### On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations

Bogachev VI, Röckner M, Shaposhnikov SV (2020)
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS.

Zeitschriftenaufsatz | E-Veröff. vor dem Druck | Englisch

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Autor*in
Bogachev, Vladimir I.; Röckner, MichaelUniBi; Shaposhnikov, Stanislav V.
Einrichtung
Abstract / Bemerkung
We prove a generalization of the known result of Trevisan on the Ambrosio-FigalliTrevisan superposition principle for probability solutions to the Cauchy problem for the Fokker-Planck-Kolmogorov equation, according to which such a solution {mu t} with initial distribution. is represented by a probability measure P. on the path space such that P. solves the corresponding martingale problem and mu t is the one-dimensional distribution of P. at time t. The novelty is that in place of the integrability of the diffusion and drift coefficients A and b with respect to the solution we require the integrability of ( A(t, x)+|b(t, x), x |)/(1+|x|2). Therefore, in the casewhere there are no a priori global integrability conditions the function A(t, x) + |b(t, x), x | can 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 bound b(t, x), x = C+ C|x|2 log |x| along with A(t, x) = C + C|x|2 log vertical bar x vertical bar.
Stichworte
Fokker-Planck-Kolmogorov equation; Martingale problem; Superposition; principle
Erscheinungsjahr
2020
Zeitschriftentitel
JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS
ISSN
1040-7294
eISSN
1572-9222
Page URI
https://pub.uni-bielefeld.de/record/2941889

### Zitieren

Bogachev VI, Röckner M, Shaposhnikov SV. On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. 2020.
Bogachev, V. I., Röckner, M., & Shaposhnikov, S. V. (2020). On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. doi:10.1007/s10884-020-09828-5
Bogachev, V. I., Röckner, M., and Shaposhnikov, S. V. (2020). On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS.
Bogachev, V.I., Röckner, M., & Shaposhnikov, S.V., 2020. On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS.
V.I. Bogachev, M. Röckner, and S.V. Shaposhnikov, “On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations”, JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS, 2020.
Bogachev, V.I., Röckner, M., Shaposhnikov, S.V.: On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS. (2020).
Bogachev, Vladimir I., Röckner, Michael, and Shaposhnikov, Stanislav V. “On the Ambrosio-Figalli-Trevisan Superposition Principle for Probability Solutions to Fokker-Planck-Kolmogorov Equations”. JOURNAL OF DYNAMICS AND DIFFERENTIAL EQUATIONS (2020).

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