Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866909771988205568 |
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| author | Bogso, Antoine-Marie Dieye, Moustapha Pamen, Olivier Menoukeu Proske, Frank |
| author_facet | Bogso, Antoine-Marie Dieye, Moustapha Pamen, Olivier Menoukeu Proske, Frank |
| contents | 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, Int. Math. Res. Not., Vol. 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 equivalent of the pioneering works by [Zvonkin, Math. URSS Sbornik, 22:129-149] and [Veretennikov, Theory Probab. Appl., 24:354-366] in the case of one-parameter SDEs with singular drift. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2212_08466 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields Bogso, Antoine-Marie Dieye, Moustapha Pamen, Olivier Menoukeu Proske, Frank Probability Primary 60H07, 60H50, 60H17, Secondary 60H15 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, Int. Math. Res. Not., Vol. 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 equivalent of the pioneering works by [Zvonkin, Math. URSS Sbornik, 22:129-149] and [Veretennikov, Theory Probab. Appl., 24:354-366] in the case of one-parameter SDEs with singular drift. |
| title | Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields |
| topic | Probability Primary 60H07, 60H50, 60H17, Secondary 60H15 |
| url | https://arxiv.org/abs/2212.08466 |