Christian Olivera

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.

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.

Well-posedness of the vector advection equations by stochastic perturbation

A linear stochastic vector advection equation is considered. The equation may model a passive magnetic field in a random fluid. The driving velocity field is a integrable to a certain power, and the noise is infinite dimensional. We prove that, thanks to the noise, the equation is well posed in a suitable sense, opposite to what may happen without noise.

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.

Regularization by Noise in One-Dimensional Continuity Equation

A linear stochastic continuity equation with non-regular coefficients is considered. We prove existence and uniqueness of strong solution, in the probabilistic sense, to the Cauchy problem when the vector field has low regularity, in which the classical DiPerna-Lions-Ambrosio theory of uniqueness of distributional solutions does not apply. We solve partially the open problem that is the case when the vector-field has random dependence. In addition, we prove a stability result for the solutions.

Well-Posedness of the Stochastic Transport Equation with Unbounded Drift

The Cauchy problem for a multidimensional linear transport equation with unbounded drift is investigated. Provided the drift is Holder continuous , existence, uniqueness and strong stability of solutions are obtained. The proofs are based on a careful analysis of the associated stochastic flow of characteristics and techniques of stochastic analysis.

Non-local Conservation Law from Stochastic Particle Systems

We consider an interacting particle system in

On the numerical integration of a random integral equation arising in the simulation of stochastic transport equations

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