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Tusheng Zhang

Publications and source records attributed to Tusheng Zhang.

At least 37 records · Page 2Linked to original sources
arXiv2024

Ergodicity of Stochastic two-phase Stefan problem driven by pure jump Lévy noise

In this paper, we consider stochastic two-phase Stefan problem driven by general jump Lévy noise. We first obtain the existence and uniqueness of the strong solution and then establish the ergodicity of the stochastic Stefan problem. Moreover, we give a precise characterization of the support of the invariant measures which provides the regularities of the stationary solutions of the stochastic free boundary problems.

math.PR↗
arXiv2024

Ergodicity for 2D Navier-Stokes equations with a degenerate pure jump noise

In this paper, we establish the ergodicity for stochastic 2D Navier-Stokes equations driven by a highly degenerate pure jump Lévy noise. The noise could appear in as few as four directions. This gives an affirmative anwser to a longstanding problem. The case of Gaussian noise was treated in Hairer and Mattingly [\emph{Ann. of Math.}, 164(3):993--1032, 2006]. To obtain the uniqueness of invariant measure, we use Malliavin calculus and anticipating stochastic calculus to establish the equi-continuity of the semigroup, the so-called {\em e-property}, and prove some weak irreducibility of the solution process.

math.PR↗
arXiv2024

Wong-Zakai approximations and support theorems for SDEs under Lyapunov conditions

In this paper, we establish the Stroock-Varadhan type support theorems for stochastic differential equations (SDEs) under Lyapunov conditions, which significantly improve the existing results in the literature where the coefficients of the SDEs are required to be globally Lipschitz and of linear growth. Our conditions are very mild to include many important models, e.g. Threshold Ornstein-Ulenbeck process, Stochastic SIR model, Stochastic Lotka-Volterra systems, Stochastic Duffing-van der Pol oscillator model, which have polynomial the coefficients. To obtain the support theorem, we prove a new Wong-Zakai approximation problem, which is of independent interest.

math.PR↗
arXiv2024

Uniform large deviations and metastability of random dynamical systems

In this paper, we first provide a criterion on uniform large deviation principles (ULDP) of stochastic differential equations under Lyapunov conditions on the coefficients, which can be applied to stochastic systems with coefficients of polynomial growth and possible degenerate driving noises. In the second part, using the ULDP criterion we preclude the concentration of limiting measures of invariant measures of stochastic dynamical systems on repellers and acyclic saddle chains and extend Freidlin and Wentzell's asymptotics theorem to stochastic systems with unbounded coefficients. Of particular interest, we determine the limiting measures of the invariant measures of the famous stochastic van der Pol equation and van der Pol Duffing equation whose noises are naturally degenerate. We also construct two examples to match the global phase portraits of Freidlin and Wentzell's unperturbed systems and to explicitly compute their transition difficulty matrices. Other applications include stochastic May-Leonard system and random systems with infinitely many equivalent classes.

math.PR↗
arXiv2024

Large deviations of fully local monotone stochastic partial differential equations driven by gradient-dependent noise

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H\subseteq V^*$ $$ \left\{ \begin{align} &dX_t=A(t,X_t)dt+B(t,X_t)dW_t,\ t\in (0,T]\\\\& X_0=x\in H, \end{align} \right. $$ where $$A: [0,T] \times V\rightarrow V^*,\ \ B:[0,T]\times V\rightarrow\ L_2(U,H)$$ are measurable maps, $L_2(U,H)$ is the space of Hilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wiener process.\par In this paper, we establish a small noise large deviation principle(LDP) for the solutions {$u^\varepsilon$}$_{\varepsilon>0}$ of the above SPDEs. The main contribution of this paper is the much more generality of our framework than that of the existing results. In particular, the diffusion coefficient $B(t,\cdot)$ may depend on the gradient of the solutions, which is of great interest in the field of SPDEs, but there are few existing results on the topic of LDP. The broader scope of the fully local monotone setting leads us to use different strategies and techniques. A combination of the pseudomonotone technique and compactness arguement plays a crucial role in the whole paper. Our framework is very general to include many interesting models that could not be covered by existing work, including stochastic quasilinear SPDEs, stochastic convection diffusion equation, stochastic 2D Liquid crystal equation, stochastic $p$-Laplace equation with gradient-dependent noise, stochastic 2D Navier-Stokes equation with gradient-dependent noise etc.

math.PR↗
arXiv2023

Large deviations of conservative stochastic partial differential equations

In this paper, we establish a large deviation principle for the conservative stochastic partial differential equations, whose solutions are related to stochastic differential equations with interaction. The weak convergence method and the contraction principle in the theory of large deviations play an important role.

math.PR↗
arXiv2023

Transportation cost inequalities for stochastic reaction diffusion equations on the whole line $\mathbb{R}$

In this paper, we established quadratic transportation cost inequalities for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise on the whole line $\mathbb{R}$. Since the space variable is defined on the unbounded domain $\mathbb{R}$, the inequalities are proved under a weighted $L^2$-norm and a weighted uniform metric in the so called $L^2_{tem}$, $C_{tem}$ spaces. The new moments estimates of the stochastic convolution with respect to space-time white noise play an important role. In addition, the transportation cost inequalities are also obtained for the stochastic reaction diffusion equations with random initial values.

math.PR↗
arXiv2023

Large deviations of reflected weakly interacting particle systems

In this paper, we prove a large deviation principle for the empirical measures of a system of weakly interacting diffusion with reflection. We adopt the weak convergence approach. To make this approach work, we show that the sequence of empirical measures of the controlled reflected system will converge to the weak solution of an associated reflected McKean--Vlasov equation.

math.PR↗
arXiv2023

Stochastic heat equations on moving domains

In this paper, we establish the well-posedness of stochastic heat equations on moving domains, which amounts to a study of infinite dimensional interacting systems. The main difficulty is to deal with the problems caused by the time-varying state spaces and the interaction of the particle systems. The interaction still occurs even in the case of additive noise. This is in contrast to stochastic heat equations in a fixed domain.

math.PR↗
arXiv2023

Approximation of the ergodic measure of SDEs with singular drift by Euler-Maruyama scheme

We study the approximation of the ergodic measure of the following stochastic differential equation (SDE) on $\mathbb{R}^d$: \begin{eqnarray}\label{e:SDEE} d X_t &=& (b_1(X_t)+b_2(X_t)) d t+σ(X_t) d W_t, \end{eqnarray} where $W_t$ is a $d$-dimensional standard Brownian motion, and $b_1: \mathbb{R}^d \mapsto \mathbb{R}^d$, $b_2: \mathbb{R}^d \mapsto \mathbb{R}^d$ and $σ: \mathbb{R}^d \mapsto \mathbb{R}^{d\times d}$ are the functions to be specified in Assumption 2.1 below. In particular, $b_1$ satisfies $b_1\in \mathbb{L}^\infty(\mathbb{R}^d)\cap \mathbb{L}^1(\mathbb{R}^d)$ or $b_1 \in \mathcal{C}_b^α(\mathbb{R}^d)$ with $α\in (0,1)$, which makes the standard numerical schemes not work or fail to give a good convergence rate. In order to overcome these two difficulties, we first apply a Zvonkin's transform to SDE and obtain a new SDE which has coefficients with nice properties and admits a unique ergodic measure $\widehat μ$, then discretize the new equation by Euler-Maruyama scheme to approximate $\widehat μ$, and finally use the inverse Zvonkin's transform to get an approximation of the ergodic measure of SDE, denoted by $μ$. Our approximation method is inspired by Xie and Zhang [22]. The proof of our main result is based on the method of introducing a stationary Markov chain, a key ingredient in this method is establishing the regularity of a Poisson equation, which is done by combining the classical PDE local regularity and a nice extension trick introduced by Gurvich [10].

math.PR↗
arXiv2022

Accessibility of SPDEs driven by pure jump noise and its applications

In this paper, we develop a new method to obtain the accessibility of stochastic partial differential equations driven by additive pure jump noise. An important novelty of this paper is to allow the driving noises to be degenerate. As an application, for the first time, we obtain the accessibility of a class of stochastic equations driven by pure jump degenerate noise, which cover 2D stochastic Navier-Stokes equations, stochastic Burgers type equations, singular stochastic p-Laplace equations, stochastic fast diffusion equations, etc. As a further application, we establish the ergodicity of singular stochastic p-Laplace equations and stochastic fast diffusion equations driven by additive pure jump noise, and we remark that the driving noises could be Levy processes with heavy tails.

math.PR↗
arXiv2022

Large deviations of stochastic heat equations with logarithmic nonlinearity

In this paper, we establish a large deviation principle for the solutions to the stochastic heat equations with logarithmic nonlinearity driven by Brownian motion, which is neither locally Lipschitz nor locally monotone. Nonlinear versions of Gronwall's inequalities and Log-Sobolev inequalities play an important role.

math.PR↗
arXiv2021

Large deviation principles for SDEs under locally weak monotonicity conditions

This paper establishes a Freidlin-Wentzell large deviation principle for stochastic differential equations(SDEs) under locally weak monotonicity conditions and Lyapunov conditions. We illustrate the main result of the paper by showing that it can be applied to SDEs with non-Lipschitzian coefficients, which can not be covered in the existing literature. These include the interesting biological models like stochastic Duffing-van der Pol oscillator model, stochastic SIR model, etc.

math.PR↗
arXiv2021

Global well-posedness to stochastic reaction-diffusion equations on the real line $\mathbb{R}$ with superlinear drifts driven by multiplicative space-time white noise

Consider the stochastic reaction-diffusion equation with logarithmic nonlinearity driven by space-time white noise: \begin{align}\label{1.a} \left\{ \begin{aligned} & \mathrm{d}u(t,x) = \frac{1}{2}Δu(t,x)\,\mathrm{d}t+ b(u(t,x)) \,\mathrm{d}t \nonumber\\ & ~~~~~~~~~~~~~~~~ + σ(u(t,x)) \,W(\mathrm{d}t,\mathrm{d}x), \ t>0, x\in I , \\ & u(0,x)=u_0(x), \quad x\in I .\nonumber \end{aligned} \right. \end{align} When $I$ is a compact interval, say $I=[0,1]$, the well-posedness of the above equation was established in [DKZ] (Ann. Prob. 47:1,2019). The case where $I=\mathbb{R}$ was left open. The essential obstacle is caused by the explosion of the supremum norm of the solution, $\sup_{x\in\mathbb{R}}|u(t,x)|=\infty$, making the usual truncation procedure invalid. In this paper, we prove that there exists a unique global solution to the stochastic reaction-diffusion equation on the whole real line $\mathbb{R}$ with logarithmic nonlinearity. Because of the nature of the nonlinearity, to get the uniqueness, we are forced to work with the first order moment of the solutions on the space $C_{tem}(\mathbb{R})$ with a specially designed norm $$\sup_{t\leq T, x\in\mathbb{R}}\left(|u(t,x)|e^{-λ|x|e^{βt}}\right),$$ where, unlike the usual norm in $C_{tem}(\mathbb{R})$, the exponent also depends on time $t$ in a particular way. Our approach depends heavily on the new, precise lower order moment estimates of the stochastic convolution and a new type of Gronwall's inequalities we obtained, which are of interest on their own right.

math.PR↗
arXiv2021

Stochastic 2D Navier-Stokes equations on time-dependent domains

We establish the existence and uniqueness of solutions to stochastic 2D Navier-Stokes equations in a time-dependent domain driven by Brownian motion. A martingale solution is constructed through domain transformation and appropriate Galerkin approximations on time-dependent spaces. The probabilistic strong solution follows from the pathwise uniqueness and the Yamada-Watanable theorem.

math.PR↗