the non-causal quantum stochastic integrals defined by Lindsay and Belavkin for integrable processes.