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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.

##### MSC:
 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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