Pedro Catuogno

Lp-solutions of the stochastic transport equation

Abstract. We consider the stochastic transport linear equation and we prove existence and uniqueness of weak Lp-solutions. Moreover, we obtain a representation of the general solution and a Wong-Zakai principle for this equation. We make only minimal assumptions, similar to the deterministic problem. The proof is supported on the generalized Itô–Ventzel–Kunita formula and the theory of Lions–DiPerna on transport linear equation.

Hermite expansions and products of tempered distributions

The space of tempered distributions 𝒮′ can be realized as a sequence spaces by means of the Hermite representation theorem. In this article, we introduce and study two new products of tempered distributions based on this Hermite representation theorem. In particular, we obtain the products [H]δ=δ/2, [δ] vp(1/x)=−δ′ and [δ(r)]vp(1/x)=−(δ(r+1))/(r+1) for even r.

ON STOCHASTIC GENERALIZED FUNCTIONS

We introduced a new algebra of stochastic generalized functions which contains to the space of stochastic distributions G, [25]. As an application, we prove existence and uniqueness of the solution of a stochastic Cauchy problem involving singularities.

Strong solution of the stochastic Burgers equation

This work introduces a pathwise notion of solution for the stochastic Burgers equation, in particular, our approach encompasses the Cole–Hopf solution. The developments are based on regularization arguments from the theory of distributions.

Harmonic measures in embedded foliated manifolds

We study harmonic and totally invariant measures in a foliated compact Riemannian manifold. We construct explicitly a Stratonovich differential equation for the foliated Brownian motion. We present a characterization of totally invariant measures in terms of the flow of diffeomorphisms associated to this equation. We prove an ergodic formula for the sum of the Lyapunov exponents in terms of the geometry of the leaves.

DECOMPOSITION OF STOCHASTIC FLOWS IN MANIFOLDS WITH COMPLEMENTARY DISTRIBUTIONS

Let M be a differentiable manifold endowed locally with two complementary distributions, say horizontal and vertical. We consider the two subgroups of (local) diffeomorphisms of M generated by vector fields in each of of these distributions. Given a stochastic flow φt of diffeomorphisms of M, in a neighbourhood of an initial condition, up to a stopping time we decompose φt = ξt ◦ ψt where the first component is a diffusion in the group of horizontal diffeomorphisms and the second component is a process in the group of vertical diffeomorphisms. Further decomposition will include more than two components; it leads to a maximal cascade decomposition in local coordinates where each component acts only in the corresponding coordinate.

DECOMPOSITION OF STOCHASTIC FLOWS WITH AUTOMORPHISM OF SUBBUNDLES COMPONENT

We show that given a G-structure P on a differential manifold M, if the group G(M) of automorphisms of P is large enough, then there exists the quotient of a stochastic flows φt by G(M), in the sense that φt = ξt ◦ ρt where ξt ∈ G(M), the remainder ρt has derivative which is vertical, transversal to the fibres of P. This geometrical context generalises previous results where M is a Riemannian manifold and φt is decomposed with an isometric component, see [12] and [15], which in our context corresponds to the particular case of an SO(n)-structure on M.

Random Lie-point symmetries

We introduce the notion of a random symmetry. It consists of taking the action given by a deterministic flow that maintains the solutions of a given differential equation invariant and replacing it with a stochastic flow. This generates a random action, which we call a random symmetry.

Time-dependent tempered generalized functions and Itô’s formula

The paper introduces a novel Itô’s formula for time- dependent tempered generalized functions. As an application, we study the heat equation when initial conditions are allowed to be a generalized tempered function. A new proof of the Üstunel-Itô’s formula for tempered distributions is also provided.

Relative rotation number for stochastic systems: dynamical and topological applications

We introduce a concept of relative rotation number to unify many different approaches of rotation number in non-linear dynamical systems. We present an ergodic result of existence a.s. for stochastic systems. In higher dimension, we show that the natural idea of projecting into a plane does work well a.s. for any plane (different from deterministic systems where projections may be degenerate). A number of further properties (invariance by homotopy and by conjugacy) and applications are presented.