Regularity results for nonlinear Young equations and applications

In this paper we provide sufficient conditions which ensure that the nonlinear equation d y(t) = Ay(t) d t+ σ(y(t)) d x(t) , t∈ (0 , T] , with y(0) = ψ and A being an unbounded operator, admits a unique mild solution such that y(t) ∈ D(A) for any t∈ (0 , T] , and we compute the blow-up rate of the norm of y(t) as t→ 0 +. We stress that the regularity of y is independent of the smoothness of the initial datum ψ, which in general does not belong to D(A). As a consequence we get an integral representation of the mild solution y which allows us to prove a chain rule formula for smooth functions of y.