ON MARTINGALE SOLUTIONS OF FRACTIONAL STOCHASTIC EVOLUTION EQUATIONS

Abstract

We study the Cauchy problem for a fractional stochastic evolution equation driven by a cylindrical Wiener process, in which the time derivative is understood in the Caputo sense of order q ∈ (0, 1) and the noise enters through a Riemann–Liouville fractional integral. After collecting the required notions on progressively measurable processes, martingales and Hilbert–Schmidt operators, we establish regularity properties of the fractional Itˆo integral and give sufficient conditions for its existence, together with a Burkholder-type inequality. The solution operator families generated by the Mittag-Leffler function are used to formulate the notion of a mild solution and then of a martingale solution. The main result asserts the existence of a martingale solution of the problem under compactness and measurability assumptions on the coefficients.