The local exponential stability of evolution equation driven by Hölder-continuous paths

In this paper, the semigroup S(t) generated by unbounded linear operator A is not Hölder-continuous at zero. By assuming the regularity of initial condition, the mild solution u(t)∈Cβ([0,T];V) is obtained. Then the local exponential stability of evolution equations driven by Hölder-continuous paths with Hölder exponent H∈(1∕2,1) is established. This result can be directly applied to the evolution equations with fractional Brownian motion with Hurst parameter H∈(1∕2,1).