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2014•Illinois Journal of MathematicsOpen access

Kolmogorov operator and Fokker–Planck equation associated to a stochastic Burgers equation driven by Lévy noise

Bing Hu, Xiaobin Sun, Yingchao Xie

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Abstract

In this paper, we consider a stochastic Burgers equation driven by Lévy noise and study the transition semigroup of the solution to the initial value problem for the equation in the space of continuous functions weighted by a proper potential. We show that the infinitesimal generator is the closure of the Kolmogorov operator associated to the equation in a suitable topology. We also prove existence and uniqueness results for the associated Fokker–Planck equation.

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What this paper is about

In this paper, we consider a stochastic Burgers equation driven by Lévy noise and study the transition semigroup of the solution to the initial value problem for the equation in the space of continuous functions weighted by a proper potential. We show that the infinitesimal generator is the closure of the Kolmogorov operator associated to the equation in a suitable topology. We also prove existence and uniqueness results for the associated Fokker–Planck equation.

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Available abstract

In this paper, we consider a stochastic Burgers equation driven by Lévy noise and study the transition semigroup of the solution to the initial value problem for the equation in the space of continuous functions weighted by a proper potential. We show that the infinitesimal generator is the closure of the Kolmogorov operator associated to the equation in a suitable topology. We also prove existence and uniqueness results for the associated Fokker–Planck equation.

Key concepts: Fokker–Planck equation, Burgers' equation, Mathematics, Operator (biology), Noise (video), Mathematical physics, First-hitting-time model, Mathematical analysis

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