arXiv subjects

Zimo Hao

Publications and source records attributed to Zimo Hao.

28 records · Page 2Linked to original sources

arXiv2023

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

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 {\it 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 values are calculated, which belongs to the subcritical regime in the scaling sense. In particular, our results provide a microscopic probabilistic explanation and establish a unified treatment for many physical models such as the 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.

math.AP↗

arXiv2022

Strong and weak convergence for averaging principle of DDSDE with singular drift

In this paper, we study the averaging principle for distribution dependent stochastic differential equations with drift in localized $L^p$ spaces. Using Zvonkin's transformation and estimates for solutions to Kolmogorov equations, we prove that the solutions of the original system strongly and weakly converge to the solution of the averaged system as the time scale $\eps$ goes to zero. Moreover, we obtain rates of the strong and weak convergence that depend on $p$ respectively.

math.PR↗

arXiv2022

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

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 for the moderate interaction particle systems on the level of particle trajectories. One of the main obstacles is to establish the strong well-posedness of the SDEs for particle systems with singular interaction. To this end, we extend the results on strong well-posedness of Krylov and Röckner \cite{Kr-Ro} to 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 \cite{JW16} and Zvonkin's transformation.

math.PR↗

arXiv2021

Well-posedness of density dependent SDE driven by $α$-stable process with Hölder drifts

In this paper, we show the weak and strong well-posedness of density dependent stochastic differential equations driven by $α$-stable processes with $α\in(1,2)$. The existence part is based on Euler's approximation as \cite{HRZ20}, while, the uniqueness is based on the Schauder estimates in Besov spaces for nonlocal Fokker-Planck equations. For the existence, we only assume the drift being continuous in the density variable. For the weak uniqueness, the drift is assumed to be Lipschitz in the density variable, while for the strong uniqueness, we also need to assume the drift being $β_0$-order Hölder continuous in the spatial variable, where $β_0\in(1-α/2,1)$.

math.PR↗

arXiv2021

Singular kinetic equations and applications

In this paper we study singular kinetic equations on $\mathbb{R}^{2d}$ by the paracontrolled distribution method introduced in \cite{GIP15}. We first develop paracontrolled calculus in the kinetic setting, and use it to establish the global well-posedness for the linear singular kinetic equations under the assumptions that the products of singular terms are well-defined. We also demonstrate how the required products can be defined in the case that singular term is a Gaussian random field by probabilistic calculation. Interestingly, although the terms in the zeroth Wiener chaos of regularization approximation are not zero, they converge in suitable weighted Besov spaces and no renormalization is required. As applications the global well-posedness for a nonlinear kinetic equation with singular coefficients is obtained by the entropy method. Moreover, we also solve the martingale problem for nonlinear kinetic distribution dependent stochastic differential equations with singular drifts.

math.PR↗

arXiv2020

Euler scheme for density dependent stochastic differential equations

In this paper we show the existence and uniqueness for a class of density dependent SDEs with bounded measurable drift, where the existence part is based on Euler's approximation for density dependent SDEs and the uniqueness is based on the associated nonlinear Fokker-Planck equation. As an application, we obtain the well-posedness of a nonlinear Fokker-Planck equation.

math.PR↗

arXiv2020

Schauder's estimates for nonlocal equations with singular Lévy measures

In this paper, we establish Schauder's estimates for the following non-local equations in \mR^d : $$ \partial_tu=\mathscr L^{(α)}_{κ,σ} u+b\cdot\nabla u+f,\ u(0)=0, $$ where $α\in(1/2,2)$ and $ b:\mathbb R_+\times\mathbb R^d\to\mathbb R$ is an unbounded local $β$-order Hölder function in $ x $ uniformly in $ t $, and $\mathscr L^{(α)}_{κ,σ}$ is a non-local $α$-stable-like operator with form: \begin{align*} {\mathscr L}^{(α)}_{κ,σ}u(t,x):=\int_{\mathbb R^d}\Big(u(t,x+σ(t,x)z)-u(t,x)-σ(t,x)z^{(α)}\cdot\nabla u(t,x)\Big)κ(t,x,z)ν^{(α)}(\mathord{\rm d} z), \end{align*} where $z^{(α)}=z\mathbf{1}_{α\in(1,2)}+z\mathbf{1}_{|z|\leq 1}\mathbf{1}_{α=1}$, $ κ:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R_+ $ is bounded from above and below, $ σ:\mathbb R_+\times\mathbb R^{d}\to \mathbb R^d\otimes \mathbb R^d$ is a $ γ$-order Hölder continuous function in $ x $ uniformly in $ t $, and $ ν^{(α)} $ is a singular non-degenerate $ α$-stable Lévy measure.

math.PR↗

arXiv2020

Hölder regularity and gradient estimates Hölder regularity and gradient estimates for SDEs driven by cylindrical $α$-stable processes

We establish Hölder regularity and gradient estimates for the transition semigroup of the solutions to the following SDE: $$ {\rm d} X_t=σ(t, X_{t-}){\rm d} Z_t+b (t, X_t){\rm d} t,\ \ X_0=x\in{\mathbb R}^d, $$ where $( Z_t)_{t\geq 0}$ is a $d$-dimensional cylindrical $α$-stable process with $α\in (0, 2)$, $σ(t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d\otimes{\mathbb R}^d$ is bounded measurable, uniformly nondegenerate and Lipschitz continuous in $x$ uniformly in $t$, and $b (t, x):{\mathbb R}_+\times{\mathbb R}^d\to{\mathbb R}^d$ is bounded $β$-Hölder continuous in $x$ uniformly in $t$ with $β\in[0,1]$ satisfying $α+β>1$. Moreover, we also show the existence and regularity of the distributional density of $X (t, x)$. Our proof is based on Littlewood-Paley's theory.

math.PR↗

arXiv2019

Schauder's estimate for nonlocal kinetic equations and its applications

In this paper we develop a new method based on Littlewood-Paley's decomposition and heat kernel estimates of integral form, to establish Schauder's estimate for the following degenerate nonlocal equation in $\mathbb R^{2d}$ with Hölder coefficients: $$ \partial_tu=\mathscr L^{(α)}_{κ;{\rm v}} u+b\cdot\nabla u+f,\ u_0=0, $$ where $u=u(t,x,{\rm v})$ and $\mathscr L^{(α)}_{κ;{\rm v}}$ is a nonlocal $α$-stable-like operator with $α\in(1,2)$ and kernel function $κ$, which acts on the variable ${\rm v}$. As an application, we show the strong well-posedness to the following degenerate stochastic differential equation with Hölder drift $b$: $$ {\rm d}Z_t=b(t,Z_t){\rm d}t+(0,σ(t,Z_t){\rm d}L^{(α)}_t),\ \ Z_0=(x,{\rm v})\in\mathbb R^{2d}, $$ where $L^{(α)}_t$ is a $d$-dimensional rotationally invariant and symmetric $α$-stable process with $α\in(1,2)$, and $b:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R^{2d}$ is a $(γ,β)$-Hölder continuous function in $(x,{\rm v})$ with $γ\in\big(\frac{2+α}{2(1+α)},1\big)$ and $β\in\big(1-\fracα{2},1\big)$, $σ:\mathbb R_+\times\mathbb R^{2d}\to\mathbb R^d\otimes\mathbb R^d$ is a Lipschitz function. Moreover, we also show that for almost all $ω$, the following random transport equation has a unique $C^1_b$-solution: $$ \partial_tu(t,x,ω)+(b(t,x)+L^{(α)}_t(ω))\cdot\nabla_x u(t,x,ω)=0,\ \ u(0,x)=φ(x), $$ where $φ\in C^1_b(\mathbb R^d)$ and $b:\mathbb R_+\times\mathbb R^d\to\mathbb R^d$ is a bounded continuous function in $(t,x)$ and $γ$-order Hölder continuous in $x$ uniformly in $t$ with $γ\in\big(\frac{2+α}{2(1+α)},1\big)$.

math.AP↗

arXiv2019

Hörmander's hypoelliptic theorem for nonlocal operators

In this paper we show the Hörmander hypoelliptic theorem for nonlocal operators by a purely probabilistic method: the Malliavin calculus. Roughly speaking, under general Hörmander's Lie bracket conditions, we show the regularization effect of discontinuous Lévy noises for possibly degenerate stochastic differential equations with jumps. To treat the large jumps, we use the perturbation argument together with interpolation techniques and some short time asymptotic estimates of the semigroup. As an application, we show the existence of fundamental solutions for operator $\partial_t-\mathscr{K}$, where $\mathscr{K}$ is the nonlocal kinetic operator: $$ \mathscr{K} f(x,{\rm v}):={\rm p.v}\int_{\mathbb{R}^d}(f(x,{\rm v}+w)-f(x,{\rm v}))\frac{κ(x,{\rm v},w)}{|w|^{d+α}}{\rm d} w +{\rm v}\cdot\nabla_x f(x,{\rm v})+b(x,{\rm v})\cdot\nabla_{\rm v} f(x,{\rm v}). $$ Here $κ_0^{-1}\leq κ(x,{\rm v},w)\leqκ_0$ belongs to $C^\infty_b(\mathbb{R}^{3d})$ and is symmetric in $w$, p.v. stands for the Cauchy principal value, and $b\in C^\infty_b(\mathbb{R}^{2d};\mathbb{R}^d)$.

math.PR↗