This paper establishes the fully discrete stochastic maximal $L^p$-regularity and the accompanying sharp maximal estimate for numerical approximations of parabolic stochastic partial differential equations. We consider the spatial finite element discretization $A_h$ of a general second-order elliptic operator $A=-\nabla \cdot a\nabla +b\cdot \nabla +c$ with Dirichlet boundary conditions on a smooth, bounded, convex domain in $\mathbb{R}^3$, coupled with a broad class of temporal schemes, including rational approximations and the exponential Euler method. To obtain these optimal discrete regularity results, we establish a bounded $H^\infty$-calculus for the discrete spatial operator $A_h$, uniformly in the mesh size $h$. As a direct byproduct, we also establish the discrete-in-space stochastic maximal regularity for the corresponding spatial semi-discretizations.
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We develop a rigorous analytical framework for Riemann-Liouville fractional
stochastic differential equations (RL-FSDEs), addressing a gap in the literature where existing treatments either focus on Caputo formulations or
provide incomplete existence arguments. Our approach combines Laplace transform operational calc...
Mohsen Alhassoun, Khalil Yahya, A. Abdallah· Boletim da Sociedade Paranae...· 0 citations
This article establishes sharp strong error estimates for the fully discrete approximation of semilinear parabolic stochastic partial differential equations (SPDEs) driven by multiplicative noise and subject to nonsmooth initial data. The spatial discretization is based on a standard finite element method, coupled with...
Let $0<s<1$ and $A$ be a constant symmetric positive definite matrix. Given the quadratic form $$ Q_A(u)=\int_{\R^N} A\nabla^s u(x)\cdot \nabla^s u(x)\,dx, $$ where $\nabla^s$ denotes the Riesz fractional gradient, we establish necessary and sufficient conditions such that $Q_A$ defines a Dirichlet form. In strike cont...
We study the role of Banach space geometry in deterministic and stochastic maximal regularity. We first construct an example showing that the UMD assumption in the characterisation of maximal $L^p$-regularity in terms of $R$-sectoriality cannot be omitted. Combining this construction with an equivalence between stochas...
Emiel Lorist, J. van Neerven, M. Veraar et al.· 0 citations
We investigate the dynamical behavior of a class of semilinear stochastic wave equations on a bounded smooth domain $\Ocal\subset\R^d$ driven by additive trace-class noise, where the elastic response is governed by a \emph{mixed local--nonlocal} operator $\Acal=-\theta\Delta+\beta(-\Delta)^s$ with $s\in(0,1)$. A fundam...
Francisco Delgado-Vences, Jose Julian Pavon-Español· 0 citations
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