论文标题
用$ l^{\ infty} $ - 向量字段的双曲线随机部分微分方程的解决方案的平滑度
Smoothness of solutions of hyperbolic stochastic partial differential equations with $L^{\infty}$-vector fields
论文作者
论文摘要
在本文中，当矢量场仅界限和可测量时，我们对准线性双曲线随机微分方程（HSPDE）感兴趣。尽管这种方程式的确定性对应物可能是不适合的（从某种意义上说，唯一性甚至存在可能无效），但我们首次表明相应的HSPDE具有独特的（Malliavin可区分）的强大解决方案。我们证明此结果的方法基于：1）Malliavin微积分的工具和2）[davie，int。数学。 res。不，第1卷。 2007]通过使用算法选择某些矩形，非微不足道扩展到平面中的SDE情况。作为产品，我们还获得了解决方案相对于其初始值的Sobolev的可不同性。此处得出的结果构成了平面上有关SDE的当前文献中的结果的显着改善，并且可以被视为[Zvonkin，Math]的开拓性作品的类似等效物。 Urss Sbornik，22：129-149]和[Veretennikov，理论概率。 Appl。，24：354-366]，如果是具有单数漂移的单参数SDE。
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.
