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Tusheng Zhang

Publications and source records attributed to Tusheng Zhang.

At least 55 records · Page 3Linked to original sources
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↗
arXiv2020

Strong existence and uniqueness of solutions of SDEs with time dependent Kato class coefficients

Consider stochastic differential equations (SDEs) in $\Rd$: $dX_t=dW_t+b(t,X_t)\d t$, where $W$ is a Brownian motion, $b(\cdot, \cdot)$ is a measurable vector field. It is known that if $|b|^2(\cdot, \cdot)=|b|^2(\cdot)$ belongs to the Kato class $\K_{d,2}$, then there is a weak solution to the SDE. In this article we show that if $|b|^2$ belongs to the Kato class $\K_{d,\a}$ for some $\a \in (0,2)$ ($\a$ can be arbitrarily close to $2$), then there exists a unique strong solution to the stochastic differential equations, extending the results in the existing literature as demonstrated by examples. Furthermore, we allow the drift to be time-dependent. The new regularity estimates we established for the solutions of parabolic equations with Kato class coefficients play a crucial role.

math.PR↗
arXiv2020

Ergodicity for stochastic conservation laws with multiplicative noise

We proved that there exists a unique invariant measure for solutions of stochastic conservation laws with Dirichlet boundary condition driven by multiplicative noise. Moreover, a polynomial mixing property is established. This is done in the setting of kinetic solutions taking values in an L^1-weighted space.

math.PR↗
arXiv2020

Large deviation principles for first-order scalar conservation laws with stochastic forcing

In this paper, we established the Freidlin-Wentzell type large deviation principles for first-order scalar conservation laws perturbed by small multiplicative noise. Due to the lack of the viscous terms in the stochastic equations, the kinetic solution to the Cauchy problem for these first-order conservation laws is studied. Then, based on the well-posedness of the kinetic solutions, we show that the large deviations holds by utilising the weak convergence approach.

math.PR↗
arXiv2020

Strong Solutions to Reflecting Stochastic Differential Equations with Singular Drift

In this paper, we prove that there exists a unique strong solution to reflecting stochastic differential equations with merely measurable drift giving an affirmative answer to the longstanding problem. This is done through Zvonkin transformation and a careful analysis of the transformed reflecting stochastic differential equations on non-smooth time-dependent domains.

math.PR↗
arXiv2019

Stochastic heat equations with logarithmic nonlinearity

In this paper, we establish the existence and uniqueness of solutions to stochastic heat equations with logarithmic nonlinearity driven by Brownian motion on a bounded domain $D$ in the setting of $L^2(D)$ space. The result is valid for all initial values in $L^2(D)$. The logarithmic Sobolev inequality plays an important role.

math.PR↗
arXiv2019

Quadratic Transportation Cost Inequalities Under Uniform Distance For Stochastic Reaction Diffusion Equations Driven by Multiplicative Space-Time White Noise

In this paper, we established a quadratic transportation cost inequality for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise based on a new inequality we proved for the moments (under the uniform norm) of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.

math.PR↗
arXiv2018

Small time asymptotics for Brownian motion with singular drift

We establish a small time large deviation principle and a Varadhan type asymptotics for Brownian motion with singular drift on $\bR^d$ with $d\geq 3$ whose infinitesimal generator is $\frac12 Δ+ μ\cdot \nabla$, where each $μ_i$ of $μ= (μ_1, \cdots, μ_d)$ is a measure in some suitable Kato class.

math.PR↗
arXiv2018

Dirichlet boundary value problems for elliptic operators with measure data: a probabilistic approach

In this paper, we use probabilistic approach to prove that there exists a unique weak solution to the Dirichlet boundary value problem for second order elliptic equations whose coefficients are signed measures, and we will give a probabilistic representation of the solutions. This is the first time to study Dirichlet boundary value problems with this generality. The heat kernel estimates play a crucial role in our approach.

math.PR↗
arXiv2018

Systems of reflected quasilinear stochastic PDEs in a convex domain

This paper presents existence and uniqueness results for reflected system of quasilinear stochastic partial differential equations in a convex domain D from Rk. The method is based on the probabilistic interpretation of the solution by using the backward doubly stochastic differential equation. The solution is expressed as a pair (u,ν) where u is a predictable continuous process which takes values in a proper Sobolev space and ν is a random signed regular measure satisfying the minimal Skohorod condition.

math.PR↗
arXiv2017

Large Deviation Principles of Obstacle Problems for Quasilinear Stochastic PDEs

In this paper, we present a sufficient condition for the large deviation criteria of Budhiraja, Dupuis and Maroulas for functionals of Brownian motions. We then establish a large deviation principle for obstacle problems of quasi-linear stochastic partial differential equations. It turns out that the backward stochastic differential equations will play an important role.

math.PR↗
arXiv2017

Approximations of Stochastic Navier-Stokes Equations

In this paper we show that solutions of two-dimensional stochastic Navier-Stokes equations driven by Brownian motion can be approximated by stochastic Navier-Stokes equations forced by pure jump noise/random kicks.

math.PR↗
arXiv2017

Global solutions to reaction-diffusion equations with super-linear drift and multiplicative noise

Let $ξ(t\,,x)$ denote space-time white noise and consider a reaction-diffusion equation of the form \[ \dot{u}(t\,,x)=\tfrac12 u"(t\,,x) + b(u(t\,,x)) + σ(u(t\,,x)) ξ(t\,,x), \] on $\mathbb{R}_+\times[0\,,1]$, with homogeneous Dirichlet boundary conditions and suitable initial data, in the case that there exists $\varepsilon>0$ such that $\vert b(z)\vert \ge|z|(\log|z|)^{1+\varepsilon}$ for all sufficiently-large values of $|z|$. When $σ\equiv 0$, it is well known that such PDEs frequently have non-trivial stationary solutions. By contrast, Bonder and Groisman (2009) have recently shown that there is finite-time blowup when $σ$ is a non-zero constant. In this paper, we prove that the Bonder--Groisman condition is unimproveable by showing that the reaction-diffusion equation with noise is "typically" well posed when $\vert b(z) \vert =O(|z|\log_+|z|)$ as $|z|\to\infty$. We interpret the word "typically" in two essentially-different ways without altering the conclusions of our assertions.

math.PR↗
arXiv2017

2D Stochastic Chemotaxis-Navier-Stokes System

In this paper, we establish the existence and uniqueness of both mild(/variational) solutions and weak (in the sense of PDE) solutions of coupled system of 2D stochastic Chemotaxis-Navier-Stokes equations. The mild/variational solution is obtained through a fixed point argument in a purposely constructed Banach space. To get the weak solution we first prove the existence of a martingale weak solution and then we show that the pathwise uniqueness holds for the martingale solution.

math.PR↗