Differential equations driven by rough signals. I: An extension of an inequality of L. C. Young.

*(English)*Zbl 0835.34004Summary: L. C. Young [Acta Math. 67, 251-282 (1936)] proved that if \(x_t,y_t\) are continuous paths of finite \(p,p'\) variations in \(\mathbb{R}^d\) where \({1 \over p} + {1 \over p'} > 1\) then the integral \(\int^t_0 y_udx_u\) can be defined. It follows that if \(p = p' < 2\), and \(f\) is a vector valued and \(\alpha\)-Lipschitz function with \(\alpha > p - 1\), one may consider the nonlinear integral equation and the associated differential equation:
\[
y_t = a + \int^t_0 \sum^d_{i = 1} f^i(y_u) dx^i_u, \quad dy_t = \sum^d_{i = 1} f^i(y_t) dx^i_t,\;y_0 = a.
\]
If one fixes \(x\) one may ask about the existence and uniqueness of \(y\) with finite \(p\)-variation where to avoid triviality we assume \(d > 1\). We prove that if each \(f^i\) is \((1 + \alpha)\)-Lipschitz then a unique solution exists and that it can be recovered as a limit of Picard iterations; in consequence it varies continuously with \(x\). If each \(f^i\) is \(\alpha\)-Lipschitz, one still has existence of solutions, but examples of A. M. Davie show that they are not, in general, unique.