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Jianliang Zhai

Publications and source records attributed to Jianliang Zhai.

At least 37 records · Page 2Linked to original sources
arXiv2023

Large Deviations for Stochastic Porous Media Equation on General Measure Spaces

In this paper, we establish the large deviation principles for stochastic porous media equations driven by time-dependent multiplicative noise on $σ$-finite measure space $(E,\mathcal{B}(E),μ)$, and the Laplacian replaced by a negative definite self-adjoint operator. The coefficient is only assumed to satisfy the increasing Lipschitz nonlinearity assumption without the restrictions to its monotone behavior at infinity for $L^2(μ)$-initial data or compact embeddings in the associated Gelfand triple. Applications include fractional powers of the Laplacian, i.e. $L=-(-Δ)^α,\ α\in(0,1]$, generalized $\rm Schr\ddot{o}dinger$ operators, i.e. $L=Δ+2\frac{\nabla ρ}ρ\cdot\nabla$, and Laplacians on fractals.

math.PR↗
arXiv2023

Stochastic generalized porous media equations driven by Lévy noise with increasing Lipschitz nonlinearities

We establish the existence and uniqueness of strong solutions to stochastic porous media equations driven by Lévy noise on a $σ$-finite measure space $(E,\mathcal{B}(E),μ)$, and with the Laplacian replaced by a negative definite self-adjoint operator. The coefficient $Ψ$ is only assumed to satisfy the increasing Lipschitz nonlinearity assumption, without the restriction $rΨ(r)\rightarrow\infty$ as $r\rightarrow\infty$ for $L^2(μ)$-initial data. We also extend the state space, which avoids the transience assumption on $L$ or the boundedness of $L^{-1}$ in $L^{r+1}(E,\mathcal{B}(E),μ)$ for some $r\geq1$. Examples of the negative definite self-adjoint operators include fractional powers of the Laplacian, i.e. $L=-(-Δ)^α,\ α\in(0,1]$, generalized Schrödinger operators, i.e. $L=Δ+2\frac{\nabla ρ}ρ\cdot\nabla$, and Laplacians on fractals.

math.PR↗
arXiv2023

Large deviation principles and Malliavin derivative for mean reflected stochastic differential equations

In this paper, we consider a class of reflected stochastic differential equations for which the constraint is not on the paths of the solution but on its law. We establish a small noise large deviation principle, a large deviation for short time and the Malliavin derivative. To prove large deviation principles, a sufficient condition for the weak convergence method, which is suitable for Mckean-Vlasov stochastic differential equation, plays an important role.

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↗
arXiv2022

Irreducibility of stochastic complex Ginzburg-Landau equations driven by pure jump noise and its applications

Considering irreducibility is fundamental for studying the ergodicity of stochastic dynamical systems. In this paper, we establish the irreducibility of stochastic complex Ginzburg-Laudau equations driven by pure jump noise. Our results are dimension free and the conditions placed on the driving noises are very mild. A crucial role is played by criteria developed by the authors of this paper and T. Zhang for the irreducibility of stochastic equations driven by pure jump noise. As an application, we obtain the ergodicity of stochastic complex Ginzburg-Laudau equations. We remark that our ergodicity result covers the weakly dissipative case with pure jump degenerate noise.

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

Mckean-Vlasov stochastic differential equations with oblique reflection on non-smooth time dependent domains

In this paper, we consider a class of Mckean-Vlasov stochastic differential equation with oblique reflection over an non-smooth time dependent domain. We establish the existence and uniqueness results of this class, address the propagation of chaos and prove a Fredlin-Wentzell type large deviations principle (LDP). One of the main difficulties is raised by the setting of non-smooth time dependent domain. To prove the LDP, a sufficient condition for the weak convergence method, which is suitable for Mckean-Vlasov stochastic differential equation, plays an important role.

math.PR↗
arXiv2022

The stochastic nonlinear Schrödinger equations driven by pure jump noise

In this paper, we establish the existence and uniqueness of solutions of stochastic nonlinear Schrödinger equations with additive jump noise in $L^2(\mathbb{R}^d)$. Our results cover all either focusing or defocusing nonlinearity in the full subcritical range of exponents as in the deterministic case.

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

Well-posedness and large deviations for 2-D Stochastic Navier-Stokes equations with jumps

The aim of this paper is threefold. Firstly, we prove the existence and the uniqueness of a global strong (in both the probabilistic and the PDE senses) $\mathrm{H}^{1}_2$-valued solution to the 2D stochastic Navier-Stokes equations (SNSEs) driven by a multiplicative Lévy noise under the natural Lipschitz on balls and linear growth assumptions on the jump coefficient. Secondly, we prove a Girsanov-type theorem for Poisson random measures and apply this result to a study of the well-posedness of the corresponding stochastic controlled problem for these SNSEs. Thirdly, we apply these results to establish a Freidlin-Wentzell-type large deviation principle for the solutions of these SNSEs by employing the weak convergence method introduced in papers [16][18].

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↗
arXiv2021

Well-posedness of Stochastic 2D Hydrodynamics type Systems with Multiplicative Lévy Noises

We establish the existence and uniqueness of solutions to an abstract nonlinear equation driven by a multiplicative noise of Lévy type, which covers many hydrodynamical models including 2D Navier-Stokes equations, 2D MHD equations, the 2D Magnetic Bernard problem, and several Shell models of turbulence. In the existing literature on this topic, besides the classical Lipschitz and one sided linear growth conditions, other assumptions, which might be untypical, are also required on the coefficients of the stochastic perturbations. This paper is to get rid of these untypical assumptions. Our assumption on the coefficients of stochastic perturbations is new even for the Wiener cases, and in some sense, is shown to be quite sharp. A new cutting-off argument and energy estimation procedure play an important role in establishing the existence and uniqueness under this assumption.

math.PR↗
arXiv2020

Large and moderate deviation principles for McKean-Vlasov SDEs with jumps

In this paper, we consider McKean-Vlasov stochastic differential equations (MVSDEs) driven by Lévy noise. By identifying the right equations satisfied by the solutions of the MVSDEs with shifted driving Lévy noise, we build up a framework to fully apply the weak convergence method to establish large and moderate deviation principles for MVSDEs. In the case of ordinary SDEs, the rate function is calculated by using the solutions of the corresponding skeleton equations simply replacing the noise by the elements of the Cameron-Martin space. It turns out that the correct rate function for MVSDEs is defined through the solutions of skeleton equations replacing the noise by smooth functions and replacing the distributions involved in the equation by the distribution of the solution of the corresponding deterministic equation (without the noise). This is somehow surprising. With this approach, we obtain large and moderate deviation principles for much wider classes of MVSDEs in comparison with the existing literature.

math.PR↗
arXiv2019

Averaging principle for stochastic real Ginzburg-Landau equation driven by $α$-stable process

In this paper, we study a system of stochastic partial differential equations with slow and fast time-scales, where the slow component is a stochastic real Ginzburg-Landau equation and the fast component is a stochastic reaction-diffusion equation, the system is driven by $α$-stable process with $α\in (1,2)$. Using the classical Khasminskii approach based on time discretization and the techniques of stopping times, we show that the slow component strong converges to the solution of the corresponding averaged equation under some suitable conditions.

math.PR↗
arXiv2018

Exponential Mixing for 3D Stochastic Primitive Equations

In this paper, we prove that weak solutions of 3D stochastic primitive equations have exponential mixing property if the noise is sufficiently smooth and non-degenerate. With the help of uniqueness of strong solution of 3D stochastic primitive equations, we obtain that all weak solutions which are limitations of Galerkin approximations share the same invariant measure. In particular, the invariant measure of strong solution is unique. The coupling method plays a key role.

math.PR↗
arXiv2018

Averaging principle for one dimensional stochastic Burgers equation

In this paper, we consider the averaging principle for one dimensional stochastic Burgers equation with slow and fast time-scales. Under some suitable conditions, we show that the slow component strongly converges to the solution of the corresponding averaged equation. Meanwhile, when there is no noise in the slow component equation, we also prove that the slow component weakly converges to the solution of the corresponding averaged equation with the order of convergence $1-r$, for any $0<r<1$.

math.PR↗