arXiv subjects

Tusheng Zhang

Publications and source records attributed to Tusheng Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Small-time annealed large deviations principle for one-dimensional diffusions in a random environment

In this paper, we establish a small-time annealed path large deviation principle for one-dimensional diffusions in a random environment associated with the generator ${\mathcal L}_W f(x)=e^{-ρ(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $\{ρ(x,\cdot):x\in\mathbb R\}$ and $\{a(x,\cdot):x\in\mathbb R\}$ are random. We assume that for each fixed realization of the environment, $ρ$ and $a$ are continuous and locally exponentially integrable, and that the support of the associated intrinsic coordinates is compact and non-collapsing. This framework includes the extensively studied Brox diffusion $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion representing the environment. The Itô--McKean representation of the diffusions and the estimates of the first exit probabilities derived via Moser iteration play a crucial role.

math.PR↗

arXiv2026

Small-time asymptotics of heat kernels of one-dimensional diffusions in a random environment

We establish the Varadhan small-time asymptotics for the quenched and annealed heat kernels of one-dimensional diffusions in a random environment with generator $\mathcal L_W f(x)=e^{-ρ(x,W)}(e^{a(x,W)}f'(x))'$. The coefficients $a$ and $ρ$ are continuous in space and satisfy a local exponential moment condition. We assume that the law of the intrinsic coordinate map $Λ_W$ has compact support $\mathscr L$ consisting of strictly increasing functions. Let $q^W(t,x,y)$ and $q(t,x,y)=\mathbb E[q^W(t,x,y)]$ denote the quenched and annealed heat kernels with respect to Lebesgue measure, respectively. We prove that, for almost every environment $W$, $\lim_{t\downarrow0}t\log q^W(t,x,y)=-\frac12|Λ_W(y)-Λ_W(x)|^2$, and that $\lim_{t\downarrow0}t\log q(t,x,y)=-\frac12\min_{Λ\in\mathscr L}|Λ(y)-Λ(x)|^2$. Both limits hold uniformly on compact subsets of $\mathbb R^2$. The framework includes Brox diffusion, formally described by $dX_t=dB_t-\frac12\dot W(X_t)\,dt$, where $B$ is a standard Brownian motion and $W$ is an independent two-sided Brownian motion.

math.PR↗

arXiv2026

Non-blowup of stochastic heat equations by noise

Consider the heat equation: \begin{equation}\label{eq:00} \begin{cases} \displaystyle du(t,x)=\frac12Δu(t,x)\,dt+b(u(t,x))\,dt, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} It is well known that superlinear growth of the coefficient $b$ will result in a blowup of the solution $u$ at a finite time. In this paper, we show that a random noise will prevent the explosion of the solution. More precisely, we show that the stochastic heat equation: \begin{equation}\label{eq:01} \begin{cases} \displaystyle du(t,x)=\frac12Δu(t,x)\,dt+b(u(t,x))\,dt+u(t,x)^n\,dB_t, &t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} has a global solution if \begin{equation} z\,b(z)\leq C_b(1+z^2)+η|z|^{2n}, \qquad z\in\R. \end{equation} for some $η\in [0, \frac{1}{2})$, here $n$ is an arbitrary, but fixed integer, $B_t, t\geq 0$ is a Brownian motion. Truncation and comparison techniques play an important role.

math.PR↗

arXiv2026

Well-posedness and regularity of stochastic heat equations on moving domains

In this paper we investigate stochastic heat equations driven by multiplicative noise on moving domains. We establish the well-posedness within a nonhomogeneous variational framework. Furthermore, by combining stochastic De Giorgi iteration with Dirichlet parabolic estimates, we obtain the Hölder regularity of the solutions.

math.PR↗

arXiv2026

Stochastic Scalar Conservation Laws on Moving Hypersurfaces

We establish the well-posedness of stochastic scalar conservation laws on moving hypersurfaces driven by Brownian motion. To handle the interaction between stochastic forcing and evolving geometry, we derive an Itô formula on moving surfaces and introduce the notion of generalized entropy solutions incorporating the relevant stochastic interaction terms. A martingale entropy solution is constructed via the vanishing-viscosity method, based on a uniform $L^\infty$-bound in space and time, an $L^1$-estimate for the spatial gradient, an $L^1$-continuity estimate in time, and a suitable tightness argument. Pathwise uniqueness is established by adapting Kruzhkov's doubling-of-variables method to moving hypersurfaces, yielding an $L^1$-contraction property. Finally, together with the Yamada-Watanabe theorem, these results yield the well-posedness of the problem.

math.PR↗

arXiv2026

Transportation-cost inequalities for invariant measures of stochastic reaction-diffusion equations driven by space-time white noise

In this paper, we establish transportation-cost inequalities for invariant measures of stochastic reaction-diffusion equations driven by multiplicative space-time white noise. In particular, the transportation-cost inequalities are established with respect to the $L^1$ metric, the $L^2$ metric and the uniform metric under different dissipativity strengths of the drift. We first obtain a global-in-time and spatially uniform moment estimate for stochastic convolutions driven by space-time white noise, which is of independent interest. We then establish Lipschitz continuity of the solutions to the associated stochastic controlled equations with respect to the controls with a time-independent Lipschitz constant. Applying this Lipschitz continuity estimate, we finally prove transportation-cost inequalities for the invariant measures of stochastic reaction-diffusion equations.

math.PR↗

arXiv2026

Ergodicity of reflected stochastic reaction-diffusion equations driven by space-time white noise

We consider the reflected stochastic reaction-diffusion equation on $[0,1]$: \begin{align*} \left\{ \begin{aligned} d u(t,x) &=\frac{1}{2}\partial_{xx} u(t,x)dt +b(u(t,x))dt + σ(u(t,x)) W(dt,dx)+L(dt,dx),\\ u(t,x)&\geq 0, \quad t\geq 0, \ x\in [0,1],\\ u(0,x)&=u_0(x)\geq 0, \quad x\in [0,1],\\ u(t,0) &= u(t,1) = 0, \quad \forall\ t\geq 0, \end{aligned} \right. \end{align*} where the initial value $u_0$ is non-negative on $[0,1]$ satisfying $u_0(0)=u_0(1)=0$, and $ W(dt,dx)$ is a space-time white noise. The $L$ in the equation is a random measure on $[0,\infty)\times(0,1)$, which is a part of the solution pair $(u, L)$. In this paper, we establish the existence and uniqueness of invariant measures, as well as exponential mixing for the reflected stochastic reaction diffusion equation under the dissipative condition $$(b(x)-b(y))(x-y)\leq -α(x-y)^2,$$ which include the coefficients having polynomial, even exponential growth. The big obstacle of utilizing the dissipative condition is the lack of the Itô formula/energy equality for such equations. To circumvent the problem, we use the newly found method in our paper (arXiv:2606.26619, 2026) to fully exploit comparison principles of reflected stochastic reaction-diffusion equation.

math.PR↗

arXiv2026

Ergodicity of stochastic reaction-diffusion equations on unbounded domains driven by space-time white noise

We consider the stochastic reaction-diffusion equation on the whole space: \begin{align*} \left\{ \begin{aligned} du(t,x) &=\frac{1}{2}\partial_{xx} u(t,x) dt+b(u(t,x))dt+ σ(u(t,x)) W(dt,dx),\quad t\geq 0,\ x\in \mathbb{R},\\ u(0,x)&=u_0(x), \quad x\in \mathbb{R}, \end{aligned} \right. \end{align*} where $W(dt,dx)$ is a space-time white noise, $b$, $σ$ are measurable coefficients. We first show that the solution is not strong Feller, and then establish the existence and uniqueness of invariant measures, exponential mixing as well as irreducibility for the solutions. To overcome the difficulties caused by the unbounded domain, we design special controls and controlled equations to prove the irreducibility. To obtain the exponential mixing property under the dissipative condition $$(b(x)-b(y))(x-y)\leq -α(x-y)^2,$$ the obstacle is the lack of the Itô formula/energy equality. To circumvent the problem, we manage to find a new way to fully exploit comparison principles, which we believe could be useful for other type of stochastic partial differential equations driven by multiplicative space-time noise. We note that the dissipative condition allows the coefficients to be of polynomial, even exponential growth. There exist plenty of models that satisfy the dissipative condition, including the Allen-Cahn type equations. To the best of our knowledge, this is the first paper to establish the ergodicity, exponential mixing and irreducibility of stochastic reaction-diffusion equations (SRDEs) driven by multiplicative space-time noise on unbounded domains. The results on exponential mixing are also new for (SRDEs) driven by multiplicative space-time noise on bounded domains.

math.PR↗

arXiv2026

Reflected stochastic partial differential equations with fully local monotone coefficients in infinite dimensional domains

This paper establishes the well-posedness of stochastic partial differential equations with reflection in an infinite-dimensional ball, within the fully local monotone framework. Our result is very general, including many important models such as the stochastic Allen-Cahn equations, stochastic p-Laplacian equations and stochastic 3D tamed Navier-Stokes equations, as well as more complex systems like the stochastic Cahn-Hilliard equations and stochastic 2D liquid crystal models. The approach relies on the penalization method, pseudo-monotonicity techniques and Mazur's lemma.

math.PR↗

arXiv2026

Asymptotic Log-Harnack Inequality for Degenerate SPDEs with Reflection

By constructing a suitable coupling by change of measures, the asymptotic log- Harnack inequality is established for a class of degenerate SPDEs with reflection. This inequality implies the asymptotic heat kernel estimate, the uniqueness of the invariant probability measure, the asymptotic gradient estimate (hence, asymptotically strong Feller property), and the asymptotic irreducibility. As application, the main result is illustrated by d-dimensional degenerate stochastic Navie-Stokes equations with reflection, where the dissipative operator is the Dirichlet Laplacian with a power θ\geq 1 \vee \frac{d+2}{4}, which includes the Laplacian when d \geq 2.

math.PR↗

arXiv2025

Exponential ergodicity of Stochastic Evolution Equations with reflection

In this paper, we establish an exponential ergodicity for stochastic evolution equations with reflection in an infinite dimensional ball. As an application, we obtain the exponential ergodicity of stochastic Navier-Stokes equations with reflection. A coupling method plays an important role.

math.PR↗

arXiv2025

$L^2$-solutions to stochastic reaction-diffusion equations with superlinear drifts driven by space-time white noise^

Consider the following stochastic reaction-diffusion equation with logarithmic superlinear coefficient b, driven by space-time white noise W: $$ u_t(t,x) = (1/2)u_{xx}(t,x) + b(u(t,x)) + σ(u(t,x))W(dt,dx) $$ for $t > 0$ and $x \in [0,1]$, with initial condition $$ u(0,x) = u_0(x) $$ for $x \in [0,1]$, where $u_0 \in L^2[0,1]$. In this paper, we establish existence and uniqueness of probabilistically strong solutions in $C(R_+, L^2[0,1])$. Our result resolves a problem from [Ann. Probab. 47 (2019) 519-559] and provides an alternative proof of the non-blowup of $L^2[0,1]$ solutions from the same reference. We use new Gronwall-type inequalities. Due to nonlinearity, we work with first order moments, requiring precise estimates of the stochastic convolution.

math.PR↗

arXiv2025

Well-posedness of stochastic partial differential equations with fully local monotone coefficients

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$: \begin{align*} \left\{ \begin{aligned} dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T], X(0) & = x\in H, \end{aligned} \right. \end{align*} where \begin{align*} A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrow L_2(U,H) \end{align*} 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. Such SPDEs include many interesting models in applied fields like fluid dynamics etc. In this paper, we establish the well-posedness of the above SPDEs under fully local monotonicity condition solving a longstanding open problem. The conditions on the diffusion coefficient $B(t,\cdot)$ are allowed to depend on both the $H$-norm and $V$-norm. In the case of classical SPDEs, this means that $B(\cdot,\cdot)$ could also depend on the gradient of the solution. The well-posedness is obtained through a combination of pseudo-monotonicity techniques and compactness arguments.

math.PR↗

arXiv2025

Irreducibility and ergodicity of SPDEs driven by pure jump noise

The irreducibility is fundamental for the study of ergodicity of stochastic dynamical systems. The existing methods on the irreducibility of stochastic partial differential equations (SPDEs) and stochastic differential equations (SDEs) driven by pure jump noise are basically along the same lines as that for the Gaussian case, which are not particularly suitable for jump noise. As a result, restrictive conditions are usually placed on the driving jump noise. Basically the driving noises are additive type and more or less in the class of stable processes. In this paper, we develop a new and effective method to obtain the irreducibility of SPDEs and SDEs driven by multiplicative pure jump noise. The conditions placed on the coefficients and the driving noise are very mild, and in some sense they are necessary and sufficient. As an application of our main results, we remove all the restrictive conditions on the driving noises in the literature,and derive new irreducibility results of a large class of equations driven by pure jump noise, including SPDEs with locally monotone coefficients, SPDEs/SDEs with singular coefficients, nonlinear Schrödinger equations, etc. We emphasize that under our setting the driving noises could be compound Poisson processes, even allowed to be infinite dimensional. As further applications of the main results, we obtain the ergodicity of multi-valued, singular stochastic evolution inclusions such as stochastic 1-Laplacian evolution (total variation flow), stochastic sign fast diffusion equation, stochastic minimal surface flow, stochastic curve shortening flow, etc.

math.PR↗

arXiv2025

Strong well-posedness of the two-dimensional stochastic Navier-Stokes equation on moving domains

In this paper, we establish the strong($H^1$) well-posedness of the two dimensional stochastic Navier-Stokes equation with multiplicative noise on moving domains. Due to the nonlocality effect, this equation exhibits a ``piecewise" variational setting. Namely the global well-posedness of this equation is decomposed into the well-posedness of a family of stochastic partial differential equations(SPDEs) in the variational setting on each small time-interval. We first examine the well-posedness on each time interval, which does not have (nonhomogeneous) coercivity. Subsequently, we give an estimate of lower bound of length of the time-interval, which enables us to achieve the global well-posedness.

math.PR↗

arXiv2025

Stochastic Stefan problem on moving hypersurfaces: an approach by a new framework of nonhomogeneous monotonicity

The purpose of this paper is to establish the well-posedness of the stochastic Stefan problem on moving hypersurfaces. Through a specially designed transformation, it turns out we need to solve stochastic partial differential equations on a fixed hypersurface with a new kind of nonhomogeneous monotonicity involving a family of time-dependent operators. This new class of SPDEs is of independent interest and can also be applied to solve many other interesting models such as the stochastic $p$-Laplacian equations, stochastic Allen-Cahn equation and stochastic heat equations on time-dependent domains or hypersurfaces. (Monotone) Operator-valued calculus and geometric analysis of moving hypersurfaces play important roles in the study. Moreover, a forthcoming result on the well-posedness of stochastic 2D Navier-Stokes equation on moving domains is also based on our framework.

math.PR↗

arXiv2024

Well-Posedness of Stochastic Chemotaxis System

In this paper, we establish the existence and uniqueness of solutions of elliptic-parabolic stochastic Keller-Segel systems. The solution is obtained through a carefully designed localization procedure together with some a priori estimates. Both noise of linear growth and nonlinear noise are considered. The Lp Ito formula plays an important role.

math.PR↗

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↗