Stochastic evolution equations with fractional Brownian motion
The authors consider the equation \[ X(dt)=AX(t)\,dt+\Phi \,dB^H_t. \] Here \(B^H_t\) is infinite-dimensional fractional Brownian motion, \(A\) is an unbounded linear operator and \(\Phi\) is a bounded linear operator in the corresponding Hilbert space. The main theorems give conditions for the existence of the so-called mild solution, i.e. for \[ X(t)=e^{tA}x+\int _0^te^{(t-s)A}\Phi \,dB^H(s), \] and for the regularity of this solution. Regularity now means that \(X(t)\) belongs to the domain of definition of \((-A)^\gamma\) and is continuous in the corresponding norm. The authors use the integral representation of the fractional Brownian motion with the help of Wiener process and estimate the integral kernel for \(X.\)
scientific article; zbMATH DE number 2018319