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

Abstract

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 ) ∈ C β ( [ 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 ) .