arXiv subjects

Guangying Lv

Publications and source records attributed to Guangying Lv.

At least 19 recordsLinked to original sources

arXiv2026

Strong solutions and sharp Euler--Maruyama approximations for SDEs with Lebesgue--Dini drift

We investigate the strong approximation of stochastic differential equations whose drift is square-integrable in time and Dini continuous in space, while the diffusion coefficient is non-constant and uniformly elliptic. Using a refined It\^{o}--Tanaka trick combined with parabolic regularity estimates, we first establish strong well-posedness and the stochastic flow property. Under additional Lipschitz regularity of the diffusion matrix, we then analyze a polygonal-type Euler--Maruyama scheme and prove the strong error estimate \[ \Big\|\sup_{0\le t\le1}|X_t-X_t^n|\Big\|_{L^p(\Omega)} \le C n^{-\frac12}\log(n)^{\frac32}, \quad p\ge2. \] We further show that this rate is sharp: even under smooth and uniformly elliptic diffusion coefficients with vanishing drift, the convergence order $1/2$ cannot be improved. These results provide the first sharp quantitative strong convergence estimates in a Lebesgue--Dini drift framework.

math.PR↗

arXiv2025

The Euler-Maruyama method for SDEs with low-regularity drift

We study the strong $L^p$-convergence rates of the Euler-Maruyama method for stochastic differential equations driven by Brownian motion with low-regularity drift coefficients. Specifically, the drift is assumed to be in the Lebesgue-Hölder spaces $L^q([0,T]; {\mathcal C}_b^α({\mathbb R}^d))$ with $α\in(0,1)$ and $q\in (2/(1+α),\infty]$. For every $p\geq 2$, by using stochastic sewing and/or the Itô-Tanaka trick, we obtain the $L^p$-convergence rates: $(1+α)/2$ for $q\in [2,\infty]$ and $(1-1/q)$ for $q\in (2/(1+α),2)$. Moreover, we prove that the unique strong solution can be constructed via the Picard iteration.

math.PR↗

arXiv2024

A new maximal regularity for parabolic equations and an application

We introduce the Lebesgue--Hölder--Dini and Lebesgue--Hölder spaces $L^p(\mathbb{R};{\mathcal C}_{\vartheta,ς}^{α,ρ}({\mathbb R}^n))$ ($\vartheta\in \{l,b\}, \, ς\in \{d,s,c,w\}$, $p\in (1,+\infty]$ and $α\in [0,1)$), and then use a vector-valued Calderón--Zygmund theorem to establish the maximal Lebesgue--Hölder--Dini and Lebesgue--Hölder regularity for a class of parabolic equations. As an application, we obtain the unique strong solvability of the following stochastic differential equation \begin{eqnarray*} X_{s,t}(x)=x+\int\limits_s^tb(r,X_{s,r}(x))dr+W_t-W_{s}, \ \ t\in [s,T], \ x\in \mathbb{R}^n, \ s\in [0,T], \end{eqnarray*} for the low regularity growing drift in critical Lebesgue--Hölder--Dini spaces $L^p([0,T];{\mathcal C}^{\frac{2}{p}-1,ρ}_{l,d}({\mathbb R}^n;{\mathbb R}^n))$ ($p\in (1,2]$), where $\{W_t\}_{0\leq t\leq T}$ is a $n$-dimensional standard Wiener process. In particular, when $p=2$ we give a partially affirmative answer to a longstanding open problem, which was proposed by Krylov and Röckner for $b\in L^2([0,T];L^\infty({\mathbb R}^n;{\mathbb R}^n))$ based upon their work ({\em Probab. Theory Relat. Fields 131(2): 154--196, 2005}).

math.PR↗

arXiv2023

Continuous data assimilation for the three dimensional primitive equations with magnetic field

In this paper, the problem of continuous data assimilation of three dimensional primitive equations with magnetic field in thin domain is studied. We establish the well-posedness of the assimilation system and prove that the $H^2$-strong solution of the assimilation system converges exponentially to the reference solution in the sense of $L^2$ as $t\rightarrow \infty$. We also study the sensitivity analysis of the assimilation system and prove that a sequence of solutions of the difference quotient equation converge to the unique solution of the formal sensitivity equation.

math.AP↗

arXiv2023

A perturbation result for the energy critical Choquard equation in $\mathbb{R}^N$

We study the singularly perturbed nonlinear energy critical Choquard equation \begin{equation*} -{\Laplace u}\qty({x}) -α \int_{\R^N}\frac{u^p\qty(y)}{\abs{x-y}^λ}\odif{y} u^{p-1}\qty({x}) -\eps k\qty(x)u^{\frac{N+2}{N-2}}\qty(x)=0, \qquad x\in\R^N, \end{equation*} where $N\geq 3$, $0<λ 0$, we construct solutions of the form \begin{align*} u_{\eps}\qty(x)=U_{μ_{\eps},ξ_{\eps}}\qty(x)\qty(1+Ø\qty(\eps)), \end{align*} where $U_{μ_{\eps},ξ_{\eps}}$ is a positive solution of the unperturbed equation \begin{equation*} -{\Laplace u}\qty({x}) -α \int_{\R^N}\frac{u^p\qty(y)}{\abs{x-y}^λ}\odif{y}=0,\qquad x\in\R^N. \end{equation*}

math.AP↗

arXiv2021

Stochastic transport equation with bounded and Dini continuous drift

The results established by Flandoli, Gubinelli and Priola ({\it Invent. Math.} {\bf 180} (2010) 1--53) for stochastic transport equation with bounded and Hölder continuous drift are generalized to bounded and Dini continuous drift. The uniqueness of $L^\infty$-solutions is established by the Itô--Tanaka trick partially solving the uniqueness problem, which is still open, for stochastic transport equation with only bounded measurable drift. Moreover the existence and uniqueness of stochastic diffeomorphisms flows for a stochastic differential equation with bounded and Dini continuous drift is obtained.

math.PR↗

arXiv2020

Noise and Stability in Reaction-diffusion Equations

We study the stability of reaction-diffusion equations in presence of noise. The relationship of stability of solutions between the stochastic ordinary different equations and the corresponding stochastic reaction-diffusion equation is firstly established. Then, by using the Lyapunov method, sufficient conditions for mean square and stochastic stability are given. The results show that the multiplicative noise can make the solution stable, but the additive noise will be not.

math.PR↗

arXiv2019

A Kolmogorov type theorem for stochastic fields

We generalize the Kolmogorov continuity theorem and prove the continuity of a class of stochastic fields with the parameter. As an application, we derive the continuity of solutions for nonlocal stochastic parabolic equations driven by non-Gaussian Lévy noises.

math.PR↗

arXiv2019

Impact of noise on parabolic equations

In this short paper, we focus on the blowup phenomenon of stochastic parabolic equations. We first discuss the probability of the event that the solutions keep positive. Then, the blowup phenomenon in the whole space is considered. The probability of the event that the solutions blow up in finite time is given. Lastly, we obtain the probability of the event that blowup time of stochastic parabolic equations large than or less than the deterministic case.

math.PR↗

arXiv2019

Schauder and Sobolev Estimates of Parabolic Equations

In this note, we use the non-homogeneous Poisson stochastic process to show how knowing Schauder and Sobolev estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs. The method is probability. We generalize the result of Krylov-Priola [7].

math.AP↗

arXiv2019

Global existence and non-existence of stochastic parabolic equations

This paper is concerned with the blowup phenomenon of stochastic parabolic equations both on bounded domain and in the whole space. We introduce a new method to study the blowup phenomenon on bounded domain. Comparing with the existing results, we delete the assumption that the solutions to stochastic heat equations are non-negative. Then the blowup phenomenon in the whole space is obtained by using the properties of heat kernel. We obtain that the solutions will blow up in finite time for nontrivial initial data.

math.AP↗

arXiv2019

Blowup solutions of Grushin's operator

In this note, we consider the blowup phenomenon of Grushin's operator. By using the knowledge of probability, we first get expression of heat kernel of Grushin's operator. Then by using the properties of heat kernel and suitable auxiliary function, we get that the solutions will blow up in finite time.

math.AP↗

arXiv2018

Periodic solution of stochastic process in the distributional sense

In this paper, we aim to study a stochastic process from a macro point of view, and thus periodic solution of a stochastic process in distributional sense is introduced. We first give the definition and then establish the existence of periodic solution on bounded domain. Lastly, for the case that probability density function exists, we obtain the existence periodic solutions of the probability density function corresponding to the stochastic process by using the technique of deterministic partial differential equations.

math.PR↗

arXiv2018

The effect of noise intensity on stochastic parabolic equations

In the present paper, the effect of noise intensity on stochastic parabolic equations is discussed. We focus on the effect of noise on the energy solutions of the stochastic parabolic equations. By utilising Itô's formula and the energy estimate method, we obtain the excitation indices of the energy solutions $u$ at any finite time $t$. Furthermore, we improve certain existing results in the literature by presenting a comparably simple method to show those existing results.

math.PR↗

arXiv2018

Regularity of stochastic nonlocal diffusion equations

In this paper, we are concerned with regularity of nonlocal stochastic partial differential equations of parabolic type. By using Companato estimates and Sobolev embedding theorem, we first show the Hölder continuity (locally in the whole state space $\mathbb{R}^d$) for mild solutions of stochastic nonlocal diffusion equations in the sense that the solutions $u$ belong to the space $C^γ(D_T;L^p(Ω))$ with the optimal Hölder continuity index $γ$ (which is given explicitly), where $D_T:=[0,T]\times D$ for $T>0$, and $D\subset\mathbb{R}^d$ being a bounded domain. Then, by utilising tail estimates, we are able to obtain the estimates of mild solutions in $L^p(Ω;C^{γ^*}(D_T))$. What's more, we give an explicit formula between the two index $γ$ and $γ^*$. Moreover, we prove Hölder continuity for mild solutions on bounded domains. Finally, we present a new criteria to justify Hölder continuity for the solutions on bounded domains. The novelty of this paper is that our method are suitable to the case of time-space white noise.

math.PR↗

arXiv2018

Well-posedness of nonlinear transport equation by stochastic perturbation

We are concerned with multidimensional nonlinear stochastic transport equation driven by Brownian motions. For irregular fluxes, by using stochastic BGK approximations and commutator estimates, we gain the existence and uniqueness of stochastic entropy solutions. Besides, for $BV$ initial data, the $BV$ and Hölder regularities are also derived for the unique stochastic entropy solution. Particularly, for the transport equation, we gain a regularization result, i.e. while the existence fails for the transport equation, we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. This seems to be another explicit example (the first example is given in [22]) of a PDE of fluid dynamics that becomes well-posed under the influence of a multiplicative Brownian type noise.

math.PR↗

arXiv2017

On weak solutions of stochastic differential equations with sharp drift coefficients

We extend Krylov and Röckner's result \cite{KR} to the drift coefficients in critical Lebesgue space, and prove the existence and uniqueness of weak solutions for a class of SDEs. To be more precise, let $b: [0,T]\times{\mathbb R}^d\rightarrow{\mathbb R}^d$ be Borel measurable, where $T>0$ is arbitrarily fixed. Consider $$X_t=x+\int_0^tb(s,X_s)ds+W_t,\quad t\in[0,T], \, x\in{\mathbb R}^d,$$ where $\{W_t\}_{t\in[0,T]}$ is a $d$-dimensional standard Wiener process. If $b=b_1+b_2$ such that $b_1(T-\cdot)\in\mathcal{C}_q^0((0,T];L^p({\mathbb R}^d))$ with $2/q+d/p=1$ for $p,q\ge1$ and $\|b_1(T-\cdot)\|_{\mathcal{C}_q((0,T];L^p({\mathbb R}^d))}$ is sufficiently small, and that $b_2$ is bounded and Borel measurable, then there exits a unique weak solution to the above equation. Furthermore, we obtain the strong Feller property of the semi-group and existence of density associated with above SDE. Besides, we extend the classical partial differential equations (PDEs) results for $L^q(0,T;L^p({\mathbb R}^d))$ coefficients to $L^\infty_q(0,T;L^p({\mathbb R}^d))$ ones, and derive the Lipschitz regularity for solutions of second order parabolic PDEs (see Lemma 2.1).

math.AP↗

arXiv2017

Existence and uniqueness of $W^{1,r}_{loc}$-solutions for stochastic transport equations

We investigate a stochastic transport equation driven by a multiplicative noise. For $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient and $W^{1,r}({\mathbb R}^d)$ initial data, we obtain the existence and uniqueness of stochastic strong solutions (in $W^{1,r}_{loc}({\mathbb R}^d))$.In particular, when $r=\infty$, we establish a Lipschitz estimate for solutions and this question is opened by Fedrizzi and Flandoli in case of $L^q(0,T;L^p({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient. Moreover, opposite to the deterministic case where $L^q(0,T;W^{1,p}({\mathbb R}^d;{\mathbb R}^d))$ drift coefficient and $W^{1,p}({\mathbb R}^d)$ initial data may induce non-existence for strong solutions (in $W^{1,p}_{loc}({\mathbb R}^d)$), we prove that a multiplicative stochastic perturbation of Brownian type is enough to render the equation well-posed. It is an interesting example of a deterministic PDE that becomes well-posed under the influence of a multiplicative Brownian type noise. We extend the existing results \cite{FF2,FGP1} partially.

math.AP↗