Fractional stochastic heat equation with mixed operator and driven by fractional-type noise

Mounir Zili; Eya Zougar; Mohamed Rhaima; Mounir Zili; Eya Zougar; Mohamed Rhaima

doi:10.3934/math.20241406

AIMS Mathematics

2024, Volume 9, Issue 10: 28970-29000. doi: 10.3934/math.20241406

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Research article

Fractional stochastic heat equation with mixed operator and driven by fractional-type noise

1.
Department of Mathematics, University of Monastir, Faculty of Sciences, Avenue de l'environnement, 5019 Monastir, Tunisia
2.
Department of Mathematics, University of Polytechnique Hauts-de-France, CERAMATHS, Campus Mont Houy, 59313, Valenciennes, CEDEX 9-France
3.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia

Received: 27 May 2024 Revised: 04 October 2024 Accepted: 08 October 2024 Published: 14 October 2024
MSC : 35R11, 60G22, 60H15

We investigated a novel stochastic fractional partial differential equation (FPDE) characterized by a mixed operator that integrated the standard Laplacian, the fractional Laplacian, and the gradient operator. The equation was driven by a random noise, which admitted a covariance measure structure with respect to the time variable and behaved as a Wiener process in space. Our analysis included establishing the existence of a solution in the general case and deriving an explicit form for its covariance function. Additionally, we delved into a specific case where the noise was modeled as a generalized fractional Brownian motion (gfBm) in time, with a particular emphasis on examining the regularity of the solution's sample paths.
- stochastic fractional partial differential equations,
- fractional Brownian and sub-Brownian motions,
- mild solution,
- sample paths,
- heat equation
Citation: Mounir Zili, Eya Zougar, Mohamed Rhaima. Fractional stochastic heat equation with mixed operator and driven by fractional-type noise[J]. AIMS Mathematics, 2024, 9(10): 28970-29000. doi: 10.3934/math.20241406

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Abstract

We investigated a novel stochastic fractional partial differential equation (FPDE) characterized by a mixed operator that integrated the standard Laplacian, the fractional Laplacian, and the gradient operator. The equation was driven by a random noise, which admitted a covariance measure structure with respect to the time variable and behaved as a Wiener process in space. Our analysis included establishing the existence of a solution in the general case and deriving an explicit form for its covariance function. Additionally, we delved into a specific case where the noise was modeled as a generalized fractional Brownian motion (gfBm) in time, with a particular emphasis on examining the regularity of the solution's sample paths.

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