White Noise Perturbation

Evolutionary integral equations as appearing in the theory of linear parabolic viscoelasticity are studied in the presence of white noise. It is shown that the stochastic convolution leads to regular solutions, and that under suitable assumptions the samples are HH older-continuous. These results are put in a wider perspective by consideration of equations with fractional derivatives which are also studied in this paper. This way, known results are recovered and put into broader perspective. 1. Statement of the problem Let H be a separable Hilbert space, A a closed linear densely deened operator in H, and b 2 L 1;loc (R +) a scalar kernel. In this paper we consider the integro-diierential equations _ u(t) + Z t 0 b(t ?)Au()dd = f(t); t 0; u(0) = u 0 ; (1) on the halline, and _ v(t) + Z t ?1 b(t ?)Av()dd = g(t); t 2 R;