arXiv subjects

Xiangchan Zhu

Publications and source records attributed to Xiangchan Zhu.

54 records · Page 3Linked to original sources

arXiv2018

Stochastic Heat Equations with Values in a Manifold via Dirichlet Forms

In this paper, we prove the existence of martingale solutions to the stochastic heat equation taking values in a Riemannian manifold, which admits Wiener (Brownian bridge) measure on the Riemannian path (loop) space as an invariant measure using a suitable Dirichlet form. Using the Andersson-Driver approximation, we heuristically derive a form of the equation solved by the process given by the Dirichlet form. Moreover, we establish the log-Sobolev inequality for the Dirichlet form in the path space. In addition, some characterizations for the lower bounds of the Ricci curvature are presented related to the stochastic heat equation.

math.PR↗

arXiv2018

A remark on global solutions to random 3D vorticity equations for small initial data

In this paper, we prove that the solution constructed in \cite{BR16} satisfies the stochastic vorticity equations with the stochastic integration being understood in the sense of the integration of controlled rough path introduced in \cite{G04}. As a result, we obtain the existence and uniqueness of the global solutions to the stochastic vorticity equations in 3D case for the small initial data independent of time, which can be viewed as a stochastic version of the Kato-Fujita result (see \cite{KF62}).

math.PR↗

arXiv2017

Piecewise linear approximation for the dynamical $Φ^4_3$ model

We construct a piecewise linear approximation for the dynamical $Φ_3^4$ model on $\mathbb{T}^3$ by the theory of regularity structures in [Hai14]. For the dynamical $Φ^4_3$ model it is proved in [Hai14] that a renormalisation has to be performed in order to define the nonlinear term. Compared to the results in [Hai14] we consider piecewise linear approximations to space-time white noise and prove that the solutions to the approximating equations converge to the solution to the dynamical $Φ^4_3$ model. The renormalisation in this case corresponds to adding the solution multiplied by a function depending on $t$ in the approximating equation.

math.PR↗

arXiv2017

Strong-Feller property for Navier-Stokes equations driven by space-time white noise

In this paper we prove strong Feller property for the Markov semigroups associated to the two or three dimensional Navier-Stokes (N-S) equations driven by space-time white noise using the theory of regularity structures introduced by Martin Hairer in [Hai14]. This implies global well-posedness of 2D N-S equation driven by space-time white noise starting from every initial point in $C^η$ for $η\in (-\frac{1}{2},0)$.

math.PR↗

arXiv2017

Dirichlet form associated with the $Φ_3^4$ model

We construct the Dirichlet form associated with the dynamical $Φ^4_3$ model obtained in [Hai14, CC13] and [MW16]. This Dirichlet form on cylinder functions is identified as a classical gradient bilinear form. As a consequence, this classical gradient bilinear form is closable and then by a well-known result its closure is also a quasi-regular Dirichlet form, which means that there exists another (Markov) diffusion process, which also admits the $Φ^4_3$ field measure as an invariant (even symmetrizing) measure.

math.PR↗

arXiv2017

Stochastic Heat Equations with Values in a Riemannian Manifold

The main result of this note is the existence of martingale solutions to the stochastic heat equation (SHE) in a Riemannian manifold by using suitable Dirichlet forms on the corresponding path/loop space. Moreover, we present some characterizations of the lower bound of the Ricci curvature by functional inequalities of various associated Dirichlet forms.

math.PR↗

arXiv2017

Three-dimensional Navier-Stokes equations driven by space-time white noise

In this paper we study 3D Navier-Stokes (NS) equation driven by space-time white noise by using regularity structure theory introduced in [Hai14] and paracontrolled distribution proposed in [GIP13]. We obtain local existence and uniqueness of solutions to the 3D Navier-Stokes equation driven by space-time white noise.

math.PR↗

arXiv2016

Ergodicity for the stochastic quantization problems on the 2D-torus

In this paper we study the stochastic quantization problem on the two dimensional torus and establish ergodicity for the solutions. Furthermore, we prove a characterization of the $Φ^4_2$ quantum field on the torus in terms of its density under translation. We also deduce that the $Φ^4_2$ quantum field on the torus is an extreme point in the set of all $L$-symmetrizing measures, where $L$ is the corresponding generator.

math.PR↗

arXiv2015

Restricted Markov uniqueness for the stochastic quantization of $P(Φ)_2$ and its applications

In this paper we obtain restricted Markov uniqueness of the generator and uniqueness of probabilistically weak solutions for the stochastic quantization problem in both the finite and infinite volume case by clarifying the precise relation between the solutions to the stochastic quantization problem obtained by the Dirichlet form approach and those obtained in [DD03] and in [MW15]. We prove that the solution $X-Z$, where $X$ is obtained by the Dirichlet form approach in [AR91] and $Z$ is the corresponding O-U process, satisfies the corresponding shifted equation (see (1.4) below). Moreover, we obtain that the infinite volume $P(Φ)_2$ quantum field is an invariant measure for the $X_0=Y+Z$, where $Y$ is the unique solution to the shifted equation.

math.PR↗

arXiv2015

Lattice approximation to the dynamical $Φ_3^4$ model

We study the lattice approximations to the dynamical $Φ^4_3$ model by paracontrolled distributions proposed in [GIP13]. We prove that the solutions to the lattice systems converge to the solution to the dynamical $Φ_3^4$ model in probability, locally in time. The dynamical $Φ_3^4$ model is not well defined in the classical sense. Renormalisation has to be performed in order to define the non-linear term. Formally, this renormalisation corresponds to adding an infinite mass term to the equation which leads to adding a drift term in the lattice systems.

math.PR↗

arXiv2015

Sub and supercritical stochastic quasi-geostrophic equation

In this paper, we study the 2D stochastic quasi-geostrophic equation on $\mathbb{T}^2$ for general parameter $α\in(0,1)$ and multiplicative noise. We prove the existence of weak solutions and Markov selections for multiplicative noise for all $α\in(0,1)$. In the subcritical case $α>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions. Moreover, we prove ergodicity for the solution of the stochastic quasi-geostrophic equations in the subcritical case driven by possibly degenerate noise. The law of large numbers for the solution of the stochastic quasi-geostrophic equations in the subcritical case is also established. In the case of nondegenerate noise and $α>2/3$ in addition exponential ergodicity is proved.

math.PR↗

arXiv2014

Approximating three-dimensional Navier-Stokes equations driven by space-time white noise

In this paper we study approximations to 3D Navier-Stokes (NS) equation driven by space-time white noise by paracontrolled distribution proposed in [GIP13]. A solution theory for this equation has been developed recently in [ZZ14] based on regularity structure theory and paracontrolled distribution. In order to make the approximating equation converge to 3D NS equation driven by space-time white noise, we should subtract some drift terms in approximating equations. These drift terms, which come from renormalizations in the solution theory, converge to the solution multiplied by some constant depending on approximations.

math.PR↗

arXiv2014

Existence and uniqueness of solutions to stochastic functional differential equations in infinite dimensions

In this paper, we present a general framework for solving stochastic functional differential equations in infinite dimensions in the sense of martingale solutions, which can be applied to a large class of SPDE with finite delays, e.g. $d$-dimensional stochastic fractional Navier-Stokes equations with delays, $d$-dimensional stochastic reaction-diffusion equations with delays, $d$-dimensional stochastic porous media equations with delays. Moreover, under local monotonicity conditions for the nonlinear term we obtain the existence and uniqueness of strong solutions to SPDE with delays.

math.PR↗

arXiv2014

A note on stochastic semilinear equations and their associated Fokker-Planck equations

In this paper we treat semilinear stochastic partial differential equations by two methods. First, we extend the framework of [BDR10] from a Hilbert space to a Gelfand triple and as an application we prove the existence of solutions for the Fokker-Planck equations associated to semilinear equations with space-time white noise and both with polynomially growing nonlinearities and Burgers type nonlinearities at the same time. Second we adopt the approximation technique from [BDR10] to obtain existence of unique strong solutions to semilinear stochastic partial differential equations driven by space-time white noise, generalizing corresponding known results from the literature.

math.PR↗

arXiv2013

Local existence and non-explosion of solutions for stochastic fractional partial differential equations driven by multiplicative noise

In this paper we prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise. We also establish that for this class of equations adding linear multiplicative noise provides a regularizing effect: the solutions will not blow up with high probability if the initial data is sufficiently small, or if the noise coefficient is sufficiently large. As applications our main results are applied to various types of SPDE such as stochastic reaction-diffusion equations, stochastic fractional Burgers equation, stochastic fractional Navier-Stokes equation, stochastic quasi-geostrophic equations and stochastic surface growth PDE.

math.PR↗

arXiv2013

BV functions in a Gelfand triple for differentiable measure and its applications

In this paper, we introduce a definition of BV functions for (non-Gaussian) differentiable measure in a Gelfand triple which is an extension of the definition of BV functions in [RZZ12], using Dirichlet form theory. By this definition, we can analyze the reflected stochastic quantization problem associated with a self-adjoint operator $A$ and a cylindrical Wiener process on a convex set $Γ$ in a Banach space $E$. We prove the existence of a martingale solution of this problem if $Γ$ is a regular convex set.

math.PR↗

arXiv2013

Large deviation principles for the stochastic quasi-geostrophic equation

In this paper we establish the large deviation principle for the stochastic quasi-geostrophic equation in the subcritical case with small multiplicative noise. The proof is mainly based on the stochastic control and weak convergence approach. Some analogous results are also obtained for the small time asymptotics of the stochastic quasi-geostrophic equation.

math.PR↗

arXiv2011

Stochastic quasi-geostrophic equation

In this note we study the 2d stochastic quasi-geostrophic equation in $\mathbb{T}^2$ for general parameter $α\in (0,1)$ and multiplicative noise. We prove the existence of martingale solutions and pathwise uniqueness under some condition in the general case, i.e. for all $α\in (0,1)$. In the subcritical case $α>1/2$, we prove existence and uniqueness of (probabilistically) strong solutions and construct a Markov family of solutions. In particular, it is uniquely ergodic for $α>2/3$ provided the noise is non-degenerate. In this case, the convergence to the (unique) invariant measure is exponentially fast. In the general case, we prove the existence of Markov selections.

math.PR↗