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Stochastic evolution equations driven by Liouville fractional Brownian motion

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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 ℝd, 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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Authors and Affiliations

  1. Department of Mathematics, University of York, York, YO10 5DD, UK

    Zdzisław Brzeźniak

  2. Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Jan van Neerven

  3. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Donna Salopek

Authors
  1. Zdzisław Brzeźniak
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  2. Jan van Neerven
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  3. Donna Salopek
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Correspondence to Zdzisław Brzeźniak.

Additional information

The first named author thanks Eurandom for kind hospitality. The second named author gratefully acknowledges financial support by VICI subsidy 639.033.604 of the Netherlands Organization for Scientific Research (NWO).

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Brzeźniak, Z., van Neerven, J. & Salopek, D. Stochastic evolution equations driven by Liouville fractional Brownian motion. Czech Math J 62, 1–27 (2012). https://doi.org/10.1007/s10587-012-0011-z

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