Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations

Zhang, Xicheng

doi:10.48550/arxiv.2109.01273

Cited by 3 publications

(3 citation statements)

References 43 publications

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“…We remark that the above mixed L p -norm was first introduced in [2]. Recently, it was essentially used in [37] to study the strong convergence of propagation of chaos for particle systems with singular interactions (see also [52], [77]). We emphasize that the order of (p x , p v ) in definition (1.16) is important because the above L p -norm is invariant under the group action Γ…”

Section: Introductionmentioning

confidence: 99%

Second order fractional mean-field SDEs with singular kernels and measure initial data

Hao¹,

Röckner²,

Zhang³

2023

Preprint

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In this paper we establish the local and global well-posedness of weak and strong solutions to second order fractional mean-field SDEs with singular/distribution interaction kernels and measure initial value, where the kernel can be Newton or Coulomb potential, Riesz potential, Biot-Savart law, etc. Moreover, we also show the stability, smoothness and the short time singularity and large time decay estimates of the distribution density. Our results reveal a phenomenon that for nonlinear mean-field equations, the regularity of the initial distribution could balance the singularity of the kernel. The precise relationship between the singularity of kernels and the regularity of initial value are calculated, which belongs to the subcritical regime in scaling sense. In particular, our results provide microscopic probability explanation and establish a unified treatment for many physical models such as fractional Vlasov-Poisson-Fokker-Planck system, the vorticity formulation of 2D-fractal Navier-Stokes equations, surface quasi-geostrophic models, fractional porous medium equation with viscosity, etc.

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Section: Introductionmentioning

confidence: 99%

Second order fractional mean-field SDEs with singular kernels and measure initial data

Hao¹,

Röckner²,

Zhang³

2023

Preprint

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“…Recently, Wang [44] studied the weak and strong well-posedness for more general density-distribution dependent SDEs with singular coefficients by fixed point argument, but not mixed L p -drifts. In [16] and [51], we also studied the well-posedness of second order McKean-Vlasov SDEs with singular drifts. Nowadays, there are vast literatures about the McKean-Vlasov or mean-field SDEs.…”

Section: Introductionmentioning

confidence: 99%

Strong convergence of propagation of chaos for McKean-Vlasov SDEs with singular interactions

Hao¹,

Röckner²,

Zhang³

2022

Preprint

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In this work we show the strong convergence of propagation of chaos for the particle approximation of McKean-Vlasov SDEs with singular L p -interactions as well as the moderate interaction particle system in the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of particle system with singular interaction. In particular, we develop the theory of strong well-posedness of Krylov and Röckner [25] in the case of mixed L p -drifts, where the heat kernel estimates play a crucial role. Moreover, when the interaction kernel is bounded measurable, we also obtain the optimal rate of strong convergence, which is partially based on Jabin and Wang's entropy method [19] and Zvonkin's transformation.

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“…On the other hand, the McKean-Vlasov stochastic differential equations (SDEs), presented in [10], can be used to characterize the nonlinear Fokker-Planck-Kolmogorov equations. Recently, there are plentiful results on the study of McKean-Vlasov SDEs, including the well-posedness, Harnack inequality, Bismut's derivative formula, exponential ergodicity, estimate of heat kernel, see [1,2,6,7,8,11,13,14,15,18,20,23] and references therein for more details. For the well-posedness, the drifts can be not continuous in the measure variable under the weak topology, for instance [8,15,20] and so on.…”

Section: Introductionmentioning

confidence: 99%

Harnack Inequality for Distribution Dependent Stochastic Hamiltonian System

Huang¹,

Ma²

2022

Preprint

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By investigating the regularity of the nonlinear semigroup P * t associated to the distribution dependent stochastic Hamiltonian system, the Harnack inequality is derived when the drift is Lipschitz continuous in the measure variable under the distance induced by the functions which are β(β > 23 )-Hölder continuous on the degenerate component and square root of Dini continuous on the non-degenerate one. The results extend the ones in the case where the drifts are Lipschitz continuous in L 2 -Wasserstein distance.

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Second order McKean-Vlasov SDEs and kinetic Fokker-Planck-Kolmogorov equations

Cited by 3 publications

References 43 publications

Second order fractional mean-field SDEs with singular kernels and measure initial data

Second order fractional mean-field SDEs with singular kernels and measure initial data

Strong convergence of propagation of chaos for McKean-Vlasov SDEs with singular interactions

Harnack Inequality for Distribution Dependent Stochastic Hamiltonian System

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