The study is devoted to the interpretation and wellposedness of the stochastic NLS model (∂_t-Δ)u=|u|^2+Ḃ, u_0=0, t∈, x∈𝕋, where Ḃ stands for a space-time fractional noise with index H=(H_0,H_1) in a subset of (0,1)^2. We first establish that in the situation where 0<2H_0+H_1≤ 2, the equation cannot be interpreted in a (classical) functional sense. Our investigations then focus on the rough regime corresponding to the condition 7/4<2H_0+H_1≤ 2. In this specific case, we exhibit an explicit renormalization procedure allowing to restore the (local) convergence of the approximated solutions. We follow a pathwise-type approach emphasizing the distinction between the stochastic objects at the core of the dynamics and the general deterministic machinery.

We exhibit various restrictions about the wellposedness of the Schrödinger product ℒ:z ⟼ -∫ _0^t e^ s ∂ ^2_x ( z_s·Ψ _s ) ds where Ψ refers to the so-called linear solution of the stochastic Schrödinger problem. We focus more specifically on the case where Ψ satisfies 0.1 (∂ _t-∂ ^2_x)Ψ =Ḃ, Ψ _0=0, t∈ℝ, x∈𝕋, where Ḃ is a white noise in space with fractional time covariance of index H>1/2 . As an consequence of our analysis, we obtain that if H is close to 1/2 (that is Ḃ is close to a space-time white noise), then it is essentially impossible to treat the stochastic NLS problem (∂ _t-∂ ^2_x)u= |u|^2+Ḃ, u_0=0, t∈ℝ, x∈𝕋, using only a first-order expansion of the solution (“ u=Ψ +z ”).

This paper is concerned with a wave equation in dimension $d\in \{1,2, 3\}$, with a multiplicative space-time Gaussian noise which is fractional in time and homogeneous in space. We provide necessary and sufficient conditions on the space-time covariance of the Gaussian noise, allowing the existence and uniqueness of a mild Skorohod solution.

We study a wave equation in dimension $d\in \{1,2\}$ with a multiplicative space-time Gaussian noise. The existence and uniqueness of the Stratonovich solution is obtained under some conditions imposed on the Gaussian noise. The strategy is to develop some Strichartz-type estimates for the wave kernel in weighted Besov spaces, by which we can prove the well-posedness of an associated Young-type equation. Those Strichartz bounds are of independent interest.

We highlight a fundamental ill-posedness issue for nonlinear stochastic wave equations driven by a fractional noise. Namely, if the noise becomes too rough (i.e., the sum of its Hurst indexes becomes too small), then there is essentially no hope to provide a systematic interpretation of the model, whether directly or through a Wick-type renormalization procedure. This phenomenon can be compared with the situation of a general SDE driven by a two-dimensional fractional noise of index H < 1/4. Our results clarify and extend previous similar properties exhibited in Deya (2020) or in Oh and Okamoto (2021). (C) 2022 Elsevier B.V. All rights reserved.

We study a full discretization scheme for the stochastic linear heat equation \begin{equation*}\begin{cases}\partial_t \langle\Psi\rangle = \Delta \langle\Psi\rangle +\dot{B}\, , \quad t\in [0,1], \ x\in \mathbb{R},\\ \langle\Psi\rangle_0=0\, ,\end{cases}\end{equation*} when $\dot{B}$ is a very \emph{rough space-time fractional noise}. The discretization procedure is divised into three steps: $(i)$ regularization of the noise through a mollifying-type approach; $(ii)$ discretization of the (smoothened) noise as a finite sum of Gaussian variables over rectangles in $[0,1]\times \mathbb{R}$; $(iii)$ discretization of the heat operator on the (non-compact) domain $[0,1]\times \mathbb{R}$, along the principles of Galerkin finite elements method. We establish the convergence of the resulting approximation to $\langle\Psi\rangle$, which, in such a specific rough framework, can only hold in a space of distributions. We also provide some partial simulations of the algorithm.

We study a stochastic Schrödinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension d ≤ 3 d\leq 3 . When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schrödinger group, which allows us to cope with the strong irregularity of the solution.

The theory of regularity structures enables the definition of the following parabolic Anderson model in a very rough environment: $\partial_{t} u_{t}(x) = \frac12 \Delta u_{t}(x) + u_{t}(x) \, \dot W_{t}(x)$, for $t\in\mathbb{R}_{+}$ and $x\in \mathbb{R}^{d}$, where $\dot W_{t}(x)$ is a Gaussian noise whose space time covariance function is singular. In this rough context, we shall give some information about the moments of $u_{t}(x)$ when the stochastic heat equation is interpreted in the Skorohod as well as the Stratonovich sense. Of special interest is the critical case, for which one observes a blowup of moments for large times.

We construct a K-rough path [along the terminology of Deya (Probab Theory Relat Fields 166:1–65, 2016)] above either a space-time or a spatial fractional Brownian motion, in any space dimension d. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise.

We pursue the investigations initiated in [Aur{\'e}lien Deya: A non-linear wave equation with fractional perturbation (2017)] about a wave-equation model with quadratic perturbation and stochastic forcing given by a space-time fractional noise. We focus here on the two-dimensional situation and therein extend the results of the previous reference to a rougher noise, through the use of a third-order expansion. We also point out the limits of the Wick-renormalisation procedure in this case.

We construct a K-rough path (along the terminology of [15, Definition 2.3]) above either a space-time or a spatial fractional Brownian motion, in any space dimension d. This allows us to provide an interpretation and a unique solution for the corresponding parabolic Anderson model, understood in the renormalized sense. We also consider the case of a spatial fractional noise.

For every $d\geq 1$, we consider the $d$-dimensional Hermitian fractional Brownian motion (HfBm), that is the process with values in the space of $(d\times d)$-Hermitian matrices and with upper-diagonal entries given by complex fractional Brownian motions of Hurst index $H\in (0,1)$. We follow the approach of [A. Deya and R. Schott: On the rough paths approach to non-commutative stochastic calculus, JFA (2013)] to define a natural integral with respect to the HfBm when $H>\frac13$, and identify this interpretation with the rough integral with respect to the $d^2$ entries of the matrix. Using this correspondence, we establish a convenient It{\^o}--Stratonovich formula for the Hermitian Brownian motion. Finally, we show that at least when $H\geq \frac12$, and as the size $d$ of the matrix tends to infinity, the integral with respect to the HfBm converges (in the tracial sense) to the integral with respect to the so-called non-commutative fractional Brownian motion.

We study a stochastic Schr{o}dinger equation with a quadratic nonlinearity and a space-time fractional perturbation, in space dimension less than 3. When the Hurst index is large enough, we prove local well-posedness of the problem using classical arguments. However, for a small Hurst index, even the interpretation of the equation needs some care. In this case, a renormalization procedure must come into the picture, leading to a Wick-type interpretation of the model. Our fixed-point argument then involves some specific regularization properties of the Schr{o}dinger group, which allows us to cope with the strong irregularity of the solution.

We pursue our investigations, initiated in [8], about stochastic integration with respect to the non-commutative fractional Brownian motion (NC-fBm). Our main objective in this paper is to compare the pathwise constructions of [8] with a Skorohod-type interpretation of the integral. As a first step, we provide details on the basic tools and properties associated with non-commutative Malliavin calculus, by mimicking the presentation of Nualart's celebrated treatise [14]. Then we check that, just as in the classical (commutative) situation, Skorohod integration can indeed be considered in the presence of the NC-fBm, at least for a Hurst index H > 1 4.This finally puts us in a position to state and prove the desired comparison result, which can be regarded as an It{\^o}-Stratonovich correction formula for the NC-fBm.

We prove existence and uniqueness of the solution of a one-dimensional rough differential equation driven by a step-2 rough path and reflected at zero. In order to deal with the lack of control of the reflection measure the proof uses some ideas we introduced in a previous work dealing with rough kinetic PDEs [arXiv:1604.00437].

We investigate the problem of the rate of convergence to equilibrium for ergodic stochastic differential equations driven by fractional Brownian motion with Hurst parameter $H\in (1/3,1)$ and multiplicative noise component $\sigma$. When $\sigma$ is constant and for every $H\in (0,1)$, it was proved in [19] that, under some mean-reverting assumptions, such a process converges to its equilibrium at a rate of order $t^{-\alpha}$ where $\alpha \in (0,1)$ (depending on $H$). In [11], this result has been extended to the multiplicative case when $H\textgreater{}1/2$. In this paper, we obtain these types of results in the rough setting $H\in (1/3,1/2)$. Once again, we retrieve the rate orders of the additive setting. Our methods also extend the multiplicative results of [11] by deleting the gradient assumption on the noise coefficient $\sigma$. The main theorems include some existence and uniqueness results for the invariant distribution.

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.

We study a $d$-dimensional wave equation model ($2\leq d\leq 4$) with quadratic non-linearity and stochastic forcing given by a space-time fractional noise. Two different regimes are exhibited, depending on the Hurst parameter $H=(H_0,\ldots,H_d) \in (0,1)^{d+1}$ of the noise: if $\sum_{i=0}^d H_i > d-\frac12$, then the equation can be treated directly, while in the case $d-\frac34<\sum_{i=0}^d H_i\leq d-\frac12$, the model must be interpreted in the Wick sense, through a renormalization procedure. Our arguments essentially rely on a fractional extension of the considerations of \cite{gubinelli-koch-oh} for the two-dimensional white-noise situation, and more generally follow a series of investigations related to stochastic wave models with polynomial perturbation.

We study the issue of integration with respect to the non-commutative fractional Brownian motion, that is the analog of the standard fractional Brownian in a non-commutative probability setting.When the Hurst index $H$ of the process is stricly larger than $1/2$, integration can be handled through the so-called Young procedure. The situation where $H=1/2$ corresponds to the specific free case, for which an It{\^o}-type approach is known to be possible.When $H<1/2$, rough-path-type techniques must come into the picture, which, from a theoretical point of view, involves the use of some a-priori-defined L{\'e}vy area process. We show that such an object can indeed be \enquote{canonically} constructed for any $H\in (\frac14,\frac12)$. Finally, when $H\leq 1/4$, we exhibit a similar non-convergence phenomenon as for the non-diagonal entries of the (classical) L{\'e}vy area above the standard fractional Brownian.

We pursue the investigations initiated by Donati-Martin [9] and Effros-Popa [10] regarding the multiplication issue in the chaoses generated by the q -Brownian motion ( q ∈ ( − 1 , 1) ), along two directions: ( i ) We provide a fully-stochastic approach to the problem and thus make a clear link with the standard Brownian setting; ( ii ) We elaborate on the situation where the kernels are given by symmetric functions, with application to the study of the q -Brownian martingales. that for all t 1 , . . . , t r and Y := X t 3 · · · X t r , ϕ (cid:0) X t 1 X t 2 Y (cid:1) − ϕ (cid:0) X t 2 X t 1 Y (cid:1) = (1 − q ) P t 1 ,...,t r ( q ) , for some polynomial P t 1 ,...,t r ( q ) , which, to some illustrates the following