Uniqueness for nonlinear Fokker-Planck equations with general diffusion terms and their associated nonlinear Markov processes
Abstract
This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type ut−∑i,j=1d D2ij(aij(x)β(x,u))+ div(b(x,u)u)=0 in (0, ∞) × ℝd ,notag with initial condition u(0,x)≡ u0(x), where aij, β, and b are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup S(t): L1(ℝd) → L1(ℝd), and u(t)=S(t)u0 is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case aij ≡ δij. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new L∞ estimate for mild solutions starting from data in L1∩ L∞ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.
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BibTeX
@misc{barbu2026uniqueness,
title = {{Uniqueness for nonlinear Fokker-Planck equations with general diffusion terms and their associated nonlinear Markov processes}},
author = {Viorel Barbu and Yuqi Li and Michael Röckner},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.15903},
eprint = {2607.15903},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.15903},
}
RIS
TY - UNPB
TI - Uniqueness for nonlinear Fokker-Planck equations with general diffusion terms and their associated nonlinear Markov processes
AU - Viorel Barbu
AU - Yuqi Li
AU - Michael Röckner
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2607.15903
UR - https://arxiv.org/abs/2607.15903
AB - This work is concerned with the uniqueness of distributional solutions to nonlinear Fokker-Planck equations with non-diagonal diffusion terms of type u_t−∑_i,j=1^d D^2_ij(a_ij(x)β(x,u))+ div(b(x,u)u)=0 in (0, ∞) × ℝ^d ,notag with initial condition u(0,x)≡ u_0(x), where a_ij, β, and b are suitable functions. Under suitable assumptions, this equation generates a continuous contraction semigroup S(t): L^1(ℝ^d) → L^1(ℝ^d), and u(t)=S(t)u_0 is a mild solution to the equation. Our main contribution is to prove that this mild solution is unique in the much larger class of distributional solutions. This extends previous uniqueness results for the diagonal (also called isotropic) diffusion case a_ij ≡ δ_ij. Another key analytical result of this paper is the uniqueness for distributional solutions of the associated linearized equation. As a main application, we prove weak uniqueness for the corresponding McKean-Vlasov SDEs. Moreover, we prove that, the probabilistically weak solution to the McKean-Vlasov SDEs is also the unique probabilistically strong solution. Furthermore, we establish a new L^∞ estimate for mild solutions starting from data in L^1∩ L^∞ and this estimate is used in the construction of nonlinear Markov processes. Finally, we prove that the path laws of the solutions to the McKean-Vlasov SDEs form a nonlinear Markov process in the sense of McKean.
ER -