论文标题
反射的粗微分方程的非唯一性
Non-uniqueness for reflected rough differential equations
论文作者
论文摘要
我们举例说明反射的差异方程式，如果驾驶信号足够粗糙，则可能具有无限的解决方案（例如，无限$ p $ - 差异，对于一些$ p> 2 $）。对于此方程，我们确定了唯一性所具有的信号连续性模量的鲜明条件。莱维（Lévy）的布朗运动模量证明是一个边界案例。我们进一步表明，在我们的示例中，当驾驶信号是赫斯特索引$ h <\ frac {1} {2} $时，几乎肯定会肯定会有肯定会保持。所考虑的方程是由具有有界变化的一个组成部分的二维信号驱动的，因此不需要粗糙的路径理论来理解方程。
We give an example of a reflected diffferential equation which may have infinitely many solutions if the driving signal is rough enough (e.g. of infinite $p$-variation, for some $p>2$). For this equation, we identify a sharp condition on the modulus of continuity of the signal under which uniqueness holds. Lévy's modulus for Brownian motion turns out to be a boundary case. We further show that in our example, non-uniqueness holds almost surely when the driving signal is a fractional Brownian motion with Hurst index $H < \frac{1}{2}$. The considered equation is driven by a two-dimensional signal with one component of bounded variation, so that rough path theory is not needed to make sense of the equation.
