Unbounded rough drivers

We propose a theory of linear differential equations driven by unbounded operator-valued rough signals. As an application we consider rough linear transport equations and more general linear hyperbolic symmetric systems of equations driven by time-dependent vector fields which are only distributions in the time direction.Comment: 38 pages. some improvements and precision

Rough flows and homogenization in stochastic turbulence

We provide in this work a tool-kit for the study of homogenisation of random ordinary differential equations, under the form of a friendly-user black box based on the tehcnology of rough flows. We illustrate the use of this setting on the example of stochastic turbulence.Comment: v2, 27 pages; presentation fairly improved; extended scope for the materia

The inverse problem for rough controlled differential equations

We provide a necessary and sufficient condition for a rough control driving a differential equation to be reconstructable, to some order, from observing the resulting controlled evolution. Physical examples and applications in stochastic filtering and statistics demonstrate the practical relevance of our result.Comment: added section on rough path theor

Space-time paraproducts for paracontrolled calculus, 3d-PAM and multiplicative Burgers equations

We sharpen in this work the tools of paracontrolled calculus in order to provide a complete analysis of the parabolic Anderson model equation and Burgers system with multiplicative noise, in a $3$-dimensional Riemannian setting, in either bounded or unbounded domains. With that aim in mind, we introduce a pair of intertwined space-time paraproducts on parabolic H\"older spaces, with good continuity, that happens to be pivotal and provides one of the building blocks of higher order paracontrolled calculus.Comment: v3, 56 pages. Different points about renormalisation matters have been clarified. Typos correcte

Uniqueness of the Φ34\Phi^4_3 measures on closed Riemannian 33-manifolds

We constructed in a previous work the $\Phi^4_3$ measures on compact boundaryless $3$-dimensional Riemannian manifolds as some invariant probability measures of some Markovian dynamics. We prove in the present work that these dynamics have unique invariant probability measures. This is done by using an explicit coupling by change of measure that does not require any a priori information on the support of the law of the solution to the dynamics. The coupling can be used to see that the semigroup generated by the dynamics satisfies a Harnack-type inequality, which entails that the semigroup has the strong Feller property.Comment: 21 page