arXiv subjects

Yassine Nachit

Publications and source records attributed to Yassine Nachit.

8 recordsLinked to original sources

arXiv2026

Malliavin calculus and densities for chaos-driven stochastic differential equations

We study stochastic differential equations driven by finite-order chaos processes on abstract Wiener spaces, with pathwise Riemann-Stieltjes integration. The driving noise is an $\mathbb{R}^m$-valued chaotic process given by multiple Wiener-Itô integrals of fixed order, allowing for non-Gaussian dynamics. Under mild smoothness assumptions on the coefficients and Hölder-type regularity of the noise, we establish existence and uniqueness of solutions. We then prove Malliavin differentiability and absolute continuity of the law of the solution. Since the usual Gaussian isonormal framework is unavailable, we rely on the Kusuoka-Stroock approach to Malliavin calculus and develop a Taylor expansion for multiple integrals under Cameron-Martin shifts. Under suitable ellipticity, independence, and non-degeneracy conditions, the Bouleau-Hirsch criterion yields density results. Applications to multidimensional Hermite-driven equations are provided.

math.PR↗

arXiv2025

Quartic variation of the solution to the semilinear stochastic heat equation: limit behavior and asymptotic independence with respect to the data

This work concerns the limit behavior of the quartic variation (i.e., the power variation of order four) with respect to the time variable of the solution to the semilinear stochastic heat equation with space-time white noise. In a first step, we prove that this sequence satisfies a Central Limit Theorem and we deduce a similar result for the viscosity parameter estimator associated with the quartic variation. Then, by using a recent variant of the Stein-Malliavin calculus, we analyze the asymptotic independence between the quartic variation (as well as the associated viscosity parameter estimator) and the data used to construct it.

math.PR↗

arXiv2025

Malliavin smoothness of the Rosenblatt process

We investigate the smoothness of the densities of the finite-dimensional distributions of the Rosenblatt process. Within the Malliavin calculus framework, we prove that Rosenblatt random vectors are nondegenerate in the Malliavin sense. As a consequence, their densities belong to the Schwartz space of rapidly decreasing smooth functions. The proof relies on establishing the existence of all negative moments of the determinant of the Malliavin matrix, exploiting the specific structure of random variables in the second Wiener chaos. In addition, we derive exponential-type upper bounds for the partial derivatives of the densities of the finite-dimensional distributions of the Rosenblatt process.

math.PR↗

arXiv2025

Absolute continuity of finite-dimensional distributions of Hermite processes via Malliavin calculus

We investigate the existence of densities for finite-dimensional distributions of Hermite processes of order $q \ge 1$ and self-similarity parameter $H\in(\frac12,1)$. Whereas the Gaussian case $q=1$ (fractional Brownian motion) is well understood, the non-Gaussian situation has not yet been settled. In this work, we extend the classical three-step approach used in the Gaussian case: factorization of the determinant into conditional terms, strong local nondeterminism, and non-degeneracy. We transport this strategy to the Hermite setting using Malliavin calculus. Specifically, we establish a determinant identity for the Malliavin matrix, prove strong local nondeterminism at the level of Malliavin derivatives, and apply the Bouleau-Hirsch criterion. Consequently, for any distinct times $t_1,\dots,t_n$, the vector $(Z^{H,q}_{t_1},\dots,Z^{H,q}_{t_n})$ of a Hermite process admits a density with respect to the Lebesgue measure. Beyond the result itself, the main contribution is the methodology, which could extend to other non-Gaussian models.

math.PR↗

arXiv2025

On the uniform Besov regularity of local times of general processes

Our main purpose is to use a new condition, $α$-local nondeterminism, which is an alternative to the classical local nondeterminism usually utilized in the Gaussian framework, in order to investigate Besov regularity, in the time variable $t$ uniformly in the space variable $x$, for local times $L(x, t)$ of a class of continuous processes. We also extend the classical Adler's theorem [1, Theorem 8.7.1] to the Besov spaces case. These results are then exploited to study the Besov irregularity of the sample paths of the underlying processes. Based on similar known results in the case of the bifractional Brownian motion, we believe that our results are sharp. As applications, we get sharp Besov regularity results for some classical Gaussian processes and the solutions of systems of non-linear stochastic heat equations. The Besov regularity of their corresponding local times is also obtained.

math.PR↗

arXiv2021

On Besov regularity and local time of the stochastic heat equation

Sharp Besov regularities in time and space variables are investigated for $\left(u(t,x),\; t\in [0,T],\; x\in \mathbb{R}\right)$, the mild solution to the stochastic heat equation driven by space-time white noise. Existence, Hölder continuity, and Besov regularity of local times are established for $u(t,x)$ viewed either as a process in the space variable or time variable. Hausdorff dimensions of their corresponding level sets are also obtained.

math.PR↗

arXiv2021

Local times for systems of non-linear stochastic heat equations

We consider $u(t,x)=(u_1(t,x),\cdots,u_d(t,x))$ the solution to a system of non-linear stochastic heat equations in spatial dimension one driven by a $d$-dimensional space-time white noise. We prove that, when $d\leq 3$, the local time $L(ξ,t)$ of $\{u(t,x)\,,\;t\in[0,T]\}$ exists and $L(\bullet,t) $ belongs a.s. to the Sobolev space $ H^α(\mathbb{R}^d)$ for $α<\frac{4-d}{2}$, and when $d\geq 4$, the local time does not exist. We also show joint continuity and establish Hölder conditions for the local time of $\{u(t,x)\,,\;t\in[0,T]\}$. These results are then used to investigate the irregularity of the coordinate functions of $\{u(t,x)\,,\;t\in[0,T]\}$. Comparing to similar results obtained for the linear stochastic heat equation (i.e., the solution is Gaussian), we believe that our results are sharp. Finally, we get a sharp estimate for the partial derivatives of the joint density of $(u(t_1,x)-u(t_0,x),\cdots,u(t_n,x)-u(t_{n-1},x))$, which is a new result and of independent interest.

math.PR↗

arXiv2020

On the Besov regularity of the bifractional Brownian motion

Our aim in this paper is to improve Hölder continuity results for the bifractional Brownian motion (bBm) $(B^{α,β}(t))_{t\in[0,1] }$ with $0<α<1$ and $0<β\leq 1$. We prove that almost all paths of the bBm belong (resp. do not belong) to the Besov spaces $\mathbf{Bes}(αβ,p)$ (resp. $\mathbf{bes}(αβ,p)$) for any $\frac{1}{αβ} \frac{1}{2}$ in the Hölder spaces $\mathcal{C}^γ$, with $γ<αβ$.

math.PR↗