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Xiangchan Zhu

Bio: Xiangchan Zhu is an academic researcher from Chinese Academy of Sciences. The author has contributed to research in topics: Uniqueness & Martingale (probability theory). The author has an hindex of 12, co-authored 58 publications receiving 519 citations. Previous affiliations of Xiangchan Zhu include Beijing Jiaotong University & Peking University.


Papers
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Journal ArticleDOI
TL;DR: In this article, the authors prove existence and uniqueness of local solutions to the Navier-Stokes (N-S) equation driven by space-time white noise using two methods: the theory of regularity structures introduced by Martin Hairer in [16] and the paracontrolled distribution proposed by Gubinelli, Imkeller, Perkowski in [12].

76 citations

Journal ArticleDOI
TL;DR: In this paper, the authors prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise, and show that adding linear multiplicative noises 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.

46 citations

Journal ArticleDOI
TL;DR: In this paper, the authors established the large deviation principle for the stochastic quasi-geostrophic equation with small multiplicative noise in the subcritical case, based on the weak convergence approach.

43 citations

Posted Content
TL;DR: In this article, the authors prove the local existence and uniqueness of solutions for a class of stochastic fractional partial differential equations driven by multiplicative noise, and show that adding linear multiplicative noises 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.
Abstract: 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.

41 citations

Journal ArticleDOI
TL;DR: In this article, the authors studied the stochastic quantization problem on the two-dimensional torus and established ergodicity for the solutions, and proved a characterization of the quantum field on the torus in terms of its density under translation.
Abstract: 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 $${\Phi^4_2}$$ quantum field on the torus in terms of its density under translation. We also deduce that the $${\Phi^4_2}$$ quantum field on the torus is an extreme point in the set of all L-symmetrizing measures, where L is the corresponding generator.

34 citations


Cited by
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01 Jan 2009
TL;DR: In this paper, a criterion for the convergence of numerical solutions of Navier-Stokes equations in two dimensions under steady conditions is given, which applies to all cases, of steady viscous flow in 2D.
Abstract: A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

1,568 citations

Book ChapterDOI
31 Oct 2006

1,424 citations

Journal ArticleDOI
TL;DR: In this article, a canonical renormalization procedure for stochastic PDEs containing nonlinearities involving generalised functions is given, which is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of automorphisms.
Abstract: We give a systematic description of a canonical renormalisation procedure of stochastic PDEs containing nonlinearities involving generalised functions This theory is based on the construction of a new class of regularity structures which comes with an explicit and elegant description of a subgroup of their group of automorphisms This subgroup is sufficiently large to be able to implement a version of the BPHZ renormalisation prescription in this context This is in stark contrast to previous works where one considered regularity structures with a much smaller group of automorphisms, which lead to a much more indirect and convoluted construction of a renormalisation group acting on the corresponding space of admissible models by continuous transformations Our construction is based on bialgebras of decorated coloured forests in cointeraction More precisely, we have two Hopf algebras in cointeraction, coacting jointly on a vector space which represents the generalised functions of the theory Two twisted antipodes play a fundamental role in the construction and provide a variant of the algebraic Birkhoff factorisation that arises naturally in perturbative quantum field theory

197 citations

Posted Content
TL;DR: In this article, a general theorem on the convergence of appropriately renormalized models arising from nonlinear stochastic PDEs was proved, and the theory of regularity structures gave a fairly automated framework for studying these problems.
Abstract: We prove a general theorem on the stochastic convergence of appropriately renormalized models arising from nonlinear stochastic PDEs. The theory of regularity structures gives a fairly automated framework for studying these problems but previous works had to expend significant effort to obtain these stochastic estimates in an ad-hoc manner. In contrast, the main result of this article operates as a black box which automatically produces these estimates for nearly all of the equations that fit within the scope of the theory of regularity structures. Our approach leverages multi-scale analysis strongly reminiscent to that used in constructive field theory, but with several significant twists. These come in particular from the presence of "positive renormalizations" caused by the recentering procedure proper to the theory of regularity structure, from the difference in the action of the group of possible renormalization operations, as well as from the fact that we allow for non-Gaussian driving fields. One rather surprising fact is that although the "canonical lift" is of course typically not continuous on any Holder-type space containing the noise (which is why renormalization is required in the first place), we show that the "BPHZ lift" where the renormalization constants are computed using the formula given in arXiv:1610.08468, is continuous in law when restricted to a class of stationary random fields with sufficiently many moments.

152 citations