Differential equations driven by rough signals. I: An extension of an inequality of L. C. Young. (English) Zbl 0835.34004

Summary: 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.


34A12 Initial value problems, existence, uniqueness, continuous dependence and continuation of solutions to ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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