arXiv subjects

Rongchan Zhu

Publications and source records attributed to Rongchan Zhu.

65 records · Page 4Linked to original sources

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↗

arXiv2012

BSDE and generalized Dirichlet forms: the finite dimensional case

We consider the following quasi-linear parabolic system of backward partial differential equations: $(\partial_t+L)u+f(\cdot,\cdot,u, \nabla uσ)=0$ on $[0,T]\times \mathbb{R}^d\qquad u_T=ϕ$, where $L$ is a possibly degenerate second order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator $L$ to obtain a probabilistic representation of the solution $u$ by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution of the PDE. A further main result is the generalization of the martingale representation theorem using the stochastic calculus associated to the generalized Dirichlet form given by $L$. The nonlinear term $f$ satisfies a monotonicity condition with respect to $u$ and a Lipschitz condition with respect to $\nabla u$.

math.PR↗

arXiv2012

BSDE and generalized Dirichlet forms: the infinite dimensional case

We consider the following quasi-linear parabolic system of backward partial differential equations on a Banach space $E$: $(\partial_t+L)u+f(\cdot,\cdot,u, A^{1/2}\nabla u)=0$ on $[0,T]\times E,\qquad u_T=ϕ$, where $L$ is a possibly degenerate second order differential operator with merely measurable coefficients. We solve this system in the framework of generalized Dirichlet forms and employ the stochastic calculus associated to the Markov process with generator $L$ to obtain a probabilistic representation of the solution $u$ by solving the corresponding backward stochastic differential equation. The solution satisfies the corresponding mild equation which is equivalent to being a generalized solution of the PDE. A further main result is the generalization of the martingale representation theorem in infinite dimension using the stochastic calculus associated to the generalized Dirichlet form given by $L$. The nonlinear term $f$ satisfies a monotonicity condition with respect to $u$ and a Lipschitz condition with respect to $\nabla u$.

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↗