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<p>We show that under broad assumptions, a probability solution to the Cauchy problem for the Fokker–Planck–Kolmogorov equation with a given initial distribution enables one to uniquely determine the diffusion matrix and the drift coefficient. In particular, this can be done if the coefficients satisfy certain global integrability condition with respect to the solution and are continuous or if the diffusion matrix is nondegenerate and sufficiently regular. Actually, the main result is formulated in terms of a technical approximability condition introduced in order to cover different cases in a unified way. The suggested reconstruction method employs the superposition principle and is based on the corresponding martingale problem for a single initial condition.</p>

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On reconstruction of coefficients of Fokker–Planck–Kolmogorov equations

  • 期刊论文

  • 发表日期：

    2025年1月3日

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  • Vladimir I. Bogachev,
  • Stanislav V. Shaposhnikov

摘要

We show that under broad assumptions, a probability solution to the Cauchy problem for the Fokker–Planck–Kolmogorov equation with a given initial distribution enables one to uniquely determine the diffusion matrix and the drift coefficient. In particular, this can be done if the coefficients satisfy certain global integrability condition with respect to the solution and are continuous or if the diffusion matrix is nondegenerate and sufficiently regular. Actually, the main result is formulated in terms of a technical approximability condition introduced in order to cover different cases in a unified way. The suggested reconstruction method employs the superposition principle and is based on the corresponding martingale problem for a single initial condition.