Paths of some stochastic processes such as fractional Brownian Motion have some amazing regularization properties. It is well known that in order to have uniqueness in differential systems such as
dy_t = b(y_t) dt,
b needs to be quite regular. However, the oscillations of a stochastic process added to the system will guarantee uniqueness for really irregular b. In this talk we will show how to solve the perturbed differential system with a certain stochastic averaging operator. As an application, we show that the stochastic transport equation driven by fractional Brownian motion has a unique solution when u0∈L∞ and b is a possibly random α-Hölder continuous function for α large enough.
This is a joint work with Massimiliano Gubinelli.