Formulation and properties of a divergence used to compare probability measures without absolute continuity and its application to uncertainty quantification

Absolute continuity under flows generated by SDE with measurable drift coefficient

We consider the Itô SDE with non-degenerate diffusion coefficient and measurable drift coefficient. Under the condition that the gradient of the diffusion coefficient and the divergences of the diffusion and drift coefficients are exponentially integrable with respect to the Gaussian measure, we show that the stochastic flow leaves the reference measure absolutely continuous.

Wellposedness for stochastic continuity equations with Ladyzhenskaya-Prodi-Serrin condition

We consider the stochastic divergence-free continuity equations with Ladyzhenskaya-Prodi-Serrin condition. Wellposedness is proved meanwhile uniqueness may fail for the deterministic PDE. The main issue of uniqueness realies on stochastic characteristic method and the generalized Ito-Ventzel-Kunita formula. Moreover, we prove a stability property for the solution with respect to the initial datum.

用于随机PDE中稀疏数据引起的模型形式不确定性的稳健信息分歧

Robust information divergences for model-form uncertainty arising from sparse data in random PDE

本文提出了一种新的混合信息分歧应用，用于分析稳态地下流问题中的不确定性。这些混合信息分歧是非侵入性的、以目标为导向的不确定性量化工具，能够实现稳健的数据驱动预测，支持关键决策任务如监管评估和风险管理。通过数值实验，研究了模型形式或epistemic不确定性传播，并展示了不确定性量化界限，包括参数敏感性分析和由于稀疏数据导致的模型规格不正确。此外，还建立了混合信息分歧与某些集中不等式之间的联系，这些不等式可用于高效计算，并通过合适的统计量考虑任何可用数据。

Measures of path-based nonlinear expansion rates and Lagrangian uncertainty in stochastic flows

We develop a probabilistic characterisation of trajectorial expansion rates in non-autonomous stochastic dynamical systems that can be defined over a finite time interval and used for the subsequent uncertainty quantification in Lagrangian (trajectory-based) predictions. These expansion rates are quantified via certain divergences (pre-metrics) between probability measures induced by the laws of the stochastic flow associated with the underlying dynamics. We construct scalar fields of finite-time divergence/expansion rates, show their existence and space-time continuity for general stochastic …

近端最优传输散度

Proximal optimal transport divergences

我们提出近端最优传输散度，一种插值信息散度与最优传输距离的新差异度量，为生成建模提供近端优化基础，并建立动态平均场博弈公式，推广现有方法。

Sensitivity Analysis for Rare Events based on Rényi Divergence

Rare events play a key role in many applications and numerous algorithms have been proposed for estimating the probability of a rare event. However, relatively little is known on how to quantify the sensitivity of the probability with respect to model parameters. In this paper, instead of the direct statistical estimation of rare event sensitivities, we develop novel and general uncertainty quantification and sensitivity bounds which are not tied to specific rare event simulation methods and which apply to families of rare events. Our method is based on a recently derived variational represent…

泊松过程与点模式的信息散度与似然比

Information divergences and likelihood ratios of Poisson processes and point patterns

本文在一般可测空间上建立了泊松过程与点模式的信息散度和似然比的解析框架，给出了 KL 散度、Rényi 散度、Hellinger 距离及似然比的显式公式，并刻画了绝对连续性与相互奇异性。

Lagrangian uncertainty quantification and information inequalities for stochastic flows

We develop a systematic information-theoretic framework for quantification and mitigation of error in probabilistic Lagrangian (i.e., path-based) predictions which are obtained from dynamical systems generated by uncertain (Eulerian) vector fields. This work is motivated by the desire to improve Lagrangian predictions in complex dynamical systems based either on analytically simplified or data-driven models. We derive a hierarchy of general information bounds on uncertainty in estimates of statistical observables $\mathbb{E}^ν[f]$, evaluated on trajectories of the approximating dynamical syste…

概率测度集与φ-散度的极值点

Extreme points of sets of probability measures and $φ$-divergences

该研究刻画了凸概率测度集的极值点，关联φ-散度与绝对连续性极小性，将结果应用于多类测度集场景以补充文献结论。

Path-space information bounds for uncertainty quantification and sensitivity analysis of stochastic dynamics

Uncertainty quantification is a primary challenge for reliable modeling and simulation of complex stochastic dynamics. Such problems are typically plagued with incomplete information that may enter as uncertainty in the model parameters, or even in the model itself. Furthermore, due to their dynamic nature, we need to assess the impact of these uncertainties on the transient and long-time behavior of the stochastic models and derive corresponding uncertainty bounds for observables of interest. A special class of such challenges is parametric uncertainties in the model and in particular sensiti…

Absolute Continuity under Time Shift of Trajectories and Related Stochastic Calculus

The paper is concerned with a class of two-sided stochastic processes of the form $X=W+A$. Here $W$ is a two-sided Brownian motion with random initial data at time zero and $A\equiv A(W)$ is a function of $W$. Elements of the related stochastic calculus are introduced. In particular, the calculus is adjusted to the case when $A$ is a jump process. Absolute continuity of $(X,P_{\sbnu})$ under time shift of trajectories is investigated. For example under various conditions on the initial density with respect to the Lebesgue measure, $m=d\bnu/dx$, and on $A$ with $A_0=0$ we verify % {eqnarray*} \…

A New Family of Bounded Divergence Measures and Application to Signal Detection

We introduce a new one-parameter family of divergence measures, called bounded Bhattacharyya distance (BBD) measures, for quantifying the dissimilarity between probability distributions. These measures are bounded, symmetric and positive semi-definite and do not require absolute continuity. In the asymptotic limit, BBD measure approaches the squared Hellinger distance. A generalized BBD measure for multiple distributions is also introduced. We prove an extension of a theorem of Bradt and Karlin for BBD relating Bayes error probability and divergence ranking. We show that BBD belongs to the cla…