We prove a new uniqueness result for highly degenerate second-order parabolic equations on the whole space. A novelty is also our class of solutions in which uniqueness holds.
We prove a new uniqueness result for solutions to the Cauchy problem for highly degenerate second order parabolic Fokker-Planck-Kolmogorov equations on the whole space. A novelty is also our class of solutions in which uniqueness holds. This result considerably improves a number of previously known uniqueness theorems in the theory of parabolic equations and is of principal importance for the study of uniqueness of solutions to degenerate Fokker-Planck-Kolmogorov equations and uniquness of solutions to martingale problems.
We prove a formula representing solutions to parabolic Fokker-Planck-Kolmogorov equations with coefficients of low regularity. This formula is applied for proving the continuity of solution densities under broad assumptions and obtaining upper bounds for them. In the case of diffusion coefficients of class VMO, we show that the solution density is locally integrable to any power.