Well-posedness of stochastic KdV–BO equation driven by fractional Brownian motion

The Cauchy problem for the Korteweg–de Vries Benjamin–Ono equation driven by cylindrical fractional Brownian motion is discussed in this paper. Fractional Brownian motion is a family of processes BHBH. It is known that the smaller the value of Hurst parameter H is, the worse of the regularity of fBm is. Using Bourgain restriction method, we obtain the lower bound of the Hurst parameter H   for the driving processes BHBH. With H>38, we prove local existence results with initial value in classical Sobolev spaces of negative indices, i.e. HsHs with s⩾-18.