Space Regularity of Evolution Equations Driven by Rough Paths

In this paper, we consider the linear evolution equation dy(t)=Ay(t)dt+∑i=1dGiy(t)dxi(t), where A is a closed operator, associated to a semigroup, with good smoothing effects in a Banach space E, x is a nonsmooth Rd-path, which is η-Hölder continuous for some η∈13,12, and Gi (i=1,…,d) is a non-smoothing linear operator on E. We prove that the Cauchy problem associated with the previous equation admits a unique mild solution and we also show that the solution increases the regularity of the initial datum as soon as time evolves. Then, we show that the mild solution is also an integral solution and this allows us to prove an Itô formula.