On a modelled rough heat equation

We use the formalism of Hairer's regularity structures theory \cite{hai-14} to study a heat equation with non-linear perturbation driven by a space-time fractional noise. Different regimes are observed, depending on the global pathwise roughness of the noise. To this end, and following the procedure exhibited in \cite{hai-14}, the equation is first "lifted" into some abstract "regularity structure" and therein solved through a fixed-point argument. Then we construct a consistent "model" above the fractional noise, by relying on a smooth Fourier-type approximation of the process.Comment: Accepted versio

Rough Volterra equations 1: the algebraic integration setting

We define and solve Volterra equations driven by an irregular signal, by means of a variant of the rough path theory called algebraic integration. In the Young case, that is for a driving signal with H\"older exponent greater than 1/2, we obtain a global solution, and are able to handle the case of a singular Volterra coefficient. In case of a driving signal with H\"older exponent in (1/3,1/2], we get a local existence and uniqueness theorem. The results are easily applied to the fractional Brownian motion with Hurst coefficient H>1/3.Comment: 31 page

A few results about the hyperbolic Anderson model (Nonlinear and Random Waves)

The objective of the subsequent notes is to give an overview of the considerations and results contained in the two studies [7, 8], written in collaboration with with X. Chen, J. Song and S. Tindel. The reader is thus referred to these two publications for further details, and in particular for the proof of the assertions below

On the 1d stochastic Schr\"odinger product

We exhibit various restrictions about the wellposedness of the Schr\"odinger product $$\mathcal{L}: z \longmapsto -\imath \int_0^t e^{\imath \Delta s}\big( z_s\cdot \Psi_s\big) ds$$ where $\Psi$ refers to the so-called linear solution of the stochastic Schr\"dinger problem. We focus more specifically on the case where $\Psi$ satisfies $$(\imath \partial_t-\Delta)\Psi=\dot{B}, \quad \Psi_0=0,\quad \quad t\in \mathbb{R}, \ x\in \mathbb{T},$$where $\dot{B}$ is a white noise in space with fractional time covariance of index $H>\frac12$. As an important consequence of our analysis, we obtain that if $H$ is close to $\frac12$ (that is $\dot{B}$ is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem $$(\imath \partial_t-\Delta)u=\lambda\, u^{p} \overline{u}^q+\dot{B}, \quad u_0=0,\quad \quad \lambda \in \{-1,1\}, \ p,q\geq 1, \ t\in \mathbb{R}, \ x\in \mathbb{T},$$using only a first-order expansion of the solution ("$u=\Psi+z$")