Mathematical Research Letters

Volume 1 (1994)

Number 4

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

Pages: 451 – 464



Terry Lyons (Imperial College)


L.C. Young proved that if $x_{t}, y_{t}$ are continuous paths of finite $p$, $p^\prime$ variations in ${\Bbb R}^{d}$ where ${1\over p} + {1\over p^\prime } > 1$ then the integral $\int ^{t}_{0} y_{u}\,dx_{u}$ can be defined. It follows that if $p = p^\prime <2$, and $f$ is vector valued and $\alpha $-Lipschitz function with $\alpha > p - 1$, one may consider the non-linear integral equation and the associated differential equation: $$\align y_{t} &=a+\int_0^t\sum_{i=1}^{d}f^{i}(y_{u}) \,dx_{u}^{i} dy_{t} &= \sum_{i=1}^{d}f^{i}(y_{t}) \,dx_{t}^{i}\qquad y_{0}= a. \tag1\endalign$$ 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 in the sense of [7] 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.

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