Scholarlink

<p>In this paper we are interested in a quasi-linear hyperbolic stochastic differential equation (HSPDE) when the vector field is merely bounded and measurable. Although the deterministic counterpart of such equation may be ill-posed (in the sense that uniqueness or even existence might not be valid), we show for the first time that the corresponding HSPDE has a unique (Malliavin differentiable) strong solution. Our approach for proving this result rests on: (1) tools from Malliavin calculus and (2) variational techniques introduced in Davie (International Mathematics Research Notices, vol. 2007, Duke University Press, Durham, 2007) non-trivially extended to the case of SDEs in the plane by using an algorithm for the selection of certain rectangles. As a by-product, we also obtain the Sobolev differentiability of the solution with respect to its initial value. The results derived here constitute a significant improvement of those in the current literature on SDEs on the plane and can be regarded as an analogous of the pioneering works by Zvonkin (Math. URSS Sb. 22:129–149, 1974) and Veretennikov (Theory Probab. Appl. 24:354–366, 1979) in the case of one-parameter SDEs with singular drift.</p>

错误：搜索内容不能为空，请输入英文关键词
错误：关键词超出字数限制，请精简
高级检索

Smoothness of solutions of hyperbolic stochastic partial differential equations with \(L^{\infty }\)-vector fields

  • 期刊论文

  • 出版模式：

    开放获取

  • 发表日期：

    2025年9月5日

查看全文
  • Antoine-Marie Bogso,
  • Moustapha Dieye,
  • Olivier Menoukeu-Pamen,
  • Frank Proske

摘要

In this paper we are interested in a quasi-linear hyperbolic stochastic differential equation (HSPDE) when the vector field is merely bounded and measurable. Although the deterministic counterpart of such equation may be ill-posed (in the sense that uniqueness or even existence might not be valid), we show for the first time that the corresponding HSPDE has a unique (Malliavin differentiable) strong solution. Our approach for proving this result rests on: (1) tools from Malliavin calculus and (2) variational techniques introduced in Davie (International Mathematics Research Notices, vol. 2007, Duke University Press, Durham, 2007) non-trivially extended to the case of SDEs in the plane by using an algorithm for the selection of certain rectangles. As a by-product, we also obtain the Sobolev differentiability of the solution with respect to its initial value. The results derived here constitute a significant improvement of those in the current literature on SDEs on the plane and can be regarded as an analogous of the pioneering works by Zvonkin (Math. URSS Sb. 22:129–149, 1974) and Veretennikov (Theory Probab. Appl. 24:354–366, 1979) in the case of one-parameter SDEs with singular drift.