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Brahim Boufoussi

Publications and source records attributed to Brahim Boufoussi.

12 recordsLinked to original sources
arXiv2026

On the Besov-Orlicz path regularity of some Gaussian processes

In this paper, we rely on the additive decomposition in law satisfied by a class of stochastic processes, combined with the well-known regulariy properties of fractional Brownian motion, to establish Besov-Orlicz regularity of their sample paths. This provides a unified and direct proof for a broad class of processes, including bifractional Brownian motion with parameters $H\in (0, 1]$, $ K\in (0, 2)$ such that $HK \in (0, 1)$, subfractional Brownian motion with Hurst parameter $H\in (0, 1)$, and certain class of self-similar processes. %associated with the stochastic heat equation.

math.PR↗
arXiv2022

Approximation of a degenerate semilinear PDEs with a nonlinear Neumann boundary condition

We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform ellipticity is not required to the diffusion coefficient. We show that this problem admits a viscosity solution which can be approximated by a penalization. The Lipschitz condition is required to the coefficients of the diffusion part. The nonlinear part as well as the Neumann condition are Lipschitz. Moreover, the nonlinear part is assumed monotone in the solution variable. Note that the existence of a viscosity solution to this problem has been established in [13] then completed in [15]. In the present paper, We construct a sequence of penalized system of decoupled forward backward stochastic differential equations (FBSDEs) then we directly show its strong convergence. This allows us to deal with the case where the nonlinearity depends on both the solution and its gradient. Our work extends, in particular, the result of [4] and, in some sense, those of [1, 3]. In contrast to works [1, 3, 4], we do not pass by the weak compactness of the laws of the stochastic system associated to our problem.

math.PR↗
arXiv2021

On the uniform Besov regularity of local times of general processes

Our main purpose is to use a new condition, $\alpha$-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↗
arXiv2020

Penalization for a PDE with a Nonlinear Neumann boundary condition and measurable coefficients *

We consider a system of semi-linear partial differential equations with measurable coefficients and a nonlinear Neumann boundary condition. We then construct a sequence of penalized partial differential equations which converges to a solution of our initial problem. The solution we construct is in the L p --viscosity sense, since the coefficients can be not continuous. The method we use is based on backward stochastic differential equations and their S-tightness. The present work is motivated by the fact that many partial differential equations arising in physics have discontinuous coefficients.

math.PR↗
arXiv2018

Controllability of Neutral Stochastic Functional Integro-Differential Equations Driven by Fractional Brownian Motion with Hurst Parameter Lesser than 1/2

In this article we investigate the controllability for neutral stochastic functional integro-differential equations with finite delay, driven by a fractional Brownian motion with Hurst parameter lesser than $1/2$ in a Hilbert space. We employ the theory of resolvent operators combined with the Banach fixed point theorem to establish sufficient conditions to prove the desired result

math.PR↗
arXiv2007

Sample path properties of the local time of multifractional Brownian motion

We establish estimates for the local and uniform moduli of continuity of the local time of multifractional Brownian motion, $B^H=(B^{H(t)}(t),t\in\mathbb{R}^+)$. An analogue of Chung's law of the iterated logarithm is studied for $B^H$ and used to obtain the pointwise Hölder exponent of the local time. A kind of local asymptotic self-similarity is proved to be satisfied by the local time of $B^H$.

math.PR↗