Analysis of Differential Equations, Stochastic Processes and Geometric Measure Theory
This community develops the mathematical frameworks for solving partial differential equations, analyzing random processes, and characterizing geometric structures in high-dimensional spaces.
The work centers on the rigorous study of elliptic and parabolic partial differential equations, including free boundary problems and mean curvature flows. A substantial portion of the research addresses stochastic analysis, specifically Brownian motion, anomalous diffusion, and large deviation principles. The community also focuses on functional analysis within Banach and Sobolev spaces, as well as optimal transport theory and metric measure spaces. These mathematical tools are applied to model complex systems, from fluid dynamics to quantum field theory, providing existence and uniqueness proofs for nonlinear operators and generalized functions.
The largest share of the community's output is found in radon research, accounting for 4.3% of all radon research, and 2,052 papers here.
The community comprises 18,470 papers, published primarily in arXiv (Cornell University), the Pacific Journal of Mathematics, and the Transactions of the American Mathematical Society.
Recent work includes studies on extended state-space constructions for Monte Carlo methods, anomalous statistics in the Langevin equation, and strong Kantorovich duality for quantum optimal transport.