Functional Analytic Developments of the Kuelbs-Steadman Space: Deterministic and Stochastic Perspectives

• Grant number: 25/07107-2
• Support Opportunities: Scholarships abroad - Research
• Start date: July 04, 2025
• End date: September 04, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Márcia Cristina Anderson Braz Federson
• Grantee: Márcia Cristina Anderson Braz Federson
• Host Investigator: Tepper L Gill
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Howard University, United States

Abstract

This project aims to develop a constructive theory of integration on Kuelbs-Steadman spaces $KS^p$ over the space $\mathbb R^\infty$, extending the recent advances of T. Gill and T. Myers on Lebesgue measures in separable Banach spaces. The $KS^p$ spaces, defined via the Kurzweil-Henstock integral, are particularly well-suited for handling highly oscillatory functions, with relevant applications in mathematical physics.The proposal consists in constructing an appropriate $\sigma$-finite Lebesgue measure on these spaces, preserving the main properties of the finite-dimensional theory, and developing an integration theory compatible with the fundamental theorems of functional analysis. Furthermore, applications to differential equations in infinite-dimensional spaces will be explored.The broader goal is to expand the scope of infinite-dimensional functional analysis by offering new conceptual and operational tools for problems in mathematical modeling, theoretical physics, and stochastic systems. (AU)