Differential equations driven by variable order Holder noise and the regularizing effect of delay

In this article, we extend the framework of rough paths to processes of variable Holder exponent or variable order paths. A typical example of such paths is the multifractional Brownian motion, where the Hurst parameter of a fractional Brownian motion is considered to be a function of time. We show how a class of multiple discrete delay differential equations driven by signals of variable order are especially well suited to be studied in a pathwise sense. In fact, under some assumptions on the Holder regularity functions of the driving signal, the local Holder regularity of the variable order signal may be close to zero, without more than two times differentiable diffusion coefficients (in contrast to higher regularity requirements known from constant order rough path theory). Furthermore, we give a canonical algorithm to construct the iterated integral of variable order on the domain unit square, by constructing the iterated integrals on some well-chosen strip around the diagonal of unit square, and then extending it to the whole domain using Chen's relation. We show how we can apply this to construct the rough path for a multifractional Brownian motion. At last, we generalize some fundamental results relating to differential equations from rough paths theory to the Holder spaces of variable order.