2015/10/28 by Jae-Suk Park, Park, Jae-Suk
Mathematics · #Mathematical and Theoretical Analysis #Functional Equations Stability Results #Advanced Topology and Set Theory
paper · pdf · doi:10.48550/arxiv.1510.08289
This is the first installment of a series of papers whose aim is to lay a\nfoundation for homotopy probability theory by establishing its basic principles\nand practices. The notion of a homotopy probability space is an enrichment of\nthe notion of an algebraic probability space with ideas from algebraic homotopy\ntheory. This enrichment uses a characterization of the laws of random variables\nin a probability space in terms of symmetries of the expectation. The laws of\nrandom variables are reinterpreted as invariants of the homotopy types of\ninfinity morphisms between certain homotopy algebras. The relevant category of\nhomotopy algebras is determined by the appropriate notion of independence for\nthe underlying probability theory. This theory will be both a natural\ngeneralization and an effective computational tool for the study of classical\nalgebraic probability spaces, while keeping the same central limit. This\narticle is focused on the commutative case, where the laws of random variables\nare also described in terms of certain affinely flat structures on the formal\nmoduli space of a naturally defined family attached to the given algebraic\nprobability space. Non-commutative probability theories will be the main\nsubject of the sequels.\n (This work is a spin-off from the author's program to characterize path\nintegrals of quantum field theory in terms of the symmetries of the quantum\nexpectation which should satisfy a certain coherence with a particular weight\nfiltration generated by the Planck constant \ℏ. A similar idea is adopted\nhere in a simplified form, without the \ℏ-conditions, and, in return, many\nresults in this paper will be used as background materials in the forthcoming\nwork on homotopy theory of quantum fields).\n