论文标题
奇异分数SDE的扰动
Perturbations of singular fractional SDEs
论文作者
论文摘要
我们为一类带有单数漂移和添加剂分数噪声的ODE获得适当的结果，其右侧涉及一些有界变化项，具体取决于溶液。此类方程式的示例是反射的方程式，其中解决方案被约束以保留在矩形域，以及所谓的扰动方程，其中动力学取决于解决方案的运行极端。我们的证明是基于基于年轻的非线性整合的Catellier-Gubinelli方法，其中一些Lipschitz估计了Skorokhod类型地图的$ p $变化，这是由于Falkowski和Słominski而导致的。一个重要的步骤需要证明，当由足够规则的路径（在$ p $变化的意义上）扰动时，布朗尼运动将保留其正则化属性。这是通过应用随机缝纫引理的变体来完成的。
We obtain well-posedness results for a class of ODE with a singular drift and additive fractional noise, whose right-hand-side involves some bounded variation terms depending on the solution. Examples of such equations are reflected equations, where the solution is constrained to remain in a rectangular domain, as well as so-called perturbed equations, where the dynamics depend on the running extrema of the solution. Our proof is based on combining the Catellier-Gubinelli approach based on Young nonlinear integration, with some Lipschitz estimates in $p$-variation for maps of Skorokhod type, due to Falkowski and Słominski. An important step requires to prove that fractional Brownian motion, when perturbed by sufficiently regular paths (in the sense of $p$-variation), retains its regularization properties. This is done by applying a variant of the stochastic sewing lemma.
