arXiv subjects

Weijun Xu

Publications and source records attributed to Weijun Xu.

35 records · Page 2Linked to original sources

arXiv2018

Sharp convergence of nonlinear functionals of a class of Gaussian random fields

We present a self-contained proof of a uniform bound on multi-point correlations of trigonometric functions of a class of Gaussian random fields. It corresponds to a special case of the general situation considered in [Hairer-Xu], but with improved estimates. As a consequence, we establish convergence of a class of Gaussian fields composite with more general functions. These bounds and convergences are useful ingredients to establish weak universalities of several singular stochastic PDEs.

math.PR↗

arXiv2018

Large-scale limit of interface fluctuation models

We extend the weak universality of KPZ in [Hairer-Quastel] to weakly asymmetric interface models with general growth mechanisms beyond polynomials. A key new ingredient is a pointwise bound on correlations of trigonometric functions of Gaussians in terms of their polynomial counterparts. This enables us to reduce the problem of a general nonlinearity with sufficient regularity to that of a polynomial.

math.PR↗

arXiv2018

Global well-posedness for the mass-critical stochastic nonlinear Schrödinger equation on $\mathbb{R}$: small initial data

We prove the global existence of solution to the small data mass critical stochastic nonlinear Schrödinger equation in $d=1$. We further show the stability of the solution under perturbation of initial data. Our construction starts with the existence of the solution to the truncated subcritical problem. We then obtain uniform bounds on these solutions that enable us to reach criticality and then remove the truncation.

math.PR↗

arXiv2018

Global well-posedness for the mass-critical stochastic nonlinear Schrödinger equation on $\mathbb{R}$: general $L^2$ data

We continue our study for the stochastic defocusing mass crtical nonlinear Schrödinger equation with conservative multiplicative noise, and show that it is globally well-posed for arbitrary initial data in $L_ω^{\infty}L_{x}^{2}$. The main ingredients are several stability type results for deterministic (modified) NLS, which have their own interest. We also give some results on other stochastic NLS type models.

math.AP↗

arXiv2018

Inverting the signature of a path

The aim of this article is to develop an explicit procedure that enables one to reconstruct any $C^1$ path (at natural parametrization) from its signature. We also explicitly quantify the distance between the reconstructed path and the original path in terms of the number of terms in the signature that are used for the construction and the modulus of continuity of the derivative of the path. A key ingredient in the construction is the use of a procedure of symmetrization that separates the behavior of the path at small and large scales.

math.CA↗

arXiv2017

Moments of 2D Parabolic Anderson Model

In this note, we use the Feynman-Kac formula to derive a moment representation for the 2D parabolic Anderson model in small time, which is related to the intersection local time of planar Brownian motions.

math.PR↗

arXiv2017

Hyperbolic development and inversion of signature

We develop a simple procedure that allows one to explicitly reconstruct any piecewise linear path from its signature. The construction is based on the development of the path onto the hyperbolic space.

math.CA↗

arXiv2017

Construction of $Φ^4_3$ diagrams for pedestrians

We aim to give a pedagogic and essentially self-contained presentation of the construction of various stochastic objects appearing in the dynamical $Φ^4_3$ model. The construction presented here is based on the use of paraproducts. The emphasis is on describing the stochastic objects themselves rather than introducing a solution theory for the equation.

math.PR↗

arXiv2017

Large scale behaviour of 3D continuous phase coexistence models

We study a class of three dimensional continuous phase coexistence models, and show that, under different symmetry assumptions on the potential, the large-scale behaviour of such models near a bifurcation point is described by the dynamical $Φ^p_3$ models for $p \in \{2,3,4\}$. This result is specific to space dimension $3$ and does not hold in dimension $2$.

math-ph↗

arXiv2016

Signature inversion for monotone paths

The aim of this article is to provide a simple sampling procedure to reconstruct any monotone path from its signature. For every N, we sample a lattice path of N steps with weights given by the coefficient of the corresponding word in the signature. We show that these weights on lattice paths satisfy the large deviations principle. In particular, this implies that the probability of picking up a "wrong" path is exponentially small in N. The argument relies on a probabilistic interpretation of the signature for monotone paths.

math.PR↗

arXiv2016

Weak universality of dynamical $Φ^4_3$: non-Gaussian noise

We consider a class of continuous phase coexistence models in three spatial dimensions. The fluctuations are driven by symmetric stationary random fields with sufficient integrability and mixing conditions, but not necessarily Gaussian. We show that, in the weakly nonlinear regime, if the external potential is a symmetric polynomial and a certain average of it exhibits pitchfork bifurcation, then these models all rescale to $Φ^4_3$ near their critical point.

math-ph↗

arXiv2013

A uniform estimate for rough paths

We prove an extension to the classical continuity theorem in rough paths. We show that two $p$-rough paths are close in all levels of iterated integrals provided the first $\lfl p \rfl$ terms are close in a uniform sense. Applications include estimation of the difference of the signatures of two uniformly close paths and convergence rates of Gaussian rough paths.

math.PR↗

arXiv2012

Concentration and exact convergence rates for expected Brownian signatures

The signature of a $d$-dimensional Brownian motion is a sequence of iterated Stratonovich integrals along the Brownian paths, an object taking values in the tensor algebra over $\RR^{d}$. In this note, we derive the exact rate of convergence for the expected signatures of piecewise linear approximations to Brownian motion. The computation is based on the identification of the set of words whose coefficients are of the leading order, and the convergence is concentrated on this subset of words. Moreover, under the choice of projective tensor norm, we give the explicit value of the leading term constant.

math.PR↗

arXiv2012

A simple proof of distance bounds for Gaussian rough paths

We derive explicit distance bounds for Stratonovich iterated integrals along two Gaussian processes (also known as signatures of Gaussian rough paths) based on the regularity assumption of their covariance functions. Similar estimates have been obtained recently in [Friz-Riedel, AIHP, to appear]. One advantage of our argument is that we obtain the bound for the third level iterated integrals merely based on the first two levels, and this reflects the intrinsic nature of rough paths. Our estimates are sharp when both covariance functions have finite 1-variation, which includes a large class of Gaussian processes. Two applications of our estimates are discussed. The first one gives the a.s. convergence rates for approximated solutions to rough differential equations driven by Gaussian processes. In the second example, we show how to recover the optimal time regularity for solutions of some rough SPDEs.

math.PR↗

arXiv2011

Inversion of signature for paths of bounded variation

We develop two methods to reconstruct a path of bounded variation from its signature. The first method gives a simple and explicit expression of any axis path in terms of its signature, but it does not apply directlty to more general ones. The second method, based on an approximation scheme, recovers any tree-reduced path from its signature as the limit of a uniformly convergent sequence of lattice paths.

math.CA↗

arXiv2011

On Number of Turns in Reduced Random Lattice Paths

We consider the tree-reduced path of symmetric random walk on $\ZZ^{d}$. It is interesting to ask about the number of turns $T_n$ in the reduced path after $n$ steps. This question arises from inverting signature for lattice paths. We show that, when $n$ is large, the mean and variance of $T_n$ have the same order as $n$, while the second order terms are O(1). We then use these estimates to obtain limit theorems for $T_n$. Similar results hold for any other finite patterns as well.

math.PR↗

arXiv2010

Computationally Efficient Estimation of Factor Multivariate Stochastic Volatility Models

An MCMC simulation method based on a two stage delayed rejection Metropolis-Hastings algorithm is proposed to estimate a factor multivariate stochastic volatility model. The first stage uses kstep iteration towards the mode, with k small, and the second stage uses an adaptive random walk proposal density. The marginal likelihood approach of Chib (1995) is used to choose the number of factors, with the posterior density ordinates approximated by Gaussian copula. Simulation and real data applications suggest that the proposed simulation method is computationally much more efficient than the approach of Chib. Nardari and Shephard (2006}. This increase in computational efficiency is particularly important in calculating marginal likelihoods because it is necessary to carry out the simulation a number of times to estimate the posterior ordinates for a given marginal likelihood. In addition to the MCMC method, the paper also proposes a fast approximate EM method to estimate the factor multivariate stochastic volatility model. The estimates from the approximate EM method are of interest in their own right, but are especially useful as initial inputs to MCMC methods, making them more efficient computationally. The methodology is illustrated using simulated and real examples.

stat.CO↗