arXiv subjects

Jianliang Zhai

Publications and source records attributed to Jianliang Zhai.

At least 19 recordsLinked to original sources

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

Stochastic curve shortening flow driven by a transport-type pure jump Lévy noise

We study the existence and uniqueness, the regularity, and the long-time behavior of strong solutions to stochastic curve shortening flow driven by a transport-type pure jump Lévy noise. To obtain the existence and uniqueness of strong solutions, we transform the equation into its equivalent Itô-type stochastic partial differential equation via a transport equation, and apply the monotone method with Lyapunov-type conditions. The obstacles to investigate the long-time behavior are the weak dissipativity and singularity inherent in the equation. To this end, we establish an improved regularity and prove that these solutions converge pathwise to zero at an exponential rate.

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arXiv2025

Stochastic transport equation with Lévy noise

We study the stochastic transport equation with globally $β$-Hölder continuous and bounded vector field driven by a non-degenerate pure-jump Lévy noise of $α$-stable type. Whereas the deterministic transport equation may lack uniqueness, we prove the existence and pathwise uniqueness of a weak solution in the presence of a multiplicative pure jump noise, assuming $\fracα{2}+β>1$. Notably, our results cover the entire range $α\in (0,2)$, including the supercritical regime $α\in(0,1)$ where the driving noise exhibits notoriously weak regularization. A key step of our strategy is the development of a \emph{sharp} $C^{1+δ}$-diffeomorphism and new regularity results for the Jacobian determinant of the stochastic flow associated to its stochastic characteristic equation. These novel probabilistic results are of independent interest and constitute a substantial component of our work. Our results are the first full generalization of the celebrated paper by Flandoli, Gubinelli, and Priola [Invent. Math. 2010] from the Brownian motion to the pure jump Lévy noise. To the best of our knowledge, this appears to be the first example of a partial differential equation of fluid dynamics where well-posedness is restored by the influence of a non-degenerate pure-jump noise.

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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.

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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↗

arXiv2025

Stochastic logarithmic Schrödinger equations driven by Lévy noise

In this paper, we study the stochastic logrithmic Schrödinger equation with saturated nonlinear multiplicative Lévy noise. The global well-posedness is established for the stochastic logrithmic Schrödinger equation in an appropriate Orlicz space by construct solutions of a regularized equation converging strongly to a solution to the original equation.

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

Large deviation principles for stochastic nonlinear Schrodinger equations driven by Levy noise

In this work we establish a Freidlin-Wentzell type large deviation principle for stochastic nonlinear Schrödinger equation, with either focusing or defocusing nonlinearity, driven by nonlinear multiplicative Lévy noise in the Marcus canonical form. This task is challenging in the current setting due to the presence of the power-type nonlinear term, the lack of regularization effect of the Schrödinger operator and the absence of compactness of embeddings. To overcome these difficulties, we employ a regularization procedure based on Yosida approximations and implement techniques such as time discretization, cut-off arguments, and relative entropy estimates of sequences of probability measures. Our innovative approach circumvents the need for compactness conditions, distinguishing our work from previous studies.

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↗

arXiv2024

Large deviations for 2D Stochastic Chemotaxis-Navier-Stokes System

In this paper, we establish a large deviation principle for 2D stochastic Chemotaxis-Navier-Stokes equation perturbed by a small multiplicative noise. The main difficulties come from the lack of a suitable compact embedding into the space occupied by the solutions and the inherent complexity of equation. Finite dimensional projection arguments and introducing suitable stopping times play important roles.

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.

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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 for locally monotone stochastic partial differential equations driven by Lévy noise

We establish a Freidlin-Wentzell type large deviation principle (LDP) for a class of stochastic partial differential equations with locally monotone coefficients driven by Lévy noise. Our results essentially improve a recent work on this topic (Bernoulli, 2018) by the second named author of this paper and his collaborator, because we drop the compactness embedding assumptions, and we also make the conditions for the coefficient of the noise term more specific and weaker. To obtain our results, we utilize an improved sufficient criteria of Budhiraja, Chen, Dupuis, and Maroulas for functions of Poisson random measures, and the techniques introduced by the first and second named authors of this paper in \cite{WZSIAM} play important roles. As an application, for the first time, the Freidlin-Wentzell type LDPs for many SPDEs driven by Lévy noise in unbounded domains of $\mathbb{R}^d$, which are generally lack of compactness embeddings properties, are achieved, like e.g., stochastic $p$-Laplace equation, stochastic Burgers-type equations, stochastic 2D Navier-Stokes equations, stochastic equations of non-Newtonian fluids, etc.

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 for Stochastic Generalized Porous Media Equations driven by Lévy Noise

We establish a large deviation principle (LDP) for a class of stochastic porous media equations driven by Lévy-type noise on a $σ$-finite measure space $(E,\mathcal{B}(E),μ)$, with the Laplacian replaced by a negative definite self-adjoint operator. One of the main contributions of this paper is that we do not assume the compactness of embeddings in the corresponding Gelfand triple, and to compensate for this generalization, a new procedure is provided. This is the first paper to deal with LDPs for stochastic evolution equations with Lévy noise without compactness conditions. The coefficient $Ψ$ is assumed to satisfy nondecreasing Lipschitz nonlinearity, so an important physical problem covered by this case is the Stefan problem. Numerous examples of negative definite self-adjoint operators are applicable to our results, for example, for open $E\subset\Bbb{R}^d$, $L=$ Laplacian or fractional Laplacians, i.e., $L=-(-Δ)^α,\ α\in(0,1]$, generalized Schrödinger operators, i.e., $L=Δ+2\frac{\nabla ρ}ρ\cdot\nabla$, Laplacians on fractals is also included.

math.PR↗

arXiv2023

Large deviation principle for stochastic reaction-diffusion equations with super-linear drift on $\mathbb{R}$ driven by space-time white noise

In this paper, we consider stochastic reaction-diffusion equations with super-linear drift on the real line $\mathbb{R}$ driven by space-time white noise. A Freidlin-Wentzell large deviation principle is established by a modified weak convergence method on the space $C([0,T], C_{tem}(\mathbb{R}))$. Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space-time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the special designed norm on $C([0,T], C_{tem}(\mathbb{R}))$, one order moment estimates of the stochastic convolution and two nonlinear Gronwall-type inequalities play an important role.

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