A probabilistic approach to Volterra equations in Banach spaces (Q1198700)

The author considers the linear Volterra equation \[ u(t) = \varphi + \int_{[0,t]} Au(t-s) W(ds), \quad t \geq 0, \tag{*} \] in a real Banach space \(X\) where \(A\) is the generator of a \(C_ 0\)-semigroup. It is assumed that \(W\) satisfies \[ aW \bigl( [0,t] \bigr) = \int_ 0^ t k(t- s) W \bigl( [0,s] \bigr) ds, \] where \(a \geq 0\) and \(k\) is nonnegative and nonincreasing. It is shown how \(a\) and \(k\) determine a stochastic process \(Z\) with nondecreasing almost surely continuous paths having stationary independent increments and then it is proved that \[ W(B) = \int_ 0^ \infty \mathbb{P} \bigl( Z(s) \in B \bigr) ds. \] Using this result the author gives the relation between the solutions of \((*)\) with certain different kernels \(W\) in a formula involving the expected values of a stochastic process.