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Guohuan Zhao

Publications and source records attributed to Guohuan Zhao.

25 records · Page 2Linked to original sources
arXiv2019

Stochastic Lagrangian path for Leray solutions of 3D Navier-Stokes equations

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.

math.PR↗
arXiv2018

Dirichlet problem for supercritical nonlocal operators

Let $D$ be a bounded $C^2$-domain. Consider the following Dirichlet initial-boundary problem of nonlocal operators with a drift: $$ \partial_t u={\mathscr L}^{(α)}_κu+b\cdot \nabla u+f\ \mathrm{in}\ \mathbb R_+\times D,\ \ u|_{\mathbb R_+\times D^c}=0,\ u(0,\cdot)|_{D}=φ, $$ where $α\in(0,2)$ and $\mathscr L^{(α)}_κ$ is an $α$-stable-like nonlocal operator with kernel function $κ(x,z)$ bounded from above and below by positive constants, and $b:\mathbb R^d\to\mathbb R^d$ is a bounded $C^β$-function with $α+β>1$, $f: \mathbb R_+\times D\to\mathbb R$ is a $C^γ$-function in $D$ uniformly in $t$ with $γ\in((1-α)\vee 0,β]$, $φ\in C^{α+γ}(D)$. Under some Hölder assumptions on $κ$, we show the existence of a unique classical solution $u\in L^\infty_{loc}(\mathbb R_+; C^{α+γ}_{loc}(D))\times C(\mathbb R_+; C_b(D))$ to the above problem. Moreover, we establish the following probabilistic representation for $u$ $$ u(t,x)=\mathbb E_x \Big(φ(X_{t}){\bf 1}_{τ_{D}>t}\Big)+\mathbb E_x\left(\int^{t\wedgeτ_{D}}_0f(t-s,X_s){\rm d} s\right),\ t\geq 0,\ x\in D, $$ where $((X_t)_{t\geq 0},\mathbb P_x; x\in\mathbb R^d)$ is the Markov process associated with the operator $\mathscr L^{(α)}_κ+b\cdot \nabla$, and $τ_D$ is the first exit time of $X$ from $D$. In the sub and critical case $α\in[1,2)$, the kernel function $κ$ can be rough in $z$. In the supercritical case $α\in(0,1)$, we classify the boundary points according to the sign of $b(z)\cdot\vec{n}(z)$, where $z\in\partial D$ and $\vec{n}(z)$ is the unit outward normal vector. Finally, we provide an example and simulate it by Monte-Carlo method to show our results.

math.AP↗
arXiv2018

Heat kernel and ergodicity of SDEs with distributional drifts

In this paper we consider the following SDE with distributional drift $b$: $$ {\rm d} X_t=σ(X_t){\rm d} B_t+b(X_t){\rm d} t,\ X_0=x\in{\mathbb R}^d, $$ where $σ$ is a bounded continuous and uniformly non-degenerate $d\times d$-matrix-valued function, $B$ is a $d$-dimensional standard Brownian motion. Let $α\in(0,\frac{1}{2}]$, $p\in(\frac{d}{1-α},\infty)$ and $β\in[α,1]$, $q\in(\frac{d}β,\infty)$. Assume $\|({\mathbb I}-Δ)^{-α/2}b\|_p+\|(-Δ)^{β/2}σ\|_q<\infty$. We show the existence and uniqueness of martingale solutions to the above SDE, and obtain sharp two-sided and gradient estimates of the heat kernel associated to the above SDE. Moreover, we study the ergodicity and global regularity of the invariant measures of the associated semigroup under some dissipative assumptions.

math.PR↗
arXiv2017

Moderate maximal inequalities for the Ornstein-Uhlenbeck process

The maximal inequalities for diffusion processes have drawn increasing attention in recent years. However, the existing proof of the $L^p$ maximum inequalities for the Ornstein-Uhlenbeck process was dubious. Here we give a rigorous proof of the moderate maximum inequalities for the Ornstein-Uhlenbeck process, which include the $L^p$ maximum inequalities as special cases and generalize the remarkable $L^1$ maximum inequalities obtained by Graversen and Peskir [P. Am. Math. Soc., 128(10):3035-3041, 2000]. As a corollary, we also obtain a new moderate maximal inequality for continuous local martingales, which can be viewed as a supplement of the classical Burkholder-Davis-Gundy inequality.

math.PR↗
arXiv2017

Well-posedness of supercritical SDE driven by Lévy processes with irregular drifts

In this paper, we study the following time-dependent stochastic differential equation (SDE) in ${\bf R}^d$: $$ d X_{t}= σ_t(X_{t-}) d Z_t + b_t(X_{t})d t, \quad X_{0}=x\in {\bf R}^d, $$ where $Z$ is a $d$-dimensioanl nondegenerate $α$-stable-like process with $α\in(0,2)$ (including cylindrical case), and uniform in $t\geq 0$, $x\mapsto σ_t(x): {\bf R}^d\to {\bf R}^d\otimes {\bf R}^d$ is Lipchitz and uniformly elliptic and $x\mapsto b_t (x)$ is $β$-order Hölder continuous with $β\in(1-α/2,1)$. Under these assumptions, we show the above SDE has a unique strong solution for every starting point $x \in {\bf R}^d$. When $σ_t (x)={\bf I}_{d\times d}$, the $d\times d$ identity matrix, our result in particular gives an affirmative answer to the open problem of Priola (2015).

math.PR↗