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Weijun Xu

Publications and source records attributed to Weijun Xu.

At least 19 recordsLinked to original sources

TempJail: Temporal Jailbreak Attacks against Image-to-Video Generation Models

In recent years, image-to-video (I2V) generation models have made remarkable progress in subject consistency and temporal coherence, enabling high quality video synthesis. However, these advances also introduce new safety risks. Existing studies mainly focus on jailbreak attacks involving single frame violations, while largely overlooking the temporal dimension unique to video generation models. In this paper, we investigate three attack scenarios and uncover a temporal vulnerability in I2V systems: unsafe semantics may emerge not from a single frame, but from semantic composition over time. We further identify two key challenges in such attacks: temporal abstraction and semantic camouflage. To address these issues, we propose TempJail, a novel temporal jailbreak framework for I2V systems. For temporal abstraction, we decompose a target malicious caption into an initial frame visual condition and a temporal text instruction. For semantic camouflage, on the image side we model semantic injection as controlled latent perturbation in diffusion sampling and introduce gradient guidance from pretrained encoders. On the text side, we rewrite the caption into an innocuous ``subject-action-scene'' template that bypasses safety filters while preserving temporal guidance. In the black-box inference phase, these two modalities jointly enable malicious semantics to be gradually triggered over time. Experiments on closed-source commercial models, including Kling, Seedance, Veo and PixVerse, show that TempJail improves attack success rate over prior state-of-the-art methods by 23.3\% under GPT-5.2 evaluation and 22.0\% under human evaluation. Our codes are available at \href{https://github.com/luqi-glory/TempJail}{GitHub}.

cs.CV

Generalised stochastic curvature flow in $d \geq 2$, and sharp interface limit for the stochastic Allen-Cahn equation with nonlinear diffusion

We construct the local-in-time solution of the generalised anisotropic direction-dependent curvature flow in dimension $d \geq 2$ forced by a white-in-time and smooth-in-space Gaussian noise. This seems to be the first construction with a white-in-time noise which also allows spatial dependence, even in the simpler case of isotropic stochastic mean curvature flow. The main difficulty is that the stochastic PDE describing the flow has a multiplicative noise depending nonlinearly on both the solution and its gradient. The key technique is a transform developed in \cite{BKMZ20} based on rough characteristics that removes this rough multiplicative term. We also illustrate the relationship of this transform with previously known special situations. As an application of the construction, we show that in a short time interval, the sharp interface limit of the stochastic Allen-Cahn equation with nonlinear diffusion and the same noise (slightly smoothened in time) is given by the above direction-dependent stochastic curvature flow.

math.PR

Periodic homogenisation for two dimensional generalised parabolic Anderson model

We consider the periodic homogenisation problem for the generalised parabolic Anderson model on the two dimensional torus. We show that, for the renormalisation that respects Wick ordering, the homogenisation and renormalisation procedures commute. The main novelty is to identify a suitable solution ansatz beyond the usual para-controlled ansatz to set up a fixed point problem uniform in the homogenisation parameter. After that, one further utilises cancellations and resonances from the homogenisation oscillations to show convergences of both the solution and flux to the right limits. At a technical level, we frequently use integration by parts as well as "completing the products" to circumvent the incompatibility between para-products and variable coefficients. As a byproduct, we also show that the standard two dimensional generalised parabolic Anderson model can be constructed with para-controlled calculus without using commutator estimates.

math.AP

Parabolic Anderson Model in the Hyperbolic Space. Part II: Quenched Asymptotics

We establish the exact quenched asymptotic growth of the solution to the parabolic Anderson model (PAM) in the hyperbolic space with a regular, stationary, time-independent Gaussian potential. More precisely, we show that with probability one, the solution $u$ to PAM with constant initial data has pointwise growth asymptotics \[ u(t,x)\sim e^{L^{*}t^{5/3}+o(t^{5/3})} \] as $t \rightarrow +\infty$. Both the power $t^{5/3}$ on the exponential and the exact value of $L^*$ are different from their counterparts in the Euclidean situation. They are determined through an explicit optimisation procedure. Our proof relies on certain fine localisation techniques, which also reveals a stronger non-Euclidean localisation mechanism.

math.PR

Interface fluctuations for $1$D stochastic Allen-Cahn equation -- singular regime

We study interface fluctuations for the $1$D stochastic Allen-Cahn equation perturbed by half a spatial derivative of the spacetime white noise. This half derivative makes the solution distribution-valued, so that proper renormalization is needed to make sense of the solution. We show that if the noise is sufficiently small, then an analogue of the classical results by \cite{Fun95,BBDMP98} holds in this singular regime. More precisely, for initial data close to the traveling wave solution of the deterministic equation, under proper long time scaling, the solution still stays close to the family of traveling waves, and the interface location moves according to an approximate diffusion process. There is one interesting difference between our singular regime and the classical situation: even if the solution and its approximate phase separation point are both well defined, the intended diffusion describing the movement of the canonical candidate of the phase point is not (even for fixed $\eps$). Two infinite quantities arise from the derivation of such an SDE, one due to singularity of the noise, and the other from renormalization. Magically, it turns out that they cancel out each other, thus making the derivation of the interface SDE valid in the $\eps \rightarrow 0$ limit.

math.PR

Operator dependent para-controlled calculus, periodic homogenisation and singular PDEs

We develop a variant of the para-controlled distributions framework based on operator dependent (generalised) Besov spaces. These spaces were introduced by Kerkyacharian-Petrushev ([KP15]). In contrast to the Fourier decomposition in the classical setting, they are built from spectral decomposition of general self-adjoint elliptic operators. A major difference is that products of two spectrally localised functions in general have ``frequencies" spread over the whole spectrum. We obtain a decay estimate on ``high frequency" component of the product of two spectrally-localised functions. This enables us to obtain uniform bounds for the naturally associated para-product and commutator operations over a class of operators. This in particular includes the family of periodic homogenisation operators uniform in the oscillation parameter. Next, we show convergence properties of these operations in the periodic homogenisation setting. A key ingredient is the convergence of the generalised Littlewood-Paley block operators with quantitative dependence on the spectrum level. The proof combines a novel clustering argument to group nearby eigenvalues, Weyl's asymptotic formula that implies quantitative bounds on the sizes and locations of the eigenvalue clusters, together with a result by Kenig-Lin-Shen ([KLS13]) on convergence of individual eigenvalues. Finally, we apply this framework to periodic homogenisation problems for dynamical $Φ^4_3$ and KPZ equations. Assuming the convergence of stochastic objects to their homogenised limit, we show that the solutions and fluxes also converge to the corresponding limits. The convergence of the explicit stochastic terms with suitable renormalisations will be treated in a separate note.

math.AP

Parabolic Anderson Model in the Hyperbolic Space. Part I: Annealed Asymptotics

We establish the second-order moment asymptotics for a parabolic Anderson model $\partial_{t}u=(Δ+ξ)u$ in the hyperbolic space with a regular, stationary Gaussian potential $ξ$. It turns out that the growth and fluctuation asymptotics both are identical to the Euclidean situation. As a result, the solution exhibits the same moment intermittency property as in the Euclidean case. An interesting point here is that the fluctuation exponent is determined by a variational problem induced by the Euclidean (rather than hyperbolic) Laplacian. Heuristically, this is due to a curvature dilation effect: the geometry becomes asymptotically flat after suitable renormalisation in the derivation of the second-order asymptotics.

math.PR

Interface fluctuations for $1$D stochastic Allen-Cahn equation revisited

We revisit the interface fluctuation problem for the $1$D Allen-Cahn equation perturbed by a small space-time white noise. We show that if the initial data is a standing wave solution to the deterministic equation, then under proper long time scale, the solution is still close to the family of traveling wave solutions. Furthermore, the motion of the interface converges to an explicit stochastic differential equation. This extends the classical result in \cite{Fun95} to full small noise regime, and recovers the result in \cite{BBDMP98}. The proof builds on the analytic framework in \cite{Fun95}. Our main novelty is the construction of a series of functional correctors that are designed to recursively cancel potential divergences. Moreover, to show these correctors are well-behaved, we develop a systematic decomposition of Fréchet derivatives of the deterministic Allen-Cahn flow of all orders. This decomposition is of its own interest, and may be useful in other situations as well.

math.PR

Hairer-Quastel universality for KPZ -- polynomial smoothing mechanisms, general nonlinearities and Poisson noise

We consider a class of weakly asymmetric continuous microscopic growth models with polynomial smoothing mechanisms, general nonlinearities and a Poisson type noise. We show that they converge to the KPZ equation after proper rescaling and re-centering, where the coupling constant depends nontrivially on all details of the smoothing and growth mechanisms in the microscopic model. This confirms some of the predictions in [HQ18], and provides a first example of Hairer-Quastel type with both a generic nonlinearity (non-polynomial) and a non-Gaussian noise. The proof builds on the general discretisation framework of regularity structures ([EH19]), and employs the idea of using the spectral gap inequality to control stochastic objects as developed and systematised in [LOTT21,HS24], together with a new observation on the specific structure of the (discrete) Malliavin derivatives in our situation. This structure enables us to reduce the control of mixed $L^p$ spacetime norms (of arbitrarily large $p$) by certain $L^2$-norms in spacetime.

math.PR

Periodic homogenisation for $P(ϕ)_2$

We consider the periodic homogenisation problem for dynamical $P(ϕ)_2$, a toy model that combines both renormalisation in singular stochastic PDEs and homogenisation. Our result shows that the two limiting procedures commute in this case.

math.AP

A voltage-conductance kinetic system from neuroscience: probabilistic reformulation and exponential ergodicity

The voltage-conductance kinetic equation for an ensemble of neurons has been studied by many scientists and mathematicians, while its rigorous analysis is still at a premature stage. In this work, we obtain for the first time the exponential convergence to the steady state of this kinetic model in the linear setting. Our proof is based on a probabilistic reformulation, which allows us to investigate microscopic trajectories and bypass the difficulties raised by the special velocity field and boundary conditions in the macroscopic equation. We construct an associated stochastic process, for which proving the minorization condition becomes tractable, and the exponential ergodicity is then proved using Harris' theorem.

math.PR

Weak universality results for a class of nonlinear wave equations

We study the weak universality of the two-dimensional fractional nonlinear wave equation. For a sequence of Hamiltonians of high-degree potentials scaling to the fractional $Φ_2^4$, we first establish a \emph{sufficient and almost necessary} criteria for the convergence of invariant measures to the fractional $Φ_2^4$. Then we prove the convergence result for the sequence of associated wave dynamics to the (renormalized) cubic wave equation. Our constraint on the fractional index is independent of the degree of the nonlinearity. This extends the result of Gubinelli-Koch-Oh [Renormalisation of the two-dimensional stochastic nonlinear wave equations, Trans. Amer. Math. Soc. 370 (2018)] to a situation where we do not have a local Cauchy theory with highly supercritical nonlinearities.

math.AP

Long time behavior of stochastic NLS with a small multiplicative noise

We prove the global space-time bound for the mass critical nonlinear Schrödinger equation perturbed by a small multiplicative noise in dimension three. The associated scattering behavior are also obtained. We also prove a global Strichartz space-time bound for the linear stochastic model, which is new itself and serves a prototype model for the nonlinear case. The proof combines techniques from \cite{fan2018global}, \cite{fan2020long} as well as local smoothing estimates for linear Schrödinger operators.

math.AP

A Wong-Zakai theorem for the stochastic mass critical NLS

We prove a Wong-Zakai theorem for the defocusing mass-critical stochastic nonlinear Schrödinger equation (NLS) on $\mathbb{R}$. The main ingredient are careful mixtures of bootstrapping arguments at both deterministic and stochastic levels. Several subtleties arising from the proof mark the difference between the dispersive case and corresponding situations in SDEs and parabolic stochastic PDEs, as well as the difference between the large-$n$ case and the limiting ($n=\infty$) case.

math.AP

Weak universality of $Φ^4_3$: polynomial potential and general smoothing mechanism

We consider a class of stochastic reaction-diffusion equations on the three dimensional torus. The non-linearities are odd polynomials in the weakly non-linear regime, and the smoothing mechanisms are very general higher order perturbations of the Laplacian. The randomness is the space-time white noise without regularisation. We show that these processes converge to the dynamical $Φ^4_3 (λ)$ model, where the coupling constant $λ$ has an explicit expression involving non-trivial interactions between all details of the smoothing mechanism and the non-linearity.

math.PR

Subcritical approximations to stochastic defocusing mass-critical nonlinear Schrödinger equation on $\mathbb{R}$

We show robustness of various truncated and subcritical approximations to the stochastic defocusing mass-critical nonlinear Schrödinger equation (NLS) in dimension $d=1$, whose solution was constructed in [FX18] with one particular such approximation. The key ingredient in the proof is a uniform bound of the solutions to the family of deterministic mass-subcritical defocusing NLS.

math.AP

Global well-posedness for the defocusing mass-critical stochastic nonlinear Schrödinger equation on $\mathbb{R}$ at $L^2$ regularity

We prove global existence and stability of solution to the mass-critical stochastic nonlinear Schrödinger equation in $d=1$ at $L^2$ regularity. Our construction starts with the existence of solution to the truncated subcritical problem. With the presence of truncation, we construct the solution to the critical equation as the limit of subcritical solutions. We then obtain uniform bounds on the solutions to the truncated critical problems that allow us to remove truncation in the limit.

math.AP