arXiv subjects

Shijie Shang

Publications and source records attributed to Shijie Shang.

At least 19 recordsLinked to original sources

arXiv2026

Transportation-cost inequalities for invariant measures of stochastic reaction-diffusion equations driven by space-time white noise

In this paper, we establish transportation-cost inequalities for invariant measures of stochastic reaction-diffusion equations driven by multiplicative space-time white noise. In particular, the transportation-cost inequalities are established with respect to the $L^1$ metric, the $L^2$ metric and the uniform metric under different dissipativity strengths of the drift. We first obtain a global-in-time and spatially uniform moment estimate for stochastic convolutions driven by space-time white noise, which is of independent interest. We then establish Lipschitz continuity of the solutions to the associated stochastic controlled equations with respect to the controls with a time-independent Lipschitz constant. Applying this Lipschitz continuity estimate, we finally prove transportation-cost inequalities for the invariant measures of stochastic reaction-diffusion equations.

math.PR↗

arXiv2026

Ergodicity of reflected stochastic reaction-diffusion equations driven by space-time white noise

We consider the reflected stochastic reaction-diffusion equation on $[0,1]$: \begin{align*} \left\{ \begin{aligned} d u(t,x) &=\frac{1}{2}\partial_{xx} u(t,x)dt +b(u(t,x))dt + \sigma(u(t,x)) W(dt,dx)+L(dt,dx),\\ u(t,x)&\geq 0, \quad t\geq 0, \ x\in [0,1],\\ u(0,x)&=u_0(x)\geq 0, \quad x\in [0,1],\\ u(t,0) &= u(t,1) = 0, \quad \forall\ t\geq 0, \end{aligned} \right. \end{align*} where the initial value $u_0$ is non-negative on $[0,1]$ satisfying $u_0(0)=u_0(1)=0$, and $ W(dt,dx)$ is a space-time white noise. The $L$ in the equation is a random measure on $[0,\infty)\times(0,1)$, which is a part of the solution pair $(u, L)$. In this paper, we establish the existence and uniqueness of invariant measures, as well as exponential mixing for the reflected stochastic reaction diffusion equation under the dissipative condition $$(b(x)-b(y))(x-y)\leq -\alpha (x-y)^2,$$ which include the coefficients having polynomial, even exponential growth. The big obstacle of utilizing the dissipative condition is the lack of the It\^{o} formula/energy equality for such equations. To circumvent the problem, we use the newly found method in our paper (arXiv:2606.26619, 2026) to fully exploit comparison principles of reflected stochastic reaction-diffusion equation.

math.PR↗

arXiv2026

Ergodicity of stochastic reaction-diffusion equations on unbounded domains driven by space-time white noise

We consider the stochastic reaction-diffusion equation on the whole space: \begin{align*} \left\{ \begin{aligned} du(t,x) &=\frac{1}{2}\partial_{xx} u(t,x) dt+b(u(t,x))dt+ \sigma(u(t,x)) W(dt,dx),\quad t\geq 0,\ x\in \mathbb{R},\\ u(0,x)&=u_0(x), \quad x\in \mathbb{R}, \end{aligned} \right. \end{align*} where $W(dt,dx)$ is a space-time white noise, $b$, $\sigma$ are measurable coefficients. We first show that the solution is not strong Feller, and then establish the existence and uniqueness of invariant measures, exponential mixing as well as irreducibility for the solutions. To overcome the difficulties caused by the unbounded domain, we design special controls and controlled equations to prove the irreducibility. To obtain the exponential mixing property under the dissipative condition $$(b(x)-b(y))(x-y)\leq -\alpha (x-y)^2,$$ the obstacle is the lack of the It\^{o} formula/energy equality. To circumvent the problem, we manage to find a new way to fully exploit comparison principles, which we believe could be useful for other type of stochastic partial differential equations driven by multiplicative space-time noise. We note that the dissipative condition allows the coefficients to be of polynomial, even exponential growth. There exist plenty of models that satisfy the dissipative condition, including the Allen-Cahn type equations. To the best of our knowledge, this is the first paper to establish the ergodicity, exponential mixing and irreducibility of stochastic reaction-diffusion equations (SRDEs) driven by multiplicative space-time noise on unbounded domains. The results on exponential mixing are also new for (SRDEs) driven by multiplicative space-time noise on bounded domains.

math.PR↗

arXiv2026

Stochastic curve shortening flow driven by a transport-type pure jump L\'evy noise

We study the existence and uniqueness, the regularity, and the long-time behavior of strong solutions to stochastic curve shortening flow driven by a transport-type pure jump L\'evy noise. To obtain the existence and uniqueness of strong solutions, we transform the equation into its equivalent It\^{o}-type stochastic partial differential equation via a transport equation, and apply the monotone method with Lyapunov-type conditions. The obstacles to investigate the long-time behavior are the weak dissipativity and singularity inherent in the equation. To this end, we establish an improved regularity and prove that these solutions converge pathwise to zero at an exponential rate.

math.PR↗

arXiv2026

STEP3-VL-10B Technical Report

We present STEP3-VL-10B, a lightweight open-source foundation model designed to redefine the trade-off between compact efficiency and frontier-level multimodal intelligence. STEP3-VL-10B is realized through two strategic shifts: first, a unified, fully unfrozen pre-training strategy on 1.2T multimodal tokens that integrates a language-aligned Perception Encoder with a Qwen3-8B decoder to establish intrinsic vision-language synergy; and second, a scaled post-training pipeline featuring over 1k iterations of reinforcement learning. Crucially, we implement Parallel Coordinated Reasoning (PaCoRe) to scale test-time compute, allocating resources to scalable perceptual reasoning that explores and synthesizes diverse visual hypotheses. Consequently, despite its compact 10B footprint, STEP3-VL-10B rivals or surpasses models 10$\times$-20$\times$ larger (e.g., GLM-4.6V-106B, Qwen3-VL-235B) and top-tier proprietary flagships like Gemini 2.5 Pro and Seed-1.5-VL. Delivering best-in-class performance, it records 92.2% on MMBench and 80.11% on MMMU, while excelling in complex reasoning with 94.43% on AIME2025 and 75.95% on MathVision. We release the full model suite to provide the community with a powerful, efficient, and reproducible baseline.

cs.CV↗

arXiv2026

PaCoRe: Learning to Scale Test-Time Compute with Parallel Coordinated Reasoning

We introduce Parallel Coordinated Reasoning (PaCoRe), a training-and-inference framework designed to overcome a central limitation of contemporary language models: their inability to scale test-time compute (TTC) far beyond sequential reasoning under a fixed context window. PaCoRe departs from the traditional sequential paradigm by driving TTC through massive parallel exploration coordinated via a message-passing architecture in multiple rounds. Each round launches many parallel reasoning trajectories, compacts their findings into context-bounded messages, and synthesizes these messages to guide the next round and ultimately produce the final answer. Trained end-to-end with large-scale, outcome-based reinforcement learning, the model masters the synthesis abilities required by PaCoRe and scales to multi-million-token effective TTC without exceeding context limits. The approach yields strong improvements across diverse domains, and notably pushes reasoning beyond frontier systems in mathematics: an 8B model reaches 94.5% on HMMT 2025, surpassing GPT-5's 93.2% by scaling effective TTC to roughly two million tokens. We open-source model checkpoints, training data, and the full inference pipeline to accelerate follow-up work.

cs.LG↗

arXiv2025

$L^2$-solutions to stochastic reaction-diffusion equations with superlinear drifts driven by space-time white noise^

Consider the following stochastic reaction-diffusion equation with logarithmic superlinear coefficient b, driven by space-time white noise W: $$ u_t(t,x) = (1/2)u_{xx}(t,x) + b(u(t,x)) + \sigma(u(t,x))W(dt,dx) $$ for $t > 0$ and $x \in [0,1]$, with initial condition $$ u(0,x) = u_0(x) $$ for $x \in [0,1]$, where $u_0 \in L^2[0,1]$. In this paper, we establish existence and uniqueness of probabilistically strong solutions in $C(R_+, L^2[0,1])$. Our result resolves a problem from [Ann. Probab. 47 (2019) 519-559] and provides an alternative proof of the non-blowup of $L^2[0,1]$ solutions from the same reference. We use new Gronwall-type inequalities. Due to nonlinearity, we work with first order moments, requiring precise estimates of the stochastic convolution.

math.PR↗

arXiv2025

StepFun-Formalizer: Unlocking the Autoformalization Potential of LLMs through Knowledge-Reasoning Fusion

Autoformalization aims to translate natural-language mathematical statements into a formal language. While LLMs have accelerated progress in this area, existing methods still suffer from low accuracy. We identify two key abilities for effective autoformalization: comprehensive mastery of formal-language domain knowledge, and reasoning capability of natural language problem understanding and informal-formal alignment. Without the former, a model cannot identify the correct formal objects; without the latter, it struggles to interpret real-world contexts and map them precisely into formal expressions. To address these gaps, we introduce ThinkingF, a data synthesis and training pipeline that improves both abilities. First, we construct two datasets: one by distilling and selecting large-scale examples rich in formal knowledge, and another by generating informal-to-formal reasoning trajectories guided by expert-designed templates. We then apply SFT and RLVR with these datasets to further fuse and refine the two abilities. The resulting 7B and 32B models exhibit both comprehensive formal knowledge and strong informal-to-formal reasoning. Notably, StepFun-Formalizer-32B achieves SOTA BEq@1 scores of 40.5% on FormalMATH-Lite and 26.7% on ProverBench, surpassing all prior general-purpose and specialized models.

cs.CL↗

arXiv2025

StepFun-Prover Preview: Let's Think and Verify Step by Step

We present StepFun-Prover Preview, a large language model designed for formal theorem proving through tool-integrated reasoning. Using a reinforcement learning pipeline that incorporates tool-based interactions, StepFun-Prover can achieve strong performance in generating Lean 4 proofs with minimal sampling. Our approach enables the model to emulate human-like problem-solving strategies by iteratively refining proofs based on real-time environment feedback. On the miniF2F-test benchmark, StepFun-Prover achieves a pass@1 success rate of $70.0\%$. Beyond advancing benchmark performance, we introduce an end-to-end training framework for developing tool-integrated reasoning models, offering a promising direction for automated theorem proving and Math AI assistant.

cs.AI↗

arXiv2025

Moderate deviation principles for stochastic 2D hydrodynamics-type systems with multiplicative jump noise

This paper establishes a moderate deviation principle for a class of stochastic 2D hydrodynamical-type systems driven by multiplicative jump noise. The proof does not require compactness of the embedding in the associated Gelfand triple, so the result applies to both bounded and unbounded domains. The combination of finite-dimensional projections and integration by parts is used to prove the strong continuity of the skeleton solution map with respect to weakly convergent controls. This approach avoids the time discretization and intricate jump estimates used in earlier treatments of noncompact settings.

math.PR↗

arXiv2024

Ergodicity of Stochastic two-phase Stefan problem driven by pure jump Lévy noise

In this paper, we consider stochastic two-phase Stefan problem driven by general jump Lévy noise. We first obtain the existence and uniqueness of the strong solution and then establish the ergodicity of the stochastic Stefan problem. Moreover, we give a precise characterization of the support of the invariant measures which provides the regularities of the stationary solutions of the stochastic free boundary problems.

math.PR↗

arXiv2024

Large deviations of fully local monotone stochastic partial differential equations driven by gradient-dependent noise

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H\subseteq V^*$ $$ \left\{ \begin{align} &dX_t=A(t,X_t)dt+B(t,X_t)dW_t,\ t\in (0,T]\\\\& X_0=x\in H, \end{align} \right. $$ where $$A: [0,T] \times V\rightarrow V^*,\ \ B:[0,T]\times V\rightarrow\ L_2(U,H)$$ are measurable maps, $L_2(U,H)$ is the space of Hilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wiener process.\par In this paper, we establish a small noise large deviation principle(LDP) for the solutions {$u^\varepsilon$}$_{\varepsilon>0}$ of the above SPDEs. The main contribution of this paper is the much more generality of our framework than that of the existing results. In particular, the diffusion coefficient $B(t,\cdot)$ may depend on the gradient of the solutions, which is of great interest in the field of SPDEs, but there are few existing results on the topic of LDP. The broader scope of the fully local monotone setting leads us to use different strategies and techniques. A combination of the pseudomonotone technique and compactness arguement plays a crucial role in the whole paper. Our framework is very general to include many interesting models that could not be covered by existing work, including stochastic quasilinear SPDEs, stochastic convection diffusion equation, stochastic 2D Liquid crystal equation, stochastic $p$-Laplace equation with gradient-dependent noise, stochastic 2D Navier-Stokes equation with gradient-dependent noise etc.

math.PR↗

arXiv2023

Large deviation principle for stochastic reaction-diffusion equations with super-linear drift on $\mathbb{R}$ driven by space-time white noise

In this paper, we consider stochastic reaction-diffusion equations with super-linear drift on the real line $\mathbb{R}$ driven by space-time white noise. A Freidlin-Wentzell large deviation principle is established by a modified weak convergence method on the space $C([0,T], C_{tem}(\mathbb{R}))$. Obtaining the main result in this paper is challenging due to the setting of unbounded domain, the space-time white noise, and the superlinear drift term without dissipation. To overcome these difficulties, the special designed norm on $C([0,T], C_{tem}(\mathbb{R}))$, one order moment estimates of the stochastic convolution and two nonlinear Gronwall-type inequalities play an important role.

math.PR↗

arXiv2023

Transportation cost inequalities for stochastic reaction diffusion equations on the whole line $\mathbb{R}$

In this paper, we established quadratic transportation cost inequalities for solutions of stochastic reaction diffusion equations driven by multiplicative space-time white noise on the whole line $\mathbb{R}$. Since the space variable is defined on the unbounded domain $\mathbb{R}$, the inequalities are proved under a weighted $L^2$-norm and a weighted uniform metric in the so called $L^2_{tem}$, $C_{tem}$ spaces. The new moments estimates of the stochastic convolution with respect to space-time white noise play an important role. In addition, the transportation cost inequalities are also obtained for the stochastic reaction diffusion equations with random initial values.

math.PR↗

arXiv2023

Hard Lefschetz theorems for free line bundles

We introduce a partial positivity notion for algebraic maps via the defect of semismallness. This positivity notion is modeled on $m$-positivity in the analytic setting and $m$-ampleness in the geometric setting. Using this positivity condition for algebraic maps, we establish Kähler packages, that is, Hard Lefschetz theorems and Hodge-Riemann bilinear relations, for the complete intersections of Chern classes of free line bundles.

math.AG↗

arXiv2022

Large deviations of stochastic heat equations with logarithmic nonlinearity

In this paper, we establish a large deviation principle for the solutions to the stochastic heat equations with logarithmic nonlinearity driven by Brownian motion, which is neither locally Lipschitz nor locally monotone. Nonlinear versions of Gronwall's inequalities and Log-Sobolev inequalities play an important role.

math.PR↗

arXiv2022

Well-posedness of stochastic partial differential equations with fully local monotone coefficients

Consider stochastic partial differential equations (SPDEs) with fully local monotone coefficients in a Gelfand triple $V\subseteq H \subseteq V^*$: \begin{align*} \left\{ \begin{aligned} dX(t) & = A(t,X(t))dt + B(t,X(t))dW(t), \quad t\in (0,T], X(0) & = x\in H, \end{aligned} \right. \end{align*} where \begin{align*} A: [0,T]\times V \rightarrow V^* , \quad B: [0,T]\times V \rightarrow L_2(U,H) \end{align*} are measurable maps, $L_2(U,H)$ is the space of Hilbert-Schmidt operators from $U$ to $H$ and $W$ is a $U$-cylindrical Wiener process. Such SPDEs include many interesting models in applied fields like fluid dynamics etc. In this paper, we establish the well-posedness of the above SPDEs under fully local monotonicity condition solving a longstanding open problem. The conditions on the diffusion coefficient $B(t,\cdot)$ are allowed to depend on both the $H$-norm and $V$-norm. In the case of classical SPDEs, this means that $B(\cdot,\cdot)$ could also depend on the gradient of the solution. The well-posedness is obtained through a combination of pseudo-monotonicity techniques and compactness arguments.

math.PR↗

arXiv2022

A remark on the Castelnuovo-Mumford regularity of powers of ideal sheaves

We show that a bound of the Castelnuovo-Mumford regularity of any power of the ideal sheaf of a smooth projective complex variety $X\subseteq\mathbb{P}^r$ is sharp exactly for complete intersections, provided the variety $X$ is cut out scheme-theoretically by several hypersurfaces in $\mathbb{P}^r$. This generalizes a result of Bertram-Ein-Lazarsfeld.

math.AG↗