arXiv subjects

Rongchan Zhu

Publications and source records attributed to Rongchan Zhu.

At least 55 records · Page 3Linked to original sources

arXiv2020

Singular HJB equations with applications to KPZ on the real line

This paper is devoted to studying the Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which is not well-defined in the classical sense and shall be understood by using paracontrolled distribution method introduced in \cite{GIP15}. By a new characterization of weighted Hölder space and Zvonkin's transformation we prove some new a priori estimates, and therefore, establish the global well-posedness for singular HJB equations. As an application, the global well-posedness for KPZ equations on the real line in polynomial weighted Hölder spaces is obtained without using Cole-Hopf's transformation. In particular, we solve the conjecture posed in \cite[Remark 1.1]{PR18}.

math.PR↗

arXiv2020

Conservative stochastic 2-dimensional Cahn-Hilliard equation

We consider the stochastic 2-dimensional Cahn-Hilliard equation which is driven by the derivative in space of a space-time white noise. We use two different approaches to study this equation. First we prove that there exists a unique solution $Y$ to the shifted equation (see (1.4) below), then $X:=Y+{Z}$ is the unique solution to stochastic Cahn-Hilliard equaiton, where ${Z}$ is the corresponding O-U process. Moreover, we use Dirichlet form approach in \cite{Albeverio:1991hk} to construct the probabilistically weak solution the the original equation (1.1) below. By clarifying the precise relation between the solutions obtained by the Dirichlet forms aprroach and $X$, we can also get the restricted Markov uniquness of the generator and the uniqueness of martingale solutions to the equation (1.1).

math.PR↗

arXiv2020

Stochastic mSQG equations with multiplicative transport noises: white noise solutions and scaling limit

We consider the modified Surface Quasi-Geostrophic (mSQG) equation on the 2D torus $\mathbb{T}^2$, perturbed by multiplicative transport noise. The equation admits the white noise measure on $\mathbb{T}^2$ as the invariant measure. We first prove the existence of white noise solutions to the stochastic equation via the method of point vortex approximation, then, under a suitable scaling limit of the noise, we show that the solutions converge weakly to the unique stationary solution of the dissipative mSQG equation driven by space-time white noise. The weak uniqueness of the latter equation is also proved by following Gubinelli and Perkowski's approach in \cite{GP-18}.

math.PR↗

arXiv2019

Sharp interface limit of stochastic Cahn-Hilliard equation with singular noise

We study the the sharp interface limit of $\varepsilon$-dependent two dimensional stochastic Cahn-Hilliard equation driven by space-time white noise and conservative noise as $\varepsilon\to 0$. In the case when the noise is sufficiently small, by comparing the solutions to equation (1.1) with the approximation solution constructed in [ABC94], we show that the limit of the solutions is also solutions to the deterministic Hele-Shaw problem.

math.PR↗

arXiv2019

Well-posedness of Backward Stochastic Partial Differential Equations with Lyapunov Condition

In this paper we show the existence and uniqueness of strong solutions for a large class of backward SPDE where the coefficients satisfy a specific type Lyapunov condition instead of the classical coercivity condition. Moreover, based on the generalized variational framework, we also use the local monotonicity condition to replace the standard monotonicity condition, which is applicable to various quasilinear and semilinear BSPDE models.

math.PR↗

arXiv2019

Convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations on 2D torus

In this paper we discuss the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations driven by space-time white noise on $\T$. First we prove that the convergence rate for stochastic 2D heat equation is of order $α-δ$ in Besov space $\C^{-α}$ for $α\in(0,1)$ and $δ>0$ arbitrarily small. Then we obtain the convergence rate for Galerkin approximation of the stochastic Allen-Cahn equations of order $α-δ$ in $\C^{-α}$ for $α\in(0,2/9)$ and $δ>0$ arbitrarily small.

math.PR↗

arXiv2019

Stochastic Heat Equations for infinite strings with Values in a Manifold

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb{R}^+$ or $\mathbb{R}$ with values in a general Riemannian maifold, which is only assumed to be complete and stochastic complete. This work is an extension of the previous paper \cite{RWZZ17} on finite volume case. Moveover, we also obtain some functional inequalities associated to these Markov processes. This implies that on infinite volume case, the exponential ergodicity of the solution if the Ricci curvature is strictly positive and the non-ergodicity of the process if the sectional curvature is negative.

math.PR↗

arXiv2019

Weak solutions to the sharp interface limit of stochastic Cahn-Hilliard equations

We study the asymptotic limit, as $\varepsilon\searrow 0$, of solutions of the stochastic Cahn-Hilliard equation: $$ \partial_t u^\varepsilon=Δ\left(-\varepsilonΔu^\varepsilon+\frac{1}{\varepsilon}f(u^\varepsilon)\right)+\dot{\mathcal{W}}^\varepsilon_t, \\ $$ where $\mathcal{W}^\varepsilon=\varepsilon^σW$ or $\mathcal{W}^\varepsilon=\varepsilon^σW^\varepsilon$, $W$ is a $Q$-Wiener process and $W^\varepsilon$ is smooth in time and converges to $W$ as $\varepsilon\searrow 0$. In the case that $\mathcal{W}^\varepsilon=\varepsilon^σW$, we prove that for all $σ>\frac{1}{2}$, the solution $u^\varepsilon$ converges to a weak solution to an appropriately defined limit of the deterministic Cahn-Hilliard equation. In radial symmetric case we prove that for all $σ\geq\frac{1}{2}$, $u^\varepsilon$ converges to the deterministic Hele-Shaw model. In the case that $\mathcal{W}^\varepsilon=\varepsilon^σW^\varepsilon$, we prove that for all $σ>0$, $u^\varepsilon$ converges to the weak solution to the deterministic limit Cahn-Hilliard equation. In radial symmetric case we prove that $u^\varepsilon$ converges to deterministic Hele-Shaw model when $σ>0$ and converges to a stochastic model related to stochastic Hele-Shaw model when $σ=0$.

math.PR↗

arXiv2018

Weak universality of the dynamical $Φ_3^4$ model on the whole space

We prove the large scale convergence of a class of stochastic weakly nonlinear reaction-diffusion models on $\mathbb{R}^3$ to the dynamical $Φ^4_3$ model by paracontrolled distributions on weighted Besov space. Our approach depends on the delicate choice of the weight, the localization operator technique and a modification version of the maximal principle from [GH18].

math.PR↗

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

Determinstic and stochastic 2d Navier-Stokes equations with anisotropic viscosity

In this paper, we investigate both deterministic and stochastic 2D Navier Stokes equations with anisotropic viscosity. For the deterministic case, we prove the global well-posedness of the system with initial data in the anisotropic Sobolev space $\tilde{H}^{0,1}.$ For the stochastic case, we obtain existence of the martingale solutions and pathwise uniqueness of the solutions, which imply existence of the probabilistically strong solution to this system by the Yamada-Watanabe Theorem.

math.AP↗

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↗