On the Ambrosio–Figalli–Trevisan Superposition Principle for Probability Solutions to Fokker–Planck–Kolmogorov Equations
On the Ambrosio–Figalli–Trevisan Superposition Principle for Probability Solutions to Fokker–Planck–Kolmogorov Equations
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DOI：
10.1007/s10884-020-09828-5
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发表时间：
2019-03
影响因子：
1.3
通讯作者：
V. Bogachev;M. Röckner;S. V. Shaposhnikov
中科院分区：
文献类型：
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作者：
V. Bogachev;M. Röckner;S. V. Shaposhnikov
我们证明了对于Fokker-Planck-Kolmogorov方程Cauchy问题概率解的ambrosio - figallii - Trevisan叠加原理的Trevisan已知结果的推广，根据该原理，具有初始分布的解可以用路径空间的一个概率度量来表示，该路径空间解决了相应的鞅问题，并且是时间的一维分布。新颖之处在于我们要求的是的可积性，而不是扩散系数和漂移系数对解的可积性。因此，在不存在先验全局可积条件的情况下，函数可以是二次增长的。这是这个方向上第一个适用于无界系数的结果，没有任何先验的全局可积条件。此外，我们还证明了在初始分布的温和条件下，伴随有单侧边界是充分的。
We prove a generalization of the known result of Trevisan on the Ambrosio–Figalli–Trevisan superposition principle for probability solutions to the Cauchy problem for the Fokker–Planck–Kolmogorov equation, according to which such a solutionwith initial distributionis represented by a probability measureon the path space such thatsolves the corresponding martingale problem andis the one-dimensional distribution ofat timet. The novelty is that in place of the integrability of the diffusion and drift coefficientsAandbwith respect to the solution we require the integrability of. Therefore, in the case where there are no a priori global integrability conditions the functioncan be of quadratic growth. This is the first result in this direction that applies to unbounded coefficients without any a priori global integrability conditions. Moreover, we show that under mild conditions on the initial distribution it is sufficient to have the one-sided boundalong with.