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Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise

Author

Listed:
  • Kaiyuqi Guan

    (School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430062, China)

  • Yu Shi

    (School of Mathematics and Statistics, Wuhan University of Technology, Wuhan 430062, China)

Abstract

Reaction–diffusion equations can model complex systems where randomness plays a role, capturing the interaction between diffusion processes and random fluctuations. The Kolmogorov equations associated with these systems play an important role in understanding the long-term behavior, stability, and control of such complex systems. In this paper, we investigate the existence of a classical solution for the Kolmogorov equation associated with a stochastic reaction–diffusion equation driven by nonlinear multiplicative trace-class noise. We also establish the existence of an invariant measure ν for the corresponding transition semigroup P t , where the infinitesimal generator in L 2 ( H , ν ) is identified as the closure of the Kolmogorov operator K 0 .

Suggested Citation

  • Kaiyuqi Guan & Yu Shi, 2025. "Kolmogorov Equation for a Stochastic Reaction–Diffusion Equation with Multiplicative Noise," Mathematics, MDPI, vol. 13(10), pages 1-19, May.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:10:p:1561-:d:1652429
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    References listed on IDEAS

    as
    1. Qiyuan Zhou & Shuhua Gong, 2012. "The Existence and Uniqueness of Periodic Solutions for Some Nonlinear th-Order Differential Equations," Abstract and Applied Analysis, Hindawi, vol. 2012, pages 1-17, May.
    2. Qiyuan Zhou & Shuhua Gong, 2012. "The Existence and Uniqueness of Periodic Solutions for Some Nonlinear nth‐Order Differential Equations," Abstract and Applied Analysis, John Wiley & Sons, vol. 2012(1).
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