A priori estimates for rough PDEs with application to rough conservation laws

We introduce a general weak formulation for PDEs driven by rough paths, as well as a new strategy to prove well-posedness. Our procedure is based on a combination of fundamental a priori estimates with (rough) Gronwall-type arguments. In particular this approach does not rely on any sort of transformation formula (flow transformation, Feynman–Kac representation formula etc.) and is therefore rather flexible. As an application, we study conservation laws driven by rough paths establishing well–posedness for the corresponding kinetic formulation.     

On the Brownian-directed polymer in a Gaussian random environment

In this paper, we introduce a model of Brownian polymer in a continuous random environment. The asymptotic behavior of the partition function associated to this polymer measure is studied, and we are able to separate a weak and strong disorder regime under some reasonable assumptions on the spatial covariance of the environment. Some further developments, concerning some concentration inequalities for the partition function, are given for the weak disorder regime.     

Young Integrals and SPDEs

In this note, we study the non-linear evolution problem $dY_t = -A Y_t dt + B(Y_t) dX_t,$ where $XIn this note, we study the non-linear evolution problem where is a -H?lder continuous function of the time parameter, with values in a distribution space, and the generator of an analytical semigroup. Then, we will give some sharp conditions on in order to solve the above equation in a function space, first in the linear case (for any value of in ), and then when satisfies some Lipschitz type conditions (for ). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.     

LAN property for stochastic differential equations with additive fractional noise and continuous time observation

We consider a stochastic differential equation with additive fractional noise with Hurst parameter $H>1∕2$, and a non-linear drift depending on an unknown parameter. We show the Local Asymptotic Normality property (LAN) of this parametric model with rate $\sqrt{\tau }$ as $\tau \to \infty$, when the solution is observed continuously on the time interval $[0,\tau ]$. The proof uses ergodic properties of the equation and a Girsanov-type transform. We analyze the particular case of the fractional Ornstein–Uhlenbeck process and show that the Maximum Likelihood Estimator is asymptotically efficient in the sense of the Minimax Theorem.     

Smooth Density for Some Nilpotent Rough Differential Equations

In this note, we provide a nontrivial example of a differential equation driven by a fractional Brownian motion with Hurst parameter 1/3<H<1/2 whose solution admits a smooth density with respect to Lebesgue measure. The result is obtained through the use of an explicit representation of the solution when the vector fields of the equation are nilpotent, plus a Norris-type lemma in the rough paths context.     

Non-linear rough heat equations

This article is devoted to define and solve an evolution equation of the form dy _t ?=?Δy _t dt?+ dX _t (y _t ), where Δ stands for the Laplace operator on a space of the form ${L^p(\mathbb R^n)}$ , and X is a finite dimensional noisy nonlinearity whose typical form is given by ${X_t(\varphi)=\sum_{i=1}^N \, x^{i}_t f_i(\varphi)}$ , where each x?=?(x ⁽¹⁾, … , x ^(N)) is a γ-H?lder function generating a rough path and each f _i is a smooth enough function defined on ${L^p(\mathbb R^n)}$ . The generalization of the usual rough path theory allowing to cope with such kind of system is carefully constructed.     

Malliavin Calculus for Fractional Delay Equations

In this paper we study the existence of a unique solution to a general class of Young delay differential equations driven by a H?lder continuous function with parameter greater that 1/2 via the Young integration setting. Then some estimates of the solution are obtained, which allow to show that the solution of a delay differential equation driven by a fractional Brownian motion (fBm) with Hurst parameter H>1/2 has a C ^??-density. To this purpose, we use Malliavin calculus based on the Fréchet differentiability in the directions of the reproducing kernel Hilbert space associated with fBm.     

The 1-d stochastic wave equation driven by a fractional Brownian sheet

In this paper, we develop a Young integration theory in dimension 2 which will allow us to solve a non-linear one- dimensional wave equation driven by an arbitrary signal whose rectangular increments satisfy some Hölder regularity conditions, for some Hölder exponent greater than 1/2. This result will be applied to the fractional Brownian sheet.     

Sharp Gaussian regularity on the circle, and applications to the fractional stochastic heat equation

A sharp regularity theory is established for homogeneous Gaussian fields on the unit circle. Two types of characterizations for such a field to have a given almost-sure uniform modulus of continuity are established in a general setting. The first characterization relates the modulus to the field's canonical metric; the full force of Fernique's zero-one laws and Talagrand's theory of majorizing measures is required. The second characterization ties the modulus to the field's random Fourier series representation. As an application, it is shown that the fractional stochastic heat equation has, up to a non-random constant, a given spatial modulus of continuity if and only if the same property holds for a fractional antiderivative of the equation's additive noise; a random Fourier series characterization is also given.