Generalized Feynman path integrals and applications to higher-order heat-type equations
type: Item
instance of: scholarly article
title: Generalized Feynman path integrals and applications to higher-order heat-type equations
zbMATH Open document ID: 1409.58003
author: N. Cangiotti
author: S. Albeverio
author: Sonia Mazzucchi
published in: Expositiones Mathematicae
publication date: 2019-01-11T00:00:00Z
review text: From the introduction: One of the aims of the present paper is precisely to give a detailed account of the theory and applications of infinite dimensional Fresnel integrals to the construction of a functional integral representation of the solution of PDEs of the form \[ \frac{\partial}{\partial t}u(x,t) = (-i)^p\alpha \frac{\partial^p}{\partial x^p}u(x,t) +V(x)u(x,t), \] where \(t\in [0,+\infty), x\in \mathbb{R}\), \(p\in \mathbb{N}\), \(\alpha\in \mathbb{C}\) and \(V:\mathbb{R} \to \mathbb{C}\) is a continuous bounded function. In Section 2 we review the theory of infinite dimensional Fresnel integrals as well as their applications to the mathematical definition of Feynman path integrals. Section 3 presents a recent generalization of infinite dimensional Fresnel integrals which covers the case of polynomial phase functions and the application to the representation of the solution of higher-order equations of the above form. In Section 4 we prove a Girsanov type formula for the higher-order equation above form. Eventually Section 5 presents a new phase space path integral solution formula for Eq. above form.
Mathematics Subject Classification ID: 58D30
Mathematics Subject Classification ID: 35C15
Mathematics Subject Classification ID: 35G05
Mathematics Subject Classification ID: 28C20
Mathematics Subject Classification ID: 47D06
zbMATH DE Number: 7000847
zbMATH Keywords: infinite dimensional integration
zbMATH Keywords: partial differential equations
zbMATH Keywords: representations of solutions
reviewed by: Nicolai K. Smolentsev
MaRDI profile type: Publication
full work available at URL: j.exmath.2018.09.001
OpenAlex ID: W2889610191
cites work: Finite dimensional approximation approach to oscillatory integrals and stationary phase in infinite dimensions
cites work: Q3742397
cites work: Phase space Feynman path integrals
cites work: A REPRESENTATION OF THE BELAVKIN EQUATION VIA PHASE SPACE FEYNMAN PATH INTEGRALS
cites work: Oscillatory integrals and the method of stationary phase in infinitely many dimensions, with applications to the classical limit of quantum mechanics. I
cites work: Mathematical theory of Feynman path integrals. An introduction
cites work: Infinite dimensional oscillatory integrals as projective systems of functionals
cites work: Feynman path integrals for polynomially growing potentials
cites work: Generalized Fresnel integrals
cites work: The time-dependent quartic oscillator --- a Feynman path integral approach
cites work: An Introduction to Infinite-dimensional Oscillatory and Probabilistic Integrals
cites work: A unified approach to infinite-dimensional integration
cites work: A unified approach to infinite dimensional integrals of probabilistic and oscillatory type with applications to Feynman path integrals
cites work: Singularities of differentiable maps, Volume 1. Classification of critical points, caustics and wave fronts. Transl. from the Russian by Ian Porteous, edited by V. I. Arnol'd
cites work: Conditional maximal distributions of processes related to higher-order heat-type equations
cites work: A white noise approach to phase space Feynman path integrals
cites work: An Itô calculus for a class of limit processes arising from random walks on the complex plane
cites work: High order heat-type equations and random walks on the complex plane
cites work: A Family of Integrals Serving to Connect the Wiener and Feynman Integrals
cites work: Functional integration
cites work: Functional Integration
cites work: Quasi-invariance of measures under translation
cites work: A Feynman-Kac-type formula for the deterministic and stochastic wave equations and other p.d.e.'s
cites work: Generalized Measures in Function Spaces
cites work: Wiener measures for path integrals with affine kinematic variables
cites work: Q5822308
cites work: Sur une résolution stochastique de l'équation de Schrödinger à coefficients analytiques
cites work: Oscillatory integrals, lagrange immersions and unfolding of singularities
cites work: Q3594454
cites work: Q3699979
cites work: Open quantum systems and Feynman integrals
cites work: Space-Time Approach to Non-Relativistic Quantum Mechanics
cites work: Q5200147
cites work: Functional Integration and Partial Differential Equations. (AM-109)
cites work: Rigorous time slicing approach to Feynman path integrals
cites work: Probabilistic construction of the solution of some higher order parabolic differential equation
cites work: Q3234008
cites work: Gibbs measures on Brownian currents
cites work: Stochastic calculus with infinitesimals
cites work: A signed measure on path space related to Wiener measure
cites work: Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations
cites work: The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis.
cites work: Q5508009
cites work: Q5700324
cites work: Q4954180
cites work: Q3741605
cites work: Q4002114
cites work: Q4692254
cites work: Semiclassical analysis for diffusions and stochastic processes
cites work: Q4831838
cites work: Probability representations of solutions of the Cauchy problem for quantum mechanical equations
cites work: Q3273541
cites work: Phase space Feynman path integrals via piecewise bicharacteristic paths and their semiclassical approximations
cites work: Phase space Feynman path integrals of higher order parabolic type with general functional as integrand
cites work: Stochastic analysis without probability: study of some basic tools
cites work: A signed measure on rough paths associated to a PDE of high order: results and conjectures
cites work: Feynman path integrals for the inverse quartic oscillator

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Q3424691

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Remarks on the operator-norm convergence of the Trotter product formula

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Feynman Integrals and the Schrödinger Equation

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Convergence in \(L^{p}\) for Feynman path integrals

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Iterated processes and their applications to higher order differential equations

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Q3160267

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Hamiltonian Feynman path integrals via the Chernoff formula

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Generalized Brownian functionals and the Feynman integral

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: The Doss trick on symmetric spaces

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Projective limits of complex measures and martingale convergence

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: On the Product of Semi-Groups of Operators

Generalized Feynman path integrals and applications to higher-order heat-type equations (cites work)
cites work: Q4200435

Generalized Feynman path integrals and applications to higher-order heat-type equations (DOI)
DOI: 10.1016/J.EXMATH.2018.09.001

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: Phase space Feynman path integrals of higher order parabolic type with general functional as integrand
Similarity Score: 0.9010149
Recommender Run: Recommender Run 3

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: Q4009300
Similarity Score: 0.8873309
Recommender Run: Recommender Run 3

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: Heat kernel asymptotics, path integrals and infinite-dimensional determinants
Similarity Score: 0.88682854
Recommender Run: Recommender Run 3

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: Q3396677
Similarity Score: 0.88647556
Recommender Run: Recommender Run 3

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: Q4373506
Similarity Score: 0.88620436
Recommender Run: Recommender Run 3

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: Q3414554
Similarity Score: 0.8853366
Recommender Run: Recommender Run 3

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: Path integral formulae for heat kernels and their derivatives
Similarity Score: 0.8850031
Recommender Run: Recommender Run 3

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: A mathematical theory of the Feynman path integral for the generalized Pauli equations
Similarity Score: 0.8836435
Recommender Run: Recommender Run 3

Generalized Feynman path integrals and applications to higher-order heat-type equations (Recommended article)
Recommended article: Mathematical Feynman Path Integrals and Their Applications
Similarity Score: 0.87838644
Recommender Run: Recommender Run 3