arXiv subjects

Xiangchan Zhu

Publications and source records attributed to Xiangchan Zhu.

At least 19 recordsLinked to original sources

arXiv2026

Dynamical classical-field limit of Bosonic Gibbs states: Renormalized Hartree NLS correlations in 2D and 3D

In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus $\mathbb T^d$, $d=2,3$, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.

math-ph↗

arXiv2026

$\mathcal{P}(Φ)_2$ Theory from many-body quantum Gibbs states

We derive the $\mathcal{P}(Φ)_2$ measure on the two-dimensional unit torus as the rigorous limit of many-body quantum Gibbs states. In particular, the quantum model corresponding to the $Φ^{2p}_2$ measure is formulated in the grand-canonical ensemble with a general symmetric $p$-body interaction potential which is not required to have a factorized form. Unlike in the $Φ^4_2$ case investigated by Fröhlich--Knowles--Schlein--Sohinger in \cite{FKSS25}, in the treatment of higher-order interactions, Wick renormalization generates a full hierarchy of lower-order interactions which we organize systematically using a graphical formalism. One key ingredient of our analysis is a logarithmic stability estimate for both the nonlocal Hartree functional and the many-body Hamiltonian that is uniform in the interaction range, allowing us to use the positivity of the leading $p$-body interaction to control all lower-order terms generated by the Wick counterterms.

math-ph↗

arXiv2026

Global well-posedness for generalized parabolic Anderson model on the whole plane

For every $0<κ<\sqrt{5}-2$, we prove global existence for the two-dimensional generalized parabolic Anderson model on the whole plane $\mathbb R^2$ with nonlinearity $F\in C_b^2(\mathbb R)$, driven by an enhanced noise $(η,Ψ)$. The noise $η$ has polynomially weighted spatial Besov--Hölder regularity $-1-κ$, and $Ψ$ is the corresponding renormalized second-order object. If $F''$ is globally Lipschitz, the solution is unique. The proof combines a weight-compatible annular high--low decomposition with a paracontrolled transport representation. The final remainder is estimated simultaneously in a weighted $L^\infty$ norm and in a higher-order weighted parabolic Hölder norm, using two strictly different polynomial weights. This weight gap absorbs the polynomial losses generated by the enhanced noise, the localization procedure, and the transport coefficient. Several refinements of earlier work allow the maximum-principle and Schauder estimates to yield a global a priori bound for a larger range of $κ$. Uniqueness is proved in a time-dependent exponentially weighted topology.

math.AP↗

arXiv2026

Derivation of Gibbs measure from Gibbs state with the fractional Bessel interaction in Two Dimensions

We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_β(k)=\langle k\rangle^{-β}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<β\leq2$, where $\widehat{v}_β(k)$ is not summable and the quantum model cannot be written in the usual density-square form, as the associated self-energy diverges. We therefore need to renormalize the zero mode by a centered number-fluctuation term and then develop a detailed analysis for the high-frequency remainders. All this allows us to implement a low-frequency localization and obtain the convergence of the quantum relative free energy to the classical fractional-Bessel free energy, as well as the convergence of the reduced density matrices to the limiting Gibbs measure.

math-ph↗

arXiv2025

Langevin dynamics of lattice Yang-Mills-Higgs and applications

In this paper, we investigate the Langevin dynamics of various lattice formulations of the Yang--Mills--Higgs model, with an inverse Yang--Mills coupling $β$ and a Higgs parameter $κ$. The Higgs component is either a bounded field taking values in a compact target space, or an unbounded field taking values in a vector space in which case the model also has a Higgs mass parameter $m$. We study the regime where $(β,κ)$ are small in the first case or $(β,κ/m)$ are small in the second case. We prove the exponential ergodicity of the dynamics on the whole lattice via functional inequalities. We establish exponential decay of correlations for a broad class of observables, namely, the infinite volume measure exhibits a strictly positive mass gap. Moreover, when the target space of the Higgs field is compact, appropriately rescaled observables exhibit factorized correlations in the large $N$ limit. These extend the earlier results \cite{SZZ22} on pure lattice Yang--Mills to the case with a coupled Higgs field. Unlike pure lattice Yang--Mills where the field is always bounded, in the case where the coupled Higgs component is unbounded, the control of its behavior is much harder and requires new techniques. Our approach involves a disintegration argument and a delicate analysis of correlations to effectively control the unbounded Higgs component.

math.PR↗

arXiv2025

Gaussian Fluctuations for the Stochastic Landau-Lifshitz Navier-Stokes Equation in Dimension $D\geq2$

We revisit the large-scale Gaussian fluctuations for the stochastic Landau-Lifshitz Navier-Stokes equation (LLNS) at and above criticality, using the method in \cite{CGT24}. With the classical diffusive scaling in $d\geq 3$ and weak coupling scaling in $d=2$, we obtain the convergence of the regularised LLNS to a stochastic heat equation with a non-trivially renormalized coefficient. Moreover, we obtain an asymptotic expansion of the effective coefficient when $d\geq3$, and show that the one in \cite[Conjecture 6.5]{JP24} is incorrect. The new ingredient in our proof is a case-by-case analysis to track the evolution of the vector under the action of the Leray projection, combined with the use of the anti-symmetric part of the generator and a rotational change of coordinates to derive the desired decoupled stochastic heat equation from the original coupled system.

math.PR↗

arXiv2025

$Φ^4_3$ Theory from many-body quantum Gibbs states

We derive the $Φ^4_3$ measure on the torus as a rigorous limit of the quantum Gibbs state of an interacting Bose gas. To be precise, starting from many-body quantum mechanics, where the problem is linear and regular but involving non commutative operators, we justify the emergence of the $Φ^4_3$ measure as a semiclassical limit which captures the formation of Bose--Einstein condensation just above the critical temperature. We employ and develop several tools from both stochastic quantization and many-body quantum mechanics. Since the quantum problem is typically formulated using a nonlocal interaction potential, our first key step involves approximating the $Φ^4_3$ measure through a Hartree measure with nonlocal interaction, achieved by developing new techniques in paracontrolled calculus. The connection between the quantum problem and the Hartree measure emerges through a variational interplay between classical and quantum models.

math-ph↗

arXiv2025

Non-uniqueness of (Stochastic) Lagrangian Trajectories for Euler Equations

We are concerned with the (stochastic) Lagrangian trajectories associated with Euler or Navier-Stokes equations. First, in the vanishing viscosity limit, we establish sharp non-uniqueness results for positive solutions to transport equations advected by weak solutions of the 3D Euler equations that exhibit kinetic energy dissipation with $C_{t,x}^{1/3-}$ regularity. As a corollary, in conjunction with the superposition principle, this yields the non-uniqueness of associated (deterministic) Lagrangian trajectories. Second, in dimension $d\geq2$, for any $\frac{1}{p}+\frac{1}{r}>1$ or $p\in(1,2),r=\infty$, we construct solutions to the Euler or Navier-Stokes equations in the space $L_t^rL^p\cap L_t^1W^{1,1}$, demonstrating that the associated (stochastic) Lagrangian trajectories are not unique. Our result is sharp in 2D in the sense that: (1) in the stochastic case, for any vector field $v\in C_tL^p$ with $p>2$, the associated stochastic Lagrangian trajectory associated with $v$ is unique (see \cite{KR05}); (2) in the deterministic case, the LPS condition guarantees that for any weak solution $v\in C_tL^p$ with $p>2$ to the Navier-Stokes equations, the associated (deterministic) Lagrangian trajectory is unique. Our result is also sharp in dimension $d\geq2$ in the sense that for any divergence-free vector field $v\in L_t^1W^{1,s}$ with $s>d$, the associated (deterministic) Lagrangian trajectory is unique (see \cite{CC21}).

math.AP↗

arXiv2025

An inverse potential problem for the stochastic heat equation with space-time noise

This paper investigates an inverse potential problem for the stochastic heat equation driven by space-time Gaussian noise, which is spatially colored and temporally white. The objective is to determine the covariance operator of the random potential. We establish that the covariance operator can be uniquely identified from the correlation of the mild solution to the stochastic heat equation at a final time, where the initial conditions are specified by a complete orthonormal basis. The analysis relies on characterizing a tensor product structure inherent in the problem and utilizing the monotonicity properties of the operators associated with the system.

math.PR↗

arXiv2024

Non-uniqueness of Leray-Hopf solutions for stochastic forced Navier-Stokes equations

We consider stochastic forced Navier--Stokes equations on $\mathbb{R}^{3}$ starting from zero initial condition. The noise is linear multiplicative and the equations are perturbed by an additional body force. Based on the ideas of Albritton, Brué and Colombo \cite{ABC22}, we prove non-uniqueness of local-in-time Leray--Hopf solutions as well as joint non-uniqueness in law for solutions on $\mathbb{R}^{+}$. In the deterministic setting, we show that the set of forces, for which Leray--Hopf solutions are non-unique, is dense in $L^{1}_{t}L^{2}_{x}$. In addition, by a simple controllability argument we show that for every divergence-free initial condition in $L^{2}_{x}$ there is a force so that non-uniqueness of Leray--Hopf solutions holds.

math.PR↗

arXiv2024

Non-unique ergodicity for deterministic and stochastic 3D Navier--Stokes and Euler equations

We establish the existence of infinitely many stationary solutions, as well as ergodic stationary solutions, to the three dimensional Navier--Stokes and Euler equations in both deterministic and stochastic settings, driven by additive noise. These solutions belong to the regularity class $C(\mathbb{R};H^{\vartheta})\cap C^{\vartheta}(\mathbb{R};L^{2})$ for some $\vartheta>0$ and satisfy the equations in an analytically weak sense. The solutions to the Euler equations are obtained as vanishing viscosity limits of stationary solutions to the Navier--Stokes equations. Furthermore, regardless of their construction, every stationary solution to the Euler equations within this regularity class, which satisfies a suitable moment bound, is a limit in law of stationary analytically weak solutions to Navier--Stokes equations with vanishing viscosities. Our results are based on a novel stochastic version of the convex integration method, which provides uniform moment bounds locally in the aforementioned function spaces.

math.PR↗

arXiv2024

Kolmogorov $4/5$ law for the forced 3D Navier-Stokes equations

We identify a sufficient condition under which solutions to the 3D forced Navier--Stokes equations satisfy an $L^p$-in-time version of the Kolmogorov 4/5 law for the behavior of the averaged third order longitudinal structure function along the vanishing viscosity limit. The result has a natural probabilistic interpretation: the predicted behavior is observed on average after waiting for some sufficiently generic random time. The sufficient condition is satisfied e.g. by the solutions constructed by Bruè, Colombo, Crippa, De~Lellis, and Sorella. In this particular case, our results can be applied to derive a bound for the exponent of the third order absolute structure function in accordance with the Kolmogorov turbulence theory.

math.AP↗

arXiv2024

Global well-posedness for 2D generalized Parabolic Anderson Model via paracontrolled calculus

This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on $\mathbb{R}^+\times \mathbb{T}^2$ within the framework of paracontrolled calculus \cite{GIP15}. The model is given by the equation: \begin{equation*} (\partial_t-Δ) u=F(u)η \end{equation*} where $η\in C^{-1-κ}$ with $1/6>κ>0$, and $F\in C_b^2(\mathbb{R})$. Assume that $η\in C^{-1-κ}$ and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work \cite{CFW24} by A.Chandra, G.L. Feltes and H.Weber to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument).

math.AP↗

arXiv2024

Non-unique Ergodicity for the 2D Stochastic Navier-Stokes Equations with Derivative of Space-Time White Noise

We prove existence of infinitely many stationary solutions as well as ergodic stationary solutions for the stochastic Navier-Stokes equations on $\mathbb{T}^2$ \begin{align*} \dif u+÷(u\otimes u)\dif t+\nabla p\dif t&=Δu\dif t + (-Δ)^{\fa/2}\dif B_t,\ \ \ \ ÷u=0,\notag \end{align*} driven by derivative of space-time white noise, where $\fa\in[0,\frac13)$. In this setting, the solutions are not function valued and probabilistic renormalization is required to give a meaning to the equations. Finally, we show that the stationary distributions are not Gaussian distribution $N(0,\frac12(-Δ)^{\fa-1})$. The proof relies on a time-dependent decomposition and a stochastic version of the convex integration method which provides uniform moment bounds in some function spaces.

math.PR↗

arXiv2024

Anomalous and total dissipation due to advection by solutions of randomly forced Navier-Stokes equations

We propose a novel approach to induce anomalous dissipation through advection driven by turbulent fluid flows. Specifically, we establish the existence of a velocity field $v$ satisfying randomly forced Navier-Stokes equations, leading to total dissipation of kinetic energy in finite time when advecting a passive scalar. This dissipation phenomenon is uniform across viscosity parameters and initial conditions, representing a case of anomalous dissipation. We further explore dissipation induced by individual realizations of $v$. Our results extend to scenarios where the passive scalar is replaced by solutions to two or three-dimensional deterministic Navier-Stokes equations advected by $v$.

math.AP↗

arXiv2023

Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise

We consider a family of singular surface quasi-geostrophic equations $$ \partial_{t}θ+u\cdot\nablaθ=-ν(-Δ)^{γ/2}θ+(-Δ)^{α/2}ξ,\qquad u=\nabla^{\perp}(-Δ)^{-1/2}θ, $$ on $[0,\infty)\times\mathbb{T}^{2}$, where $ν\geq 0$, $γ\in [0,3/2)$, $α\in [0,1/4)$ and $ξ$ is a space-time white noise. For the first time, we establish the existence of infinitely many non-Gaussian $\bullet$ probabilistically strong solutions for every initial condition in $C^η$, $η>1/2$ $\bullet$ ergodic stationary solutions The result presents a single approach applicable in the subcritical, critical as well as supercritical regime in the sense of Hairer (M. Hairer, A theory of regularity structures). It also applies in the particular setting $α=γ/2$ which formally possesses a Gaussian invariant measure. In our proof, we first introduce a modified Da Prato--Debussche trick which, on the one hand, permits to convert irregularity in time into irregularity in space and, on the other hand, increases the regularity of the linear solution. Second, we develop a convex integration iteration for the corresponding nonlinear equation which yields non-unique non-Gaussian solutions satisfying powerful global-in-time estimates and generating stationary as well as ergodic stationary solutions.

math.PR↗

arXiv2023

Large $N$ limit and $1/N$ expansion of invariant observables in $O(N)$ linear $σ$-model via SPDE

In this paper, we continue the study of large $N$ problems for the Wick renormalized linear sigma model, i.e. $N$-component $Φ^4$ model, in two spatial dimensions, using stochastic quantization methods and Dyson--Schwinger equations. We identify the large $N$ limiting law of a collection of Wick renormalized $O(N)$ invariant observables. In particular, under a suitable scaling, the quadratic observables converge in the large $N$ limit to a mean-zero (singular) Gaussian field denoted by $\mathcal{Q}$ with an explicit covariance; and the observables which are renormalized powers of order $2n$ converge in the large $N$ limit to suitably renormalized $n$-th powers of $\mathcal{Q}$. The quartic interaction term of the model has no effect on the large $N$ limit of the field, but has nontrivial contributions to the limiting law of the observables, and the renormalization of the $n$-th powers of $\mathcal{Q}$ in the limit has an interesting finite shift from the standard one. Furthermore, we derive the $1/N$ asymtotic expansion for the $k$-point functions of the quadratic observables by employing graph representations and analyzing the order of each graph from Dyson--Schwinger equations. Finally, turning to the stationary solutions to the stochastic quantization equations, with the Ornstein--Uhlenbeck process being the large $N$ limiting dynamic, we derive here its next order correction in stationarity, as described by an SPDE with the right-hand side having explicit fixed-time marginal law which involves the above field $\mathcal{Q}$.

math.PR↗

arXiv2023

An SPDE approach to perturbation theory of $Φ^4_2$: asymptoticity and short distance behavior

In this paper we study the perturbation theory of $Φ^4_2$ model on the whole plane via stochastic quantization. We use integration by parts formula (i.e. Dyson-Schwinger equations) to generate the perturbative expansion for the $k$-point correlation functions, and prove bounds on the remainder of the truncated expansion using PDE estimates; this in particular proves that the expansion is asymptotic. Furthermore, we derive short distance behaviors of the $2$-point function and the connected $4$-point function, also via suitable Dyson-Schwinger equations combined with PDE arguments.

math.PR↗