Weak Solutions of McKean–Vlasov SDEs with Supercritical Drifts

Abstract

Consider the following McKean–Vlasov SDE:

\begin{array}{c}\hfill \text{d}{X}_t=\sqrt{2}\text{d}{W}_t+{\int }_{{\mathbb{R}}^d}K(t,X_t-y){\mu }_{X_t}(\text{d}y)\text{d}t,X_0=x,\end{array}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \textrm{d} X_t=\sqrt{2}\textrm{d} W_t+\int _{{\mathbb {R}}^d}K(t,X_t-y)\mu _{X_t}(\textrm{d} y)\textrm{d} t,\ \ X_0=x, \end{aligned}$$\end{document}

where

{\mu }_{X_t}

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mu _{X_t}$$\end{document}

stands for the distribution of

X_t

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$X_t$$\end{document}

and

K(t,x):{\mathbb{R}}_{+}\times {\mathbb{R}}^d\to {\mathbb{R}}^d

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K(t,x): {{\mathbb {R}}}_+\times {{\mathbb {R}}}^d\rightarrow {{\mathbb {R}}}^d$$\end{document}

is a time-dependent divergence free vector field. Under the assumption

K\in L_t^q({\tilde{L}}_x^p)

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$K\in L^q_t({\widetilde{L}}_x^p)$$\end{document}

with

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{d}{p}+\frac{2}{q}<2$$\end{document}

, where

{\tilde{L}}_x^p

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widetilde{L}}^p_x$$\end{document}

stands for the localized

L^p

\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^p$$\end{document}

-space, we show the existence of weak solutions to the above SDE. As an application, we provide a new proof for the existence of weak solutions to 2D Navier–Stokes equations with measure as initial vorticity.

Keywords

McKean–Vlasov system / Supercritical drift / 2D Navier–Stokes equation / Krylov’s estimate

References

[1.] Carmona R, Delarue F. . Probabilistic Theory of Mean Field Games with Applications. II. Mean Field Games with Common Noise and Master Equations, 2018 Berlin Springer,

[2.] Cottet GH. Equations de Navier–Stokes dans le plan avec tourbillon initial mesure. C. R. Acad. Sci. Paris Sér. I Math., 1986, 303: 105-108

[3.] Funaki T. A certain class of diffusion processes associated with nonlinear parabolic equations. Prob. Theory Relat. Fields, 1984, 67(3): 331-348

[4.] Gallagher I, Gallay T. Uniqueness for the two-dimensional Navier–Stokes equation with a measure as initial vorticity. Math. Ann., 2005, 332: 287-327,

[5.] Giga Y, Miyakawa T, Osada H. Two-dimensional Naveri–Stokes flow with measures as initial vorticity. Arch. Rational Mech. Anal., 1988, 104: 223-250,

[6.] Huang X, Wang FY. Distribution dependent SDEs with singular coefficients. Stochastic Process. Appl., 2019, 129(11): 4747-4770,

[7.] Jabin PE, Wang Z. Quantitative estimates of propagation of chaos for stochastic systems with W - 1 , ∞ \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W^{-1,\infty }$$\end{document} kernels. Invent. math., 2018, 214: 523-591,

[8.] Li J, Min H. Weak solutions of mean-field stochastic differential equations and application to zero-sum stochastic differential games. SIAM J. Control Optim., 2016, 54(3): 1826-1858,

[9.] Majda AJ, Bertozzi AL. . Vorticity and Impressible Flow, 2002 Cambridge Cambridge University Press

[10.] McKean HP. A class of Markov processes associated with nonlinear parabolic equations. Proc Nat. Acad Sci USA, 1966, 56(6): 1907-1911, pmcid: 220210

[11.] Mishura, Y.S., Veretennikov, A.Y.: Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations, arXiv:1603.02212v4

[12.] Röckner M, Zhang X. Well-posedness of distribution dependent SDEs with singular drifts. Bernoulli, 2021, 27(2): 1131-1158,

[13.] Stroock DW, Varadhan SS. . Multidimensional Diffusion Processes, 1979 Berlin Springer-Verlag

[14.] Sznitman, A.S.: Topics in propagation of chaos. In: École d’Été de Prob. de Saint-Flour XIX-1989, Vol. 1464, Lecture Notes in Mathematics, pp. 165-251. Springer-Verlag (1991)

[15.] Trevisan D. Well-posedness of multidimensional diffusion processes with weakly differentiable coefficients. Electron. J. Probab., 2016, 21: 41,

[16.] Zhang, X., Zhao, G.: Singular Brownian Diffusion Processes. Communications in Mathematics and Statistics, pp. 1-49 (2018)

[17.] Zhang X, Zhao G. Stochastic Lagrangian path for Leray solutions of 3D Navier–Stokes equations. Comm. Math. Phys., 2021, 381(2): 491-525,