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We define a deterministic integral with respect to irregular paths as a limit of standard line integrals and completely describe a class of all paths for which this integral exists for functions with H\"older exponent in the range of (0,1].…
We introduce the class of "smooth rough paths" and study their main properties. Working in a smooth setting allows us to discard sewing arguments and focus on algebraic and geometric aspects. Specifically, a Maurer-Cartan perspective is the…
Calculus via regularizations and rough paths are two methods to approach stochastic integration and calculus close to pathwise calculus. The origin of rough paths theory is purely deterministic, calculus via regularization is based on…
We give here a covariant definition of the path integral formalism for the Lagrangian, which leaves a freedom to choose anyone of many possible quantum systems that correspond to the same classical limit without adding new potential terms…
Path integrals represent a powerful route to quantization: they calculate probabilities by summing over classical configurations of variables such as fields, assigning each configuration a phase equal to the action of that configuration.…
We present a new method for the consistent construction of time-continuous coherent-state path integrals using the theory of half-form quantization. Through the inversion of the quantization procedure we construct a de-quantization map…
In this paper we review some connections between harmonic analysis and the modern theory of automorphic forms. We indicate in some examples how the study of problems of harmonic analysis brings us to the important objects of the theory of…
Rough sheets are two-parameter analogs of rough paths. In this work the theory of integration over functions of two parameters is extended to cover the case of irregular functions by developing an appropriate notion of rough sheet. The main…
For description of the quantum dynamics on a curved group manifold the path integrals in a space of the group parameters is offered. The formalism is illustrated by the $H$-atom problem.
Recently a path integral formalism has been proposed by the author which gives the time evolution of moments of slow variables in a Hamiltonian statistical system. This closure relies on evaluating the informational discrepancy of a time…
We introduce a geometric approach of integral curves for functional inequalities involving directional derivatives in the general context of differentiable manifolds that are equipped with a volume form. We focus on Hardy-type inequalities…
Path integrals are a central tool when it comes to describing quantum or thermal fluctuations of particles or fields. Their success dates back to Feynman who showed how to use them within the framework of quantum mechanics. Since then, path…
We study the path integral formulation of Friedmann universe filled with a massless scalar field in loop quantum cosmology. All the isotropic models of $k=0,+1,-1$ are considered. To construct the path integrals in the timeless framework, a…
We discuss stochastic calculus for large classes of Gaussian processes, based on rough path analysis. Our key condition is a covariance measure structure combined with a classical criterion due to Jain and Monrad [Ann. Probab. 11 (1983)…
The propagator for a certain class of two time-dependent coupled and driven harmonic oscillators with time-varying angular frequencies and masses is evaluated by path integration. This is simply done through suitably chosen generalized…
Using some basic notions from the theory of Hopf algebras and quasi-shuffle algebras, we introduce rigorously a new family of rough paths: the quasi-geometric rough paths. We discuss their main properties. In particular, we will relate them…
Whenever an It\^o-Wentsel type of formula holds for composition of flows of a certain differential dynamics, there exists locally a decomposition of the corresponding flow according to complementary distributions (or foliations, in the case…
The theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in…
We develop a fundamental framework for and extend the theory of rough paths to Lipschitz-gamma manifolds.
In the context of two illustrative examples from supersymmetric quantum mechanics we show that the semi-classical analysis of the path integral requires complexification of the configuration space and action, and the inclusion of complex…