From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index 
                
                  
                
                $${{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}$$

Magnen, J. Le; Unterberger, Jérémie

doi:10.1007/s00023-011-0119-y

Cited by 6 publications

(5 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We believe the most powerful techniques for proving our conjecture in the sense of perturbative QFT are the multiscale x-space methods for BPHZ renormalization [32,33] developed by the École Polytechnique school of constructive QFT. The power of these methods resides in the fact they can adapted (albeit with tears) to the non-perturbative situation [34,78,93,8,2,80,81,107].…”

Section: And the Configuration Spacementioning

confidence: 99%

A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion

Abdesselam

2020

Commun. Math. Phys.

View full text Add to dashboard Cite

We establish a direct connection between two fundamental topics: one in probability theory and one in quantum field theory. The first topic is the problem of pointwise multiplication of random Schwartz distributions which has been the object of recent progress thanks to Hairer's theory of regularity structures and the theory of paracontrolled distributions introduced by Gubinelli, Imkeller and Perkowski. The second topic is Wilson's operator product expansion which is a general property of models of quantum field theory and a cornerstone of the bootstrap approach to conformal field theory. Our main result is a general theorem for the almost sure construction of products of random distributions by mollification and suitable additive as well as multiplicative renormalizations. The hypothesis for this theorem is the operator product expansion with precise bounds for pointwise correlations. We conjecture these bounds to be universal features of quantum field theories with gapped dimension spectrum. Our theorem can accommodate logarithmic corrections, anomalous scaling dimensions and even lack of translation invariance. However, it only applies to fields with short distance singularities that are milder than white noise. As an application, we provide a detailed treatment of a scalar conformal field theory of mean field type, i.e., the fractional massless free field also known as the fractional Gaussian field. Contents 4.3. Graph construction 33 4.4. Two-scale pin and sum argument 34 4.5. Power-counting 36 5. Proof of the main theorem 37 6. A detailed example: the fractional massless free field 38 6.1. The explicit abstract system of pointwise correlations and the BNNFB 38 6.2. The OPE structure 40 6.3. A proof of the ENNFB 44 6.4. A proof of the soft and hard hypotheses on the OPE coefficients 47 6.5. Application of the main theorem to the construction of local Wick monomials as random distributions 48 6.6. Conformal invariance 49 References 50 1.2. Abstract systems of pointwise correlations. Let A be a finite set which we will call an alphabet and will serve to label fields. Define the big diagonal

show abstract

Section: And the Configuration Spacementioning

confidence: 99%

A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion

Abdesselam

2020

Commun. Math. Phys.

View full text Add to dashboard Cite

show abstract

“…This relates, e.g., to the construction of iterated integrals of fractional Brownian motion for low Hurst exponent. Very little is known, apart from the findings from the difficult work by J. Unterberger, with help from J. Magnen [173,174,127,128]. Finally, before moving on to the p-adic fractional φ 4 3 model, it is worth remarking that if CI and OS positivity hold, then so do more exotic forms of the latter.…”

Section: Of Course [φ] Needs To Be Replaced By [φmentioning

confidence: 99%

Towards Three-Dimensional Conformal Probability

Abdesselam

2018

P-Adic Num Ultrametr Anal Appl

View full text Add to dashboard Cite

In this outline of a program, based on rigorous renormalization group theory, we introduce new definitions which allow one to formulate precise mathematical conjectures related to conformal invariance as studied by physicists in the area known as higher-dimensional conformal bootstrap which has developed at a breathtaking pace over the last few years. We also explore a second theme, intimately tied to conformal invariance for random distributions, which can be construed as a search for very general first and secondquantized Kolmogorov-Chentsov Theorems. First-quantized refers to regularity of sample paths. Secondquantized refers to regularity of generalized functionals or Hida distributions and relates to the operator product expansion. We formulate this program in both the Archimedean and p-adic situations. Indeed, the study of conformal field theory and its connections with probability provides a golden opportunity where padic analysis can lead the way towards a better understanding of open problems in the Archimedean setting. Finally, we present a summary of progress made on a p-adic hierarchical model and point out possible connections to number theory. Parts of this article were presented in author's talk at the 6th International Conference on p-adic Mathematical Physics and its Applications, Mexico 2017.conformal invariance (CI). The simplest example is provided by the time-inversion invariance of Brownian motion tB 1/t d = B t proved by Paul Lévy in [122]. Much deeper is the analogous invariance under conformal transformation in the 2D space of "time parameters" for the Ising model critical scaling limit [55,59] which represents the culmination of major effort led by Schramm, Smirnov and others. Define the lattice couplings

show abstract

“…• A close analog of the Ercolani-McLaughlin Theorem in the case of quartic interactions [27]. • The construction of the Lévy area for fractional Brownian motion with low Hurst exponent [23]. In almost every application, the only difficulty is to "see" the quantity under study as f ([1 k ]) for a suitable function f .…”

Section: 24mentioning

confidence: 99%

Counting colored planar maps free-probabilistically

Abdesselam,

Anderson

2012

Preprint

View full text Add to dashboard Cite

Our main result is an explicit operator-theoretic formula for the number of colored planar maps with a fixed set of stars each of which has a fixed set of half-edges with fixed coloration. The formula bounds the number of such colored planar maps well enough to prove convergence near the origin of generating functions arising naturally in the matrix model context. Such convergence is known but the proof of convergence proceeding by way of our main result is relatively simple. Besides Voiculescu's generalization of Wigner's semicircle law, our main technical tool is an integration identity representing the joint cumulant of several functions of a Gaussian random vector. The latter identity in the case of cumulants of order 2 reduces to one well-known as a means to prove the Poincaré inequality. We derive the identity by combining the heat equation with the so-called BKAR formula from constructive quantum field theory and rigorous statistical mechanics.

show abstract

From Constructive Field Theory to Fractional Stochastic Calculus. (II) Constructive Proof of Convergence for the Lévy Area of Fractional Brownian Motion with Hurst Index $${{\alpha}\,{\in}\,(\frac{1}{8},\frac{1}{4})}$$

Cited by 6 publications

References 50 publications

A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion

A Second-Quantized Kolmogorov–Chentsov Theorem via the Operator Product Expansion

Towards Three-Dimensional Conformal Probability

Counting colored planar maps free-probabilistically

Contact Info

Product

Resources

About