Quantitative stability estimates for Fokker–Planck equations

Abstract

We consider the Fokker-Planck equations with irregular coefficients. Two different cases are treated: in the degenerate case, the coefficients are assumed to be weakly differentiable, while in the non-degenerate case the drift satisfies only the Ladyzhenskaya-Prodi-Serrin condition. Using Trevisan's superposition principle which represents the solution as the marginal of the solution to the martingale problem of the diffusion operator, we establish quantitative stability estimates for the solutions of Fokker-Planck equations.MSC 2010: primary 35Q84; secondary 60H10

On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity

“…One can compare the rate given by Proposition 7.4 with the ones in [30] and [34]. The rate in (7.5) depends upon φ u 0 and cannot be better than O 1 | ln |εn−ε|| , but provides convergence in the strong norm C([0, T ]; L 1 (T d )).…”

“…The rate in (7.5) depends upon φ u 0 and cannot be better than O 1 | ln |εn−ε|| , but provides convergence in the strong norm C([0, T ]; L 1 (T d )). On the other hand, the rates of [30] and [34] are of order |ε n − ε| and |ε n − ε|, respectively, but they are given for a logarithmic distance which instead metrizes weak convergence.…”

Stability estimates for invariant measures of diffusion processes, with applications to stability of moment measures and Stein kernels

“…Our focus will be on the latter, and arguably more challenging, case. As we shall later discuss, the techniques can be carried over to the setting of stochastic differential equations we are interested in here, as worked out in [8,32].…”

“…The method used here is an adaptation of a technique developed by Crippa and De Lellis for transport equations. That method was introduced in the SDE setting in [8,32]. Our implementation here will be a bit different, to allow for estimates in weighted Sobolev space that behave better for large times, and will allow us to compare the invariant measures of the two processes, under suitable ergodic assumptions.…”

Lusin-type approximation of Sobolev by Lipschitz functions, in Gaussian and RCD(K,∞) spaces

“…One could also obtain quantitative bounds in terms of suitable transportation distances, following e.g. the approach in [26] (in a stochastic setting). We leave to future research possible applications in the theory of flows in RCD spaces, for which well-posedness and stability, not in quantitative form, are established respectively in [9] and [11].…”