Phase space Feynman path integrals with smooth functional derivatives by time slicing approximation

We give two general classes of functionals for which the phase space Feynman path integrals have a mathematically rigorous meaning. More precisely, for any functional belonging to each class, the time slicing approximation of the phase space path integral converges uniformly on compact subsets with respect to the starting point of momentum paths and the endpoint of position paths. Each class is closed under addition, multiplication, translation, real linear transformation and functional differentiation. Therefore, we can produce many functionals which are phase space path integrable. Furthermore, though we need to pay attention for use, the interchange of the order with the integrals with respect to time, the interchange of the order with some limits, the semiclassical approximation of Hamiltonian type, the natural property under translation, the integration by parts with respect to functional differentiation, and the natural property under orthogonal transformation are valid in the phase space path integrals.     

Smooth functional derivatives in Feynman path integrals by time slicing approximation

We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, functional differentiation, translation and real linear transformation. The integration by parts and Taylor's expansion formula with respect to functional differentiation holds in Feynman path integral. Feynman path integral is invariant under translation and orthogonal transformation. The interchange of the order with Riemann-Stieltjes integrals, the interchange of the order with a limit, the semiclassical approximation and the fundamental theorem of calculus in Feynman path integral stay valid as well as N. Kumano-go [Bull. Sci. Math. 128 (3) (2004) 197-251].     

Feynman path integrals as analysis on path space by time slicing approximation

Using the time slicing approximation, we give a mathematically rigorous definition of Feynman path integrals for a general class of functionals on the path space. As an application, we prove the interchange with Riemann-Stieltjes integrals, the interchange with a limit, the perturbation expansion formula, the semiclassical approximation, and the fundamental theorem of calculus in Feynman path integral.     

Path integrals as analysis on path space by time slicing approximation

This is a survey of our papers [3, 4]. We give a fairly general class of functionals on a path space so that Feynman path integral has a mathematically rigorous meaning. More precisely, for any functional belonging to our class, the time slicing approximation of Feynman path integral converges uniformly on compact subsets of the configuration space. Our class of functionals is closed under addition, multiplication, translation, real linear transformation and functional differentiation. The invariance under translation and orthogonal transformation, the interchange of the order with Riemann-Stieltjes integrals and some limits, the semiclassical approximation, the integration by parts and the Taylor expansion formula with respect to functional differentiation, and the fundamental theorem of calculus hold in Feynman path integral. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Generalized Fresnel integrals

A general class of (finite dimensional) oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proven as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. Their asymptotic expansion for “strong oscillations” is given. The expansion is in powers of ?1/2M, where ? is a small parameters and 2M is the order of growth of the phase function. Additional assumptions on the integrands are found which are sufficient to yield convergent, resp. Borel summable, expansions.     

Feynman path integrals as infinite-dimensional oscillatory integrals: Some new developments

We discuss a basic mathematical approach to Feynman path integrals as infinite-dimensional oscillatory integrals. We present new results on asymptotics of such integrals which exploit recently developed approximation techniques via finite dimensional oscillatory integrals. Applications are also given, namely to the study of the trace of the time evolution operator in quantum mechanics and to the interpretation of Gutzwiller's trace formula as a leading term in an asymptotic expansion around classical periodic orbits.The second named author is an Alexander von Humboldt Stiftung fellow.     

Feynman path integrals for polynomially growing potentials

A general class of infinite dimensional oscillatory integrals with polynomially growing phase functions is studied. A representation formula of the Parseval type is proved, as well as a formula giving the integrals in terms of analytically continued absolutely convergent integrals. These results are applied to provide a rigorous Feynman path integral representation for the solution of the time-dependent Schrödinger equation with a quartic anharmonic potential. The Borel summability of the asymptotic expansion of the solution in power series of the coupling constant is also proved.     

Numerical integration with Taylor truncations for the quadrilateral and hexahedral finite elements

For general quadrilateral or hexahedral meshes, the finite-element methods require evaluation of integrals of rational functions, instead of traditional polynomials. It remains as a challenge in mathematics to show the traditional Gauss quadratures would ensure the correct order of approximation for the numerical integration in general. However, in the case of nested refinement, the refined quadrilaterals and hexahedra converge to parallelograms and parallelepipeds, respectively. Based on this observation, the rational functions of inverse Jacobians can be approximated by the Taylor expansion with truncation. Then the Gauss quadrature of exact order can be adopted for the resulting integrals of polynomials, retaining the optimal order approximation of the finite-element methods. A theoretic justification and some numerical verification are provided in the paper.     

Weak convergence of Galerkin approximations for fractional elliptic stochastic PDEs with spatial white noise

The numerical approximation of the solution to a stochastic partial differential equation with additive spatial white noise on a bounded domain is considered. The differential operator is assumed to be a fractional power of an integer order elliptic differential operator. The solution is approximated by means of a finite element discretization in space and a quadrature approximation of an integral representation of the fractional inverse from the Dunford–Taylor calculus. For the resulting approximation, a concise analysis of the weak error is performed. Specifically, for the class of twice continuously Fréchet differentiable functionals with second derivatives of polynomial growth, an explicit rate of weak convergence is derived, and it is shown that the component of the convergence rate stemming from the stochasticity is doubled compared to the corresponding strong rate. Numerical experiments for different functionals validate the theoretical results.     

Infinite-dimensional stochastic differential equations obtained by subordination and related Dirichlet forms

New results related to the decomposition theorem of additive functionals associated to quasi-regular Dirichlet forms are presented. A characterization of subordinate processes associated to quasi-regular symmetric Dirichlet forms in terms of the unique solutions of the corresponding martingale problems is obtained.The subordinate of (generalized) Ornstein-Uhlenbeck processes are exhibited explicitly in terms of generators, Dirichlet forms, and unique pathwise solutions of stochastic differential equations (SDEs). In the case where the state space is infinite dimensional as, e.g. in Euclidean quantum field theory, the construction provides a characterization of the processes in terms of projections on the topological dual space, and corresponding finite-dimensional SDEs.     

Applying reproducing kernels to the evaluation and approximation of the simple and time-dependent imaginary time harmonic oscillator path integrals

Reproduction of kernel Hilbert spaces offers an attractive setting for imaginary time path integrals, since they allow to naturally define a probability on the space of paths, which is equal to the probability associated with the paths in Feynman's path integral formulation. This study shows that if the propagator is Gaussian, its variance equals the squared norm of a linear functional on the space of paths. This can be used to rederive the harmonic oscillator propagator, as well as to offer a finite-dimensional perturbative approximation scheme for the time-dependent oscillator wave function and its ground state energy, and its bound error. The error is related to the rate of decay of the Fourier coefficients of the time-dependent part of the potential. When the rate of decay increases beyond a certain threshold, the error in the approximation over a subspace of dimension n is of order (1/n ³).     

On a Chen–Fliess approximation for diffusion functionals

We show that the so-called functional derivatives, as recently introduced by Dupire (Functional Ito calculus, SSRN, 2010), can provide intuitive meaning to classic expansions of path dependent functionals that appear in control theory (work of Brockett, Fliess, Sussmann et. al). We then focus on stochastic differential equations and show that vector fields can be lifted to act as derivations on such functionals. This allows to revisit and generalize the classic stochastic Taylor expansion to arrive at a Chen–Fliess approximation for smooth, path dependent functionals of SDEs with a corresponding \(L^{2}\) -error estimate.     

Analytic Feynman Integrals of Functionals in a Banach Algebra Involving the First Variation

This paper deals with the analytic Feynman integral of functionals on a Wiener space. First the authors establish the existence of the analytic Feynman integrals of functionals in a Banach algebra S_α. The authors then obtain a formula for the first variation of integrals. Finally, various analytic Feynman integration formulas involving the first variation are established.     

Minimally supported error representations and approximation by the constants

Summary. Distribution theory is used to construct minimally supported Peano kernel type representations for linear functionals such as the error in multivariate Hermite interpolation. The simplest case is that of representing the error in approximation to f by the constant polynomial f(a) in terms of integrals of the first order derivatives of f. This is discussed in detail. Here it is shown that suprisingly there exist many representations which are not minimally supported, and involve the integration of first order derivatives over multidimensional regions. The distance of smooth functions from the constants in the uniform norm is estimated using our representations for the error. Received June 30, 1997 / Revised version received April 6, 1999 / Published online February 17, 2000     

HILBERT SPACES OF FUNCTIONALS ASSOCIATED WITH NONLINEAR OPERATORS

Abstract

In [7] the subject of reproducing kernel Hilbert spaces (RKHSs) of linear functionals associated with linear operators and, in particular, with second-order generalized stochastic processes (GSPs), is pursued. In this work these ideas are extended to nonlinear operators. As an example the characteristic operator of a GSP is pursued. The so-called nonlinear space of the process associated with the characteristic operator is investigated and the RKHS of functionals isometrically isomorphic to it is constructed. Unlike the linear space, the nonlinear analysis is not limited to second order GSPs.     

Stationary phase method in infinite dimensions by finite dimensional approximations: Applications to the Schrödinger equation

We develop an approach by finite dimensional approximations for the study of infinite dimensional oscillatory integrals and the relative method of stationary phase. We provide detailed asymptotic expansions in the nondegenerate as well as in the degenerate case. We also give applications to the derivation of detailed asymptotic expansions in Planck's constant for the Schrödinger equation.     

Path integrals on finite sets

We construct an analogue of the Feynman path integral for the case of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeyOeI0YaaS% aaaeaacaaIXaaabaGaamyAaaaadaWcaaqaaiabgkGi2cqaaiabgkGi% 2kaadshaaaqeduuDJXwAKbYu51MyVXgaiuaacqWFvpGAcaWG0bGaey% ypa0JaamisamaaBaaaleaacaGGOaaabeaakmaaBaaaleaacaGGPaaa% beaakiab-v9aQjaadshaaaa!4A8D!\[ - \frac{1}{i}\frac{\partial }{{\partial t}}\varphi t = H_( _) \varphi t\] in which H ₍₎ is a self-adjoint operator in the space L ²(M)= % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSOaHmkaaa!3744!\[\mathbb{C}\], where _M is a finite set, the paths being functions of % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaeSyhHekaaa!375D!\[\mathbb{R}\] with values in M. The path integral is a family of measures F _t,t with values in the operators on L ²(M), or equivalently, a family of complex measures corresponding to matrix coefficients.It is shown that these measures on path space are in some sense dominated by the measure of a Markov process. This implies that F _t,t is concentrated on the set of step functions S[t,t].This allows one to make sense of, and prove, the analogue of Feynman's formula for the propagator of the Hamiltonian H=H ₀+V, where V is a potential, namely the formula: % MathType!MTEF!2!1!+-% feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq-Jc9% vqaqpepm0xbba9pwe9Q8fs0-yqaqpepae9pg0FirpepeKkFr0xfr-x% fr-xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaeyzamaaCa% aaleqabaGaeyOeI0IaamyAaiaacIcacaWG0bGaai4jaiabgkHiTiaa% dshacaGGPaGaamisaaaakiabg2da9maapebabaGaaeyzamaaCaaale% qabaGaeyOeI0IaamyAamaapedabaGaamOvaiaacIcatCvAUfKttLea% ryqr1ngBPrgaiuGacqWF4baEcaGGOaGaam4CaiaacMcacaGGPaGaae% izaiaabohaaWqaaiaadshaaeaacaWG0bGaai4jaaGdcqGHRiI8aaaa% kiaadAeadaWgaaWcbaGaamiDaiaacEcacaGGSaGaamiDaaqabaGcca% GGOaGaaeizaiab-Hha4jaacMcaaSqaaiaadofacaGGBbGaamiDaiaa% cYcacaWG0bGaai4jaiaac2faaeqaniabgUIiYdaaaa!6410!\[{\text{e}}^{ - i(t' - t)H} = \int_{S[t,t']} {{\text{e}}^{ - i\int_t^{t'} {V(x(s)){\text{ds}}} } F_{t',t} ({\text{d}}x)} \]and the corresponding formulas for the matrix coefficients, in which the integral extends over the paths beginning and ending in the appropriate points. We show that the measures F _t,t are completely determined by these equations and by a certain multiplicative property.The path integral corresponding to a two-particle system without interaction is the direct product of the corresponding path integrals. The propagator for a two-particle system with interaction can be obtained by repeated integration.Finally, we show that the above integral formula can be generalized to the case where the potential is time dependent.     

Configuration space integrals and Taylor towers for spaces of knots

We describe Taylor towers for spaces of knots arising from Goodwillie-Weiss calculus of the embedding functor and extend the configuration space integrals of Bott and Taubes from spaces of knots to the stages of the towers. We show that certain combinations of integrals, indexed by trivalent diagrams, yield cohomology classes of the stages of the tower, just as they do for ordinary knots.