42 research outputs found
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 where refers to the so-called linear solution
of the stochastic Schr\"dinger problem. We focus more specifically on the case
where satisfies where is a
white noise in space with fractional time covariance of index . As
an important consequence of our analysis, we obtain that if is close to
(that is is close to a space-time white noise), then it is
essentially impossible to treat the stochastic NLS problem using only a first-order expansion of the solution ("")