Young integrals and SPDEs

摘要

In this note, we study the non-linear evolution problem dY(t) = - AY(t)dt + B(Y-t) dX(t), where X is a gamma-Holder continuous function of the time parameter, with values in a distribution space, and - A the generator of an analytical semigroup. Then, we will give some sharp conditions on X in order to solve the above equation in a function space, first in the linear case ( for any value of gamma in ( 0, 1)), and then when B satisfies some Lipschitz type conditions ( for gamma > 1/2). The solution of the evolution problem will be understood in the mild sense, and the integrals involved in that definition will be of Young type.