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Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation

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Abstract

The present work deals with the global solvability as well as absolute continuity of the law of the solution to stochastic generalized Burgers–Huxley (SGBH) equation driven by multiplicative space-time white noise in a bounded interval of \({\mathbb {R}}\). We first prove the existence of a unique local mild solution to SGBH equation with the help of a truncation argument and contraction mapping principle. Then global solvability results are obtained by using uniform bounds of the local mild solution and stopping time arguments. Later, we establish a comparison theorem for the solution of SGBH equation having higher order nonlinearities and it plays a crucial role in this work. Then, we discuss the weak differentiability of the solution to SGBH equation in the Malliavin calculus sense. Finally, we obtain the absolute continuity of the law of the solution with respect to the Lebesgue measure on \({\mathbb {R}}\), and the existence of density with the aid of comparison theorem and weak differentiability of the solution.

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Acknowledgements

The first author would like to thank Ministry of Education, Government of India - MHRD for financial assistance. M. T. Mohan would like to thank the Department of Science and Technology (DST), India for Innovation in Science Pursuit for Inspired Research (INSPIRE) Faculty Award (IFA17-MA110). The authors sincerely would like to thank the reviewers for their valuable comments and suggestions, which helped us to improve the manuscript significantly, especially Proposition 3.5.

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  1. Department of Mathematics, Indian Institute of Technology Roorkee-IIT Roorkee, Haridwar Highway, Roorkee, Uttarakhand, 247667, India

    Ankit Kumar & Manil T. Mohan

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  1. Ankit Kumar
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Kumar, A., Mohan, M.T. Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation. Stoch PDE: Anal Comp (2023). https://doi.org/10.1007/s40072-023-00308-7

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