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Hitting properties of generalized fractional kinetic equation with time-fractional noise

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Abstract

This paper investigates the hitting properties of a system of generalized fractional kinetic equations driven by Gaussian noise that is fractional in time and either white or colored in space. The considered model encompasses various examples such as the stochastic heat equation and the stochastic biharmonic heat equation. Under relatively general conditions, we derive the mean square modulus of continuity and explore certain second order properties of the solution. These are then utilized to deduce lower and upper bounds for probabilities that the path process hits bounded Borel sets in terms of the \(\mathfrak {g}_q\)-capacity and \(g_q\)-Hausdorff measure, respectively, which reveal the critical dimension for hitting points. Furthermore, by introducing the harmonizable representation of the solution and utilizing it to construct a family of approximating random fields which have certain smoothness properties, we prove that all points are polar in the critical dimension. This provides a compelling evidence supporting the conjecture raised in Hinojosa-Calleja and Sanz-Solé (Stoch Part Differ Equ Anal Comput 10(3):735–756, 2022. https://doi.org/10.1007/s40072-021-00234-6).

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Acknowledgements

The authors thank the anonymous referees for an in-depth review of the initial version of the manuscript which led to several improvements.

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Authors and Affiliations

  1. Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China

    Derui Sheng & Tau Zhou

  2. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, China

    Derui Sheng & Tau Zhou

  3. Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong

    Derui Sheng & Tau Zhou

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This work is supported by National key R &D Program of China under Grant (No. 2020YFA0713701) and National Natural Science Foundation of China (Nos. 11971470, 12031020, 12171047).

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Sheng, D., Zhou, T. Hitting properties of generalized fractional kinetic equation with time-fractional noise. Stoch PDE: Anal Comp (2023). https://doi.org/10.1007/s40072-023-00321-w

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