arXiv subjects

Rahma Yasmina Moulay Hachemi

Publications and source records attributed to Rahma Yasmina Moulay Hachemi.

6 recordsLinked to original sources

arXiv2026

The stochastic fractional transport heat equation

We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time-space white noise, in the following two cases: (i) With additive fractional time-space white noise (ii) With multiplicative time-space Brownian noise The fractional time derivative is interpreted as the Caputo derivative of order $α\in (0,2]$ and we assume that the Hurst coefficient $H=(H_0,H_1,H_2, ...,H_d)$ of the time-space fractional white noise is in $(\tfrac{1}{2},1)^{1+d}$. We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i). In the multiplicative case (ii) we derive the correct space--time convolution equation and its Wiener chaos expansion. For $1<α\leq2$ a second initial condition is required. The endpoint $α=2$ is treated separately and yields a stochastic wave equation with finite propagation speed. A solution $Y(t,x)$ is called \emph{mild} if $E[Y^2(t,x)] < \infty$ for all $t,x$. In the multiplicative case the random-field condition is expressed by the square integrability of the forcing kernel. For the classical Laplacian and space--time white noise, the endpoint $α=2$ admits a random-field solution only in space dimension $d=1$. This paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research paper, in the sense that some results are new, to the best of our knowledge.

math.PR↗

arXiv2026

Multiparameter Lévy white noise theory and applications

We construct a white noise theory and white noise calculus for the (multi-parameter) L\' evy sheet and its compensated Poisson random measures. The theory applies to stochastic partial differential equations subject to L\' evy noise.

math.PR↗

arXiv2026

A time-fractional Kalman filter

We study a linear filtering problem in which the signal process is described by a time-fractional linear stochastic differential equation driven by Brownian motion. We derive a stochastic integral equation for the conditional mean alongside a Riccati--Volterra type integral equation for the mean-square error function. As a core application, we introduce a time-fractional state-estimation framework for modelling learning trajectories in children with developmental dyscalculia.

math.PR↗

arXiv2025

The stochastic heat inclusion with fractional time driven by time-space Brownian and Lévy white noise

We study a time-fractional stochastic heat inclusion driven by additive time-space Brownian and Lévy white noise. The fractional time derivative is interpreted as the Caputo derivative of order $α\in (0,2).$ We show the following: \\ a) If a solution exists, then it is a fixed point of a specific set-valued map.\\ b) Conversely, any fixed point of this map is a solution of the heat inclusion.\\ c) Finally, we show that there is at least one fixed point of this map, thereby proving that there is at least one solution of the time-fractional stochastic heat inclusion. A solution $Y(t,x)$ is called \emph{mild} if $\E[Y^2(t,x)] < \infty$ for all $t,x$. We show that the solution is mild if\\ $α=1$ \& $d=1,$ \ or \ $α\geq 1$ \& $d\in \{1,2\}$. On the other hand, if $α< 1$ we show that the solution is not mild for any space dimension $d$.

math.PR↗

arXiv2024

The time-fractional heat equation driven by fractional time-space white noise

We give an introduction to the time-fractional stochastic heat equation driven by 1+d-parameter fractional time-space white noise, in the following two cases: (i) With additive noise (ii) With multiplicative noise. The fractional time derivative is interpreted as the Caputo derivative of order $α\in (0,2)$ and we assume that the Hurst coefficient $H=(H_0,H_1,H_2, ...,H_d)$ of the time-space fractional white noise is in $(\tfrac{1}{2},1)^{1+d}$. We find an explicit expression for the unique solution in the sense of distribution of the equation in the additive noise case (i). In the multiplicative case (ii) we show that there is a unique solution in the Hida space $(\mathcal{S})^{*}$ of stochastic distributions and we show that the solution coincides with the solution of an associated fractional stochastic Volterra equation.Then we give an explicit expression for the solution of this Volterra equation. A solution $Y(t,x)$ is called \emph{mild} if $E[Y^2(t,x)] < \infty$ for all $t,x$. For both the additive noise case and the multiplicative noise case we show that if $α\geq 1$ then the solution is mild if $d=1$ or $d=2$, while if $α< 1$ the solution is not mild for any $d$. The paper is partly a survey paper, explaining the concepts and methods behind the results. It is also partly a research paper, in the sense that some results appears to be new.

math.PR↗

arXiv2022

The time-fractional stochastic heat equation driven by time-space white noise

We study the time-fractional stochastic heat equation driven by time-space white noise with space dimension $d\in\mathbb{N}=\{1,2,...\}$ and the fractional time-derivative is the Caputo derivative of order $α\in (0,2)$. We consider the equation in the sense of distribution, and we find an explicit expression for the $\mathcal{S}'$-valued solution $Y(t,x)$, where $\mathcal{S}'$ is the space of tempered distributions. Following the terminology of Y. Hu \cite{Hu}, we say that the solution is \emph{mild} if $Y(t,x) \in L^2(\mathbb{P})$ for all $t,x$, where $\mathbb{P}$ is the probability law of the underlying time-space Brownian motion. It is well-known that in the classical case with $α= 1$, the solution is mild if and only if the space dimension $d=1$. We prove that if $α\in (1,2)$ the solution is mild if $d=1$ or $d=2$. If $α< 1$ we prove that the solution is not mild for any $d$.

math.PR↗