Analytic aspects of optimal transportation

• Applicant: Professor Dr. Michael Röckner
• Subject Area: Mathematics
• Term: from 2016 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 316972354

• Final Report Year: 2020

Final Report Abstract

During the work on the project, its main goals have been achieved and the announced programme has been successfully fulfilled. Considerable progress has been achieved in the study of optimal transportation of measures and in the investigation of a large circle of related problems in the theory of Fokker–Planck–Kolmogorov equations, infinite dimensional analysis and measure theory. A fruitful cooperation between the groups involved in the project resulted in solving several important problems and opening new perspectives of research, which forms a basis of future cooperation and continuation of the collaboration.

Publications

• Estimates of distances between solutions of Fokker–Planck–Kolmogorov equations with partially degenerate diffusion matrices. Theory of Stochastic Processes. 2018. V. 23, N 2. P. 41–54
  O.A. Manita, M.S. Romanov, S.V. Shaposhnikov
  
• Poincaré and Brunn-Minkowski inequalities on the boundary of weighted Riemannian manifolds. American Journal of Mathematics. 2018. V. 140, N 5. P. 1147–1185
  A.V. Kolesnikov, E. Milman
  (See online at https://doi.org/10.1353/ajm.2018.0027)
• The KLS isoperimetric conjecture for generalized Orlicz balls. Annals of Probability. 2018. V. 46, N 6. P. 3578–3615
  A.V. Kolesnikov, E. Milman
  (See online at https://doi.org/10.1214/18-AOP1257)
• The Poisson equation and estimates for distances between stationary distributions of diffusions. Journal of Mathematical Sciences (New York). 2018. V. 232, N 3. P. 254–282
  V.I. Bogachev, M. Röckner, S.V. Shaposhnikov
  (See online at https://doi.org/10.1007/s10958-018-3872-3)
• Convergence in variation of solutions of nonlinear Fokker–Planck–Kolmogorov equations to stationary measures. Journal of Functional Analysis. 2019. V. 276, N 12. P. 3681–3713
  V.I. Bogachev, M. Röckner, S.V. Shaposhnikov
  (See online at https://doi.org/10.1016/j.jfa.2019.03.014)
• Extremal Kähler–Einstein metric for two-dimensional convex bodies. Journal of Geometric Analysis. 2019. V. 29, N 3. P. 2347–2373
  B. Klartag, A.V. Kolesnikov
  (See online at https://doi.org/10.1007/s12220-018-0077-4)
• On the Ambrosio–Figalli–Trevisan superposition principle for probability solutions to Fokker–Planck–Kolmogorov equations. Journal of Dynamics and Differential Equations. 2020
  V.I. Bogachev, M. Röckner, S.V. Shaposhnikov
  (See online at https://doi.org/10.1007/s10884-020-09828-5)
• Representations of solutions to Fokker–Planck–Kolmogorov equations with coefficients of low regularity. Journal of Evolution Equations. 2020. V. 20, N 2
  V.I. Bogachev, S.V. Shaposhnikov
  (See online at https://doi.org/10.1007/s00028-019-00532-6)
• On the Gardner–Zvavitch conjecture: symmetry in the inequalities of Brunn–Minkowski type. Advances in Mathematics. 2021
  A.V. Kolesnikov, G. Livshyts
  (See online at https://doi.org/10.1016/j.aim.2021.107689)
• Total variation distance estimates via L2-norm for polynomials in log-concave random vectors. International Mathematics Research Notices. 2021
  E.D. Kosov
  (See online at https://doi.org/10.1093/imrn/rnz278)