change of measure

Gaussian processes (GPs) struggle to rigorously incorporate uncountably infinite-dimensional functional prior information—such as boundary conditions or global physical constraints satisfied by PDE solutions. Method: This paper proposes a unified modeling framework grounded in reproducing kernel Hilbert spaces (RKHS), establishing for the first time a rigorous equivalence between the GP conditional expectation and orthogonal projection in RKHS. This enables direct embedding of functional constraints (e.g., Dirichlet or Neumann boundary conditions) into the GP prior, bypassing conventional pseudo-point approximations. Contribution/Results: We provide theoretical guarantees on existence, uniqueness, and convergence of the constrained GP posterior. Computationally, we design a practical numerical approximation algorithm. Experiments on PDE inverse problems demonstrate substantial improvements in uncertainty quantification accuracy and posterior consistency. The framework delivers a rigorous, general, and computationally tractable paradigm for integrating domain knowledge into Bayesian modeling.

This work addresses the lack of theoretical convergence advantages—specifically, rates faster than $O(n^{-1/2})$—for kernel herding in infinite-dimensional reproducing kernel Hilbert spaces (RKHS). We propose a deterministic sampling framework grounded in Gibbs measures, wherein node configurations are selected by minimizing the worst-case integration error under a suitably constructed joint distribution. This marks the first systematic incorporation of Gibbs measure theory into deterministic numerical integration analysis. Theoretically, our method yields tighter concentration inequalities for integration error in RKHS compared to i.i.d. Monte Carlo, establishing a strictly superior worst-case error bound. Empirically, preliminary experiments demonstrate super-root-$n$ convergence rates beyond the worst-case setting. The core innovation lies in the synergistic integration of Gibbs measures with worst-case error analysis, providing the first theoretically grounded acceleration mechanism for kernel herding.

This paper addresses the lack of rigorous mathematical foundations for Local Stochastic Intensity (LSI) models in credit risk modeling. Specifically, it resolves, for the first time, the Markovian projection inverse problem for pure-jump processes: reconstructing the original jump dynamics reversibly from knowledge of only the one-dimensional marginal distribution. Methodologically, the approach integrates Itô semimartingale theory, decomposition of random measures, Lévy-driven local time change, and inversion of projection operators to achieve exact marginal distribution matching. The key contribution is the construction of the first explicitly calibratable LSI model, thereby providing theoretical grounding for jump-type Local Stochastic Volatility (LSV) frameworks in credit risk. Empirically, the model significantly improves the fit of default probability curves and constitutes the first analytically tractable framework for high-dimensional jump risk modeling.

This paper addresses the conditional distribution of Banach space-valued jointly Gaussian random variables. It establishes that the conditional distribution remains Gaussian and develops a finite-dimensional approximation scheme based on Banach space-valued martingales to compute the conditional mean and covariance operator exactly. Methodologically, it unifies nuclear norm convergence and weak convergence analyses—yielding, for the first time, a rigorously convergent Gaussian conditioning theory in general Banach spaces. The framework applies broadly, including to reproducing kernel Hilbert spaces (RKHS) and spaces of continuous functions. For continuous Gaussian process paths, it guarantees uniform convergence of the conditional mean and covariance functions, as well as weak convergence of the conditional probability measures. These results provide a rigorous, general mathematical foundation for infinite-dimensional statistical inference and Bayesian inverse problems involving Gaussian processes.

This work unifies diffusion sampling and stochastic localization under a single theoretical framework, addressing the lack of rigorous theoretical connections between them and the limited applicability of existing algorithms. Methodologically, we establish the first formal equivalence between diffusion processes and stochastic localization via stochastic process analysis and statistical mechanical modeling; we introduce a generalized stochastic localization framework wherein standard denoising diffusion is shown to be a specific instance, and extend it to broader distribution families by parameterizing the drift term with neural networks. Key contributions include: (1) a theoretical proof that multiple classes of diffusion samplers—including DDPM, DDIM, and score-based SDEs—are instantiations of stochastic localization; (2) derivation of novel, computationally efficient sampling algorithms grounded in this equivalence; and (3) a new analytical perspective on mixing properties and convergence rates via Poincaré inequality characterization, substantially deepening the understanding of the dynamical mechanisms underlying generative models.

This study investigates the intrinsic connection between Gaussian processes and Gaussian random elements in reproducing kernel Banach spaces, with a focus on the relationship between covariance operators and positive definite functions. By introducing the γ-Radonifying operator, the authors characterize the covariance structure of weakly second-order Radon Gaussian measures in reproducing kernel Banach spaces and establish a rigorous functional-analytic foundation for sample paths of Gaussian processes. The main contributions include proving that the covariance operator is uniquely determined by a positive definite function, providing an operator-theoretic characterization specific to the Gaussian setting, and successfully extending the classical Driscoll’s theorem to the Banach space framework, thereby significantly broadening its applicability.

This study addresses the quantification of posterior uncertainty in kernel density estimation within a predictive Bayesian framework. By analyzing the predictive measure induced by kernel density estimators, the authors establish, for the first time, that the associated resampling sequence—despite failing to satisfy conditional independence and identical distribution (c.i.d.) or asymptotic c.i.d. (a.c.i.d.) conditions—converges weakly almost surely. In the case of Gaussian kernels, they further derive an explicit density representation of the limiting random probability measure and construct corresponding moment estimators. Leveraging these results, the paper successfully derives Bayesian credible intervals for kernel density estimates and demonstrates their empirical validity on two real-world datasets, thereby providing a rigorous tool for uncertainty quantification in nonparametric density estimation.

This work proposes a unified mathematical framework grounded in dynamic information flow for constructing structurally rigorous models of future prediction. By integrating filtering theory, regular conditional probabilities, Markov semigroups, infinitesimal generators, and multiple information geometries—including Hilbert, Fisher–Rao, and Wasserstein—the approach conceptualizes prediction as the construction of conditional distributions governed by informational, geometric, and modeling constraints. The framework elucidates deep connections among classical results such as the tower property and semigroup laws, as well as Itô’s formula and backward equations. Explicit transition laws, spectral decompositions, term structures, and asymptotic behaviors are derived within canonical models like Ornstein–Uhlenbeck and Cox–Ingersoll–Ross, thereby establishing a compact mathematical mapping from idealized theoretical constructs to empirical forecasting.

该研究通过控制Radon-Nikodym导数来生成异常值，解决了现有方法不能显式控制生成样本似然性的问题，利用扩散模型中的分数函数修改实现。

This work addresses the absence of rigorous formalization of Itô integration and Itô’s formula in existing proof assistants, particularly the lack of machine-verified treatment of these constructs as martingale processes. Building upon Lean 4, Mathlib, and the BrownianMotion library, we develop an L²-theoretic Itô calculus on a bounded interval [0,T] by constructing the Itô integral via Hilbert space isometry, establishing it as an L²-continuous martingale, and proving Itô’s formula for C³ functions with an explicit remainder bound. To our knowledge, this is the first machine-checked verification of Itô’s formula in any proof assistant and the first formalization of the Itô integral as a martingale-valued process. A single structural identity uniformly yields adaptivity, the martingale property, contraction bounds, and both forms of Itô isometry. The entire development comprises approximately 7,200 lines of sorry-free code across 22 modules, with all main theorems validated under classical axioms.