arXiv subjects
Publications and source records attributed to Guohuan Zhao.
We establish a Schauder estimate for a nonlocal Cauchy problem with drift. The leading operator is the generator of a non-degenerate $\alpha$-stable process with $\alpha\in(0,1)$, and the drift is $\beta$-H\"older continuous with $\beta \in (1-\alpha,1)$. The proof relies on a refined conic Littlewood--Paley decomposition and a one-dimensional one-sided dyadic maximum principle.
We utilize Stein's method to establish quantitative bounds on the total variation distance between the invariant measure of a drifted nonlocal Markov operator and that of its local counterpart under minimal assumptions on the drifts. The main ingredient is a reduction via Stein's method that transforms the original problem into analyzing growth estimates for solutions to a nonlocal Poisson equation and decay estimates for the invariant measure of the local operator.
We consider a super-Brownian motion $\{X_t, t\geq 0\}$ in a random environment described by a centered Gaussian field $\{W(t,x),t\geq 0, x\in\mathbb{R}^d\}$ whose correlation function is given by $\mathcal{C} (x,y)(t \wedge s)$. The process takes values in $\mathcal{M}(\mathbb{R}^d)$, the space of Radon measures on $\mathbb{R}^d$. It can be characterized through a conditional Laplace transform by a parabolic stochastic partial differential equation driven by $W(t, x)$. Suppose that $\mathcal{C} (x, y)\leq g(x-y)$ for some bounded positive function $g$ on $\mathbb{R}^d$ and the initial distribution of process $X$ is the Lebesgue measure $m$ on $\mathbb{R}^d$. We prove that for dimension $d\geq 3$, whenever $$ \sup_{x\in \mathbb{R}^d} \int_{\mathbb{R}^d} |x-y|^{2-d} g(y)dy< \frac{8 (d-2) \pi^{d/2}}{d 2^d \Gamma \left(d/2-1\right)}, $$ the distribution of $X_t$ converges weakly as $t \to \infty$ to a non-trivial invariant probability distribution $\pi^m$ on $\mathcal{M}(\mathbb{R}^d)$ with mean measure $m$. This result in particular gives an affirmative answer to Conjecture 1.4 of Mytnik and Xiong (Electron. J. Probab. 12: 1349-1378 (2007)). We further show that given $ \Theta \in C^\beta(\mathbb{R}^d)$ $(\beta>1)$, when $\mathcal{C}(x,y)= a \Theta (x-y)$ with $a$ being large enough, the superprocess $X$ suffers local extinction.
We prove the existence and conditional uniqueness in the Krylov class for SDEs with singular divergence-free drifts in the endpoint critical Lorentz space $L^{\infty}(0,T; L^{d,\infty}(\mathbb{R}^d))$, $d \geqslant 2$, which particularly includes the $2$D Biot-Savart law. The uniqueness result is shown to be optimal in dimensions $d \geqslant 3$, by constructing different martingale solutions in the case of supercritical Lorentz drifts. As a consequence, the well-posedness of McKean-Vlasov equations and nonlinear Fokker-Planck equations with critical singular kernels is derived. In particular, this yields the uniqueness of the $2$D vorticity Navier-Stokes equations even in certain supercritical-scaling spaces. Furthermore, we prove that the path laws of solutions to McKean-Vlasov equations with critical singular kernel form a nonlinear Markov process in the sense of McKean.
Based on a compactness criterion for random fields in Wiener-Sobolev spaces, in this paper, we prove the unique strong solvability of time-inhomogeneous stochastic differential equations with drift coefficients in critical Lebesgue spaces, which gives an affirmative answer to a longstanding open problem. As an application, we also prove a regularity criterion for solutions of a stochastic system proposed by Constantin and Iyer (Comm. Pure. Appl. Math. 61(3): 330-345, 2008), which is closely related to the Navier-Stokes equations.
We consider the linear non-local operator $\mathcal{L}$ denoted by \[ \mathcal{L} u (x) = \int_{\mathbb{R}^d} \left(u(x+z)-u(x)\right) a(x,z)J(z)\,d z. \] Here $a(x,z)$ is bounded and $J(z)$ is the jumping kernel of a Lévy process, which only has a low-order singularity near the origin and does not allow for standard scaling. The aim of this work is twofold. Firstly, we introduce generalized Orlicz-Besov spaces tailored to accommodate the analysis of elliptic equations associated with $\mathcal{L}$, and establish regularity results for the solutions of such equations in these spaces. Secondly, we investigate the martingale problem associated with $\mathcal{L}$. By utilizing analytic results, we prove the well-posedness of the martingale problem under mild conditions. Additionally, we obtain a new Krylov-type estimate for the martingale solution through the use of a Morrey-type inequality for generalized Orlicz-Besov spaces.
We prove the existence and uniqueness of Stochastic Lagrangian Flows and almost everywhere Stochastic Flows for non-degenearted SDEs with rough coefficients. As an application of our main result, we show that there exists a unique Stochastic Flow corresponding to each Leray-Hopf solution of 3D Navier-Stokes equation in the DiPerna-Lions sense.
Let $d \geq 2$, $α\in (0,2)$, and $X$ be the rectilinear $α$-stable process on $\mathbb{R}^d$. We first present a geometric characterization of an open subset $D\subset \mathbb{R}^d$ so that the part process $X^D$ of $X$ in $D$ is irreducible. We then study the properties of the transition density functions of $X^D$, including the strict positivity property as well as their sharp two-sided bounds in $C^{1,1}$ domains in $\mathbb{R}^d$. Our bounds are shown to be sharp for a class of $C^{1,1}$ domains.
For the stochastic linear transport equation with $L^p$-initial data ($1<p<2$) on the full space $\mathbb{R}^d$, we provide quantitative estimates, in negative Sobolev norms, between its solutions and that of the deterministic heat equation. Similar results are proved for the stochastic 2D Euler equations with transport noise.
We investigate the mixing properties of solutions to the stochastic transport equation $d u= \circ d W \cdot\nabla u$, where the driving noise $W(t,x)$ is white in time, colored and divergence-free in space. Furthermore, we prove the dissipation enhancement in the presence of a small viscous term. Applying our results, we also derive the mixing properties for a regularized stochastic 2D Euler equation.
We investigate the well-posedness of following McKean-Vlasov equation in $\mathbb{R}^d$: \[ \mathrm{d} X_t=σ(t,X_t, μ_{X_t})\mathrm{d} W_t+b(t, X_t, μ_{X_t}) \mathrm{d} t, \] where $μ_{X_t}$ is the law of $X_t$. The existence of solutions is demonstrated when $σ$ satisfies certain non-degeneracy and continuity assumptions, and when $b$ meets some integrability conditions, and continuity requirements in the (generalized) total variation distance. Furthermore, uniqueness is established under additional continuity assumptions of a Lipschitz type.
This study focuses on approximating solutions to SDEs driven by Lévy processes with Hölder continuous drifts using the Euler-Maruyama scheme. We derive the $L^p$-error for a broad range of driven noises, including all nondegenerate $α$-stable processes ($0<α<2$).
We consider Itô uniformly nondegenerate equations with random coefficients. When the coefficients satisfy some low regularity assumptions with respect to the spatial variables and Malliavin differentiability assumptions on the sample points, the unique solvability of singular SDEs is proved by solving backward stochastic Kolmogorov equations and utilizing a modified Zvonkin type transformation.
We prove the unique weak solvability of time-inhomogeneous stochastic differential equations with additive noises and drifts in critical Lebsgue space $L^q([0,T]; L^{p}(\mathbb{R}^d))$ with $d/p+2/q=1$. The weak uniqueness is obtained by solving corresponding Kolmogorov's backward equations in some second order Sobolev spaces, which is analytically interesting in itself.
In this paper we investigate the regularity properties of strong solutions to SDEs driven by Lévy processes with irregular drift coefficients. Under some mild conditions, we show that the singular SDE has a unique strong solution for each starting point and the family of all the solutions forms a stochastic flow. Moreover, the Malliavin differentiability of the strong solutions is also obtained. As an application, we also prove a path-by-path uniqueness result for some related random ODEs.
The well-posedness of nonlocal elliptic equation with singular drift is investigated in Besov-Hölder spaces. As an application, we show the existence and uniqueness for corresponding martingale problem. Moreover, we prove that the one dimensional distribution of the martingale solution has a density in some Besov space.
In this paper we show the weak differentiability of the unique strong solution with respect to the starting point $x$ as well as Bismut-Elworthy-Li's derivative formula for the following stochastic differential equation in $\mathbb R^d$: $$ {\rm d} X_t=b(t,X_t){\rm d} t+σ(t,X_t){\rm d} W_t,\ \ X_0=x\in\mathbb R^d, $$ where $σ$ is bounded, uniformly continuous and nondegenerate, $\nablaσ\in \widetilde{\mathbb L}^{p_1}_{q_1}$ and $b\in \widetilde{\mathbb L}^{p_2}_{q_2}$ for some $p_i,q_i\in[2,\infty)$ with $\frac{d}{p_i}+\frac{2}{q_i}<1$, $i=1,2$, where $\widetilde{\mathbb L}^{p_i}_{q_i}, i=1,2$ are some localized spaces. Moreover, in the endpoint case $b\in \widetilde{\mathbb L}^{d; {\rm uni}}_\infty$, we also show the weak well-posedness.
In this paper we show the existence of stochastic Lagrangian particle trajectory for Leray's solution of 3D Navier-Stokes equations. More precisely, for any Leray's solution ${\mathbf u}$ of 3D-NSE and each $(s,x)\in\mathbb{R}_+\times\mathbb{R}^3$, we show the existence of weak solutions to the following SDE, which has a density $ρ_{s,x}(t,y)$ belonging to $\mathbb{H}^{1,p}_q$ provided $p,q\in[1,2)$ with $\frac{3}{p}+\frac{2}{q}>4$: $$ \mathrm{d} X_{s,t}={\mathbf u} (s,X_{s,t})\mathrm{d} t+\sqrt{2ν}\mathrm{d} W_t,\ \ X_{s,s}=x,\ \ t\geq s, $$ where $W$ is a three dimensional standard Brownian motion, $ν>0$ is the viscosity constant. Moreover, we also show that for Lebesgue almost all $(s,x)$, the solution $X^n_{s,\cdot}(x)$ of the above SDE associated with the mollifying velocity field ${\mathbf u}_n$ weakly converges to $X_{s,\cdot}(x)$ so that $X$ is a Markov process in almost sure sense.