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<p>In this paper, we study stochastic Rayleigh-Stokes equations (stochastic RSEs) driven by space-time white noise. The first contribution is to investigate the existence, uniqueness, and boundedness results in two cases of the initial functions. The second objective focuses on establishing Hölder continuity with respect to the time variable. In the last result, we are strongly interested in analyzing the continuity in the fractional order and diffusion coefficient (or called <i>parameter-continuity</i>). Although deterministic RSEs have been widely examined, as far as we know, no research has been conducted on the stochastic case involving white noise in both space and time. Additionally, despite the critical importance of studying parameter-continuity for modeling purposes, this behavior has not been explored in either deterministic or stochastic RSEs until now. Our present paper is the first result on RSEs driven by space-time white noise and on the parameter-continuity for its solution, inspired by recent publications on stochastic classical (and fractional) heat equations of M. Foondun, see Ann. Probab. <b>42</b>(5), 1911–1937, (2017), arXiv preprint, <a href="http://arxiv.org/abs/2510.00214">arXiv:2510.00214</a>, (2025), Fract. Calc. Appl. Anal. <b>19</b>(6), 1527–1553, (2016), Proc. Am. Math. Soc. <b>149</b>(5), 2235–2247, (2021), Math. Z. <b>287</b>, 493–519 (2017), Math. Z. <b>287</b>, 493–519 (2017), <a href="http://arxiv.org/abs/1412.2343">arXiv:1412.2343</a>, (2014).</p>

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Rayleigh-Stokes Equations with Space-Time White Noise: Existence, Hölder Regularity, and Parameter-Continuity

  • 期刊论文

  • 发表日期：

    2026年1月13日

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  • Nguyen Huy Tuan,
  • Tran Ngoc Thach

摘要

In this paper, we study stochastic Rayleigh-Stokes equations (stochastic RSEs) driven by space-time white noise. The first contribution is to investigate the existence, uniqueness, and boundedness results in two cases of the initial functions. The second objective focuses on establishing Hölder continuity with respect to the time variable. In the last result, we are strongly interested in analyzing the continuity in the fractional order and diffusion coefficient (or called parameter-continuity). Although deterministic RSEs have been widely examined, as far as we know, no research has been conducted on the stochastic case involving white noise in both space and time. Additionally, despite the critical importance of studying parameter-continuity for modeling purposes, this behavior has not been explored in either deterministic or stochastic RSEs until now. Our present paper is the first result on RSEs driven by space-time white noise and on the parameter-continuity for its solution, inspired by recent publications on stochastic classical (and fractional) heat equations of M. Foondun, see Ann. Probab. 42(5), 1911–1937, (2017), arXiv preprint, arXiv:2510.00214, (2025), Fract. Calc. Appl. Anal. 19(6), 1527–1553, (2016), Proc. Am. Math. Soc. 149(5), 2235–2247, (2021), Math. Z. 287, 493–519 (2017), Math. Z. 287, 493–519 (2017), arXiv:1412.2343, (2014).