Related papers: On the Feynman-Kac Formula
The book deals with a stochastic formulation of path integration in real time, by rotating the_space_ variables over exp(i pi/4). Preliminary chapters deal with quantum and classical mechanics, probability theory and stochastic calculus,…
Attention is given to the interface of mathematics and physics, specifically noting that fundamental principles limit the usefulness of otherwise perfectly good mathematical general integral solutions. A new set of multivector solutions to…
In this paper, we build Wiener measure for the path space on the Heisenberg group by using of the heat kernel corresponding to the sub-Laplacian and give the definition of the Wiener integral. Then we give the Feynman-Kac formula.
We design a particle interpretation of Feynman-Kac measures on path spaces based on a backward Markovian representation combined with a traditional mean field particle interpretation of the flow of their final time marginals. In contrast to…
In the absence of a satisfactory interpretation of quantum theory, physical law lacks physical basis. This paper reviews the orthodox, or Dirac-von Neumann interpretation, and makes explicit that Hilbert space describes propositions about…
The classical Feynman-Kac formula states the connection between linear parabolic partial differential equations (PDEs), like the heat equation, and expectation of stochastic processes driven by Brownian motion. It gives then a method for…
The infinite-dimensional Hilbert sphere $S^\infty$ has been widely employed to model density functions and shapes, extending the finite-dimensional counterpart. We consider the Fr\'echet mean as an intrinsic summary of the central tendency…
We give the solution of certain parabolic evolution problems (time-depending perturbations of the heat equation for the harmonic oscillator) as explicit integrals on the Wiener space.
It is shown that the complex phase of the Feynman propagator is a solution of the quantum Hamilton Jacobi equation
We prove that, under the H\"ormander criterion on an It\^{o} process, all its martingale observables are smooth. As a consequence, we also obtain a generalized Feynman-Kac formula providing smooth solutions to certain PDE boundary-value…
This paper is concerned with the relationship between forward-backward stochastic Volterra integral equations (FBSVIEs, for short) and a system of (non-local in time) path dependent partial differential equations (PPDEs, for short). Due to…
In this paper, we survey recent progress on the constructive theory of the Feynman operator calculus. (The theory is constructive in that, operators acting at different times, actually commute.) We first develop an operator version of the…
We develop the contour integral method for numerically solving the Feynman-Kac equation with two internal states [P. B. Xu and W. H. Deng, Math. Model. Nat. Phenom., 13 (2018), 10], describing the functional distribution of particle's…
We propose a natural, parameter-free, discrete-variable formulation of Feynman path integrals. We show that for discrete-variable quantum systems, Feynman path integrals take the form of walks on the graph whose weighted adjacency matrix is…
A generalized Feynman-Kac formula based on the Wiener measure is presented. Within the setting of a quantum particle in an electromagnetic field it yields the standard Feynman-Kac formula for the corresponding Schr\"odinger semigroup. In…
We {\em derive} the exact configuration space path integral, together with the way how to evaluate it, from the Hamiltonian approach for any quantum mechanical system in flat spacetime whose Hamiltonian has at most two momentum operators.…
In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulas, we…
We establish a version of the Feynman-Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the…
In the framework of stochastic functional differential equations (SFDE's) and the corresponding calculus developed in the recent years by F. Yan and S. Mohammed, we provide a series of representation formulae for a variety of highly…
Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form $\int_a^b f(x) d\mu(x) = \sum_{i=1}^n w_i f(x_i)$ where $f$ belongs to $H^1(\mu)$. Here, $\mu$ belongs to a class of continuous…