惯性聚合 高效追踪和阅读你感兴趣的博客、新闻、科技资讯
阅读原文 在惯性聚合中打开

推荐订阅源

酷 壳 – CoolShell
酷 壳 – CoolShell
G
Google Developers Blog
L
LangChain Blog
Y
Y Combinator Blog
Vercel News
Vercel News
WordPress大学
WordPress大学
大猫的无限游戏
大猫的无限游戏
博客园 - Franky
V
Visual Studio Blog
小众软件
小众软件
月光博客
月光博客
A
About on SuperTechFans
H
Hackread – Cybersecurity News, Data Breaches, AI and More
T
The Blog of Author Tim Ferriss
有赞技术团队
有赞技术团队
M
MIT News - Artificial intelligence
阮一峰的网络日志
阮一峰的网络日志
Last Week in AI
Last Week in AI
博客园 - 【当耐特】
让小产品的独立变现更简单 - ezindie.com
让小产品的独立变现更简单 - ezindie.com
MongoDB | Blog
MongoDB | Blog
Jina AI
Jina AI
美团技术团队
量子位

math.ST updates on arXiv.org

What is Learnable in Valiant's Theory of the Learnable? Learning Perturbations to Extrapolate Your LLM Byzantine-Robust Distributed Sparse Learning Revisited The Sample Complexity of Multiple Change Point Identification under Bandit Feedback A proximal gradient algorithm for composite log-concave sampling Model-based Bootstrap of Controlled Markov Chains Approximation of Maximally Monotone Operators : A Graph Convergence Perspective Posterior Contraction Rates for Sparse Kolmogorov-Arnold Networks in Anisotropic Besov Spaces MIST: Reliable Streaming Decision Trees for Online Class-Incremental Learning via McDiarmid Bound A Spectral Framework for Closed-Form Relative Density Estimation Fast Rates for Offline Contextual Bandits with Forward-KL Regularization under Single-Policy Concentrability Higher-Order Equilibrium Tracking for EM-Compressible Online Estimation Scaling Limits of Long-Context Transformers A Note on Non-Negative $L_1$-Approximating Polynomials Susceptibilities and Patterning: A Primer on Linear Response in Bayesian Learning Linear Response Estimators for Singular Statistical Models Statistical inference with belief functions: A survey Robust stochastic first order methods in heavy-tailed noise via medoid mini-batch gradient sampling Every Feedforward Neural Network Definable in an o-Minimal Structure Has Finite Sample Complexity Adaptive auditing of AI systems with anytime-valid guarantees Locally Near Optimal Piecewise Linear Regression in High Dimensions via Difference of Max-Affine Functions Risk-Controlled Post-Processing of Decision Policies Covariate Balancing and Riesz Regression Should Be Guided by the Neyman Orthogonal Score in Debiased Machine Learning A Unified Pair-GRPO Family: From Implicit to Explicit Preference Constraints for Stable and General RL Alignment Time-Inhomogeneous Preconditioned Langevin Dynamics A Fine-Grained Understanding of Uniform Convergence for Halfspaces CITE: Anytime-Valid Statistical Inference in LLM Self-Consistency Ratio-based Loss Functions Optimal Confidence Band for Kernel Gradient Flow Estimator A renormalization-group inspired lattice-based framework for piecewise generalized linear models
Colored Stochastic Multiplicative Processes with Additive...
Marco Bianucci, Mauro Bologna, Riccardo Mannella · 2024-04-22 · via math.ST updates on arXiv.org

Research on stochastic differential equations (SDE) involving both additive and multiplicative noise has been extensive. In situations where the primary process is driven by a multiplicative stochastic process, additive white noise typically represents an intrinsic and unavoidable fast factor, including phenomena like thermal fluctuations, inherent uncertainties in measurement processes, or rapid wind forcing in ocean dynamics. This work focuses on a significant class of such systems, particularly those characterized by linear drift and multiplicative noise, extensively explored in the literature. Conventionally, multiplicative stochastic processes are also treated as white noise in existing studies. However, when considering colored multiplicative noise, the emphasis has been on characterizing the far tails of the probability density function (PDF), regardless of the spectral properties of the noise. In the absence of additive noise and with a general colored multiplicative SDE, standard perturbation approaches lead to a second-order PDE known as the Fokker-Planck Equation (FPE), consistent with Fick's law. This investigation unveils a notable departure from this standard behavior when introducing additive white noise. At the leading order of the stochastic process strength, perturbation approaches yield a \textit{third-order PDE}, irrespective of the white noise intensity. The breakdown of the FPE further signifies the breakdown of Fick's law. Additionally, we derive the explicit solution for the equilibrium PDF corresponding to this third-order PDE Master Equation. Through numerical simulations, we demonstrate significant deviations from outcomes derived using the FPE obtained through the application of Fick's law.