Homotopy probability theory I

@article{DrummondCole2013HomotopyPT,
  title={Homotopy probability theory I},
  author={Gabriel C. Drummond-Cole and Jae-Suk Park and John Terilla},
  journal={Journal of Homotopy and Related Structures},
  year={2013},
  volume={10},
  pages={425-435}
}
This is the first of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. In this paper, cumulants are proved to coincide with morphisms of homotopy algebras. The sequel paper outlines how the framework presented here can assist in the development of homotopy probability theory, allowing the principles of derived mathematics to participate in classical and noncommutative probability theory. 
Homotopy probability theory II
This is the second of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. This paper outlines how the framework
Homotopy probability theory on a Riemannian manifold and the Euler equation
Homotopy probability theory is a version of probability theory in which the vector space of random variables is replaced with a chain complex. A natural example extends ordinary probability theory on
Cones in homotopy probability theory
This note defines cones in homotopy probability theory and demonstrates that a cone over a space is a reasonable replacement for the space. The homotopy Gaussian distribution in one variable is
Non-commutative Probability Theory for Topological Data Analysis
Recent developments have found unexpected connections between non-commutative probability theory and algebraic topology. In particular, Boolean cumulants functionals seem to be important for
A Non-crossing Word Cooperad for Free Homotopy Probability Theory
We construct a cooperad which extends the framework of homotopy probability theory to free probability theory. The cooperad constructed, which seems related to the sequence and cactus operads, may be
Classical independence and homotopyLie algebras
Dennis Sullivan used to ask me "What is a Quantum Field Theory?" My temporary answer, only until and in honor of his 70th birthday, is "A homotopy probability space that is enhanced with a weight
Topological Perspectives on Statistical Quantities II
In statistics cumulants are defined to be functions that measure the linear independence of random variables. In the non-communicative case the Boolean cumulants can be described as functions that
The Cumulant Bijection and Differential Forms
According to Jae Suk Park, physicists use "canonical coordinate systems" to compute correlations in perturbative quantum field theories. One may interpret these canonical coordinate systems as
Cumulants, Koszul brackets and homological perturbation theory for commutative $BV_\infty$ and $IBL_\infty$ algebras
We explore the relationship between the classical constructions of cumulants and Koszul brackets, showing that the former are an expontial version of the latter. Moreover, under some additional
Topological Perspectives on Statistical Quantities I
In statistics cumulants are defined to be functions that measure dependence of random variables. If the random variables are independent the cumulants are zero. The idea of cumulants can be
...
...

References

SHOWING 1-10 OF 32 REFERENCES
Homotopy probability theory II
This is the second of two papers that introduce a deformation theoretic framework to explain and broaden a link between homotopy algebra and probability theory. This paper outlines how the framework
Deformation Theory of Algebras and Their Diagrams
This book brings together both the classical and current aspects of deformation theory. The presentation is mostly self-contained, assuming only basic knowledge of commutative algebra, homological
Gauge Algebra and Quantization
Notes on Free Probability Theory
These notes are from a 4-lecture mini-course taught by the author at the conference on von Neumann algebras as part of the ``Geometrie non commutative en mathematiques et physique'' month at CIRM in
Algebraic Principles of Quantum Field Theory I: Foundation and an exact solution of BV QFT
This is the first in a series of papers on an attempt to understand quantum field theory mathematically. In this paper we shall introduce and study BV QFT algebra and BV QFT as the proto-algebraic
Period integrals of smooth projective hypersurfaces and homotopy Lie algebras
The goal of this paper is to reveal hidden structures on the Griffiths period integrals of differential forms on smooth projective hypersurfaces, studied extensively in [6], in terms of period
Algebraic Principles of Quantum Field Theory II: Quantum Coordinates and WDVV Equation
This paper is about algebro-geometrical structures on a moduli space $\CM$ of anomaly-free BV QFTs with finite number of inequivalent observables or in a finite superselection sector. We show that
Cumulants in noncommutative probability theory II
AbstractWe continue the investigation of noncommutative cumulants. In this paper various characterizations of generalized Gaussian random variables are proved.
Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems
Abstract.Cumulants linearize convolution of measures. We use a formula of Good to define noncommutative cumulants in a very general setting. It turns out that the essential property needed is
Renormalization and Effective Field Theory
This book tells mathematicians about an amazing subject invented by physicists and it tells physicists how a master mathematician must proceed in order to understand it. Physicists who know quantum
...
...