Stochastic evolution equations driven by Liouville fractional Brownian motion

Abstract

Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ½ we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations $$dU(t) = AU(t)dt + B dW_H^\beta (t)$$ driven by an H-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space E, the operators A: D(A) → E and B: H → E, and the Hurst parameter. As an application it is shown that second-order parabolic SPDEs on bounded domains in ℝ d , driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if ¼d < β < 1.

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