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Journal of Physics A Mathematical and Theoretical

Fundamentals of higher order stochastic equations

P. D. Drummond

Published Aug 4, 2014• 7 citations
Publisher

Abstract

Stochastic processes that correspond to equations of arbitrary differential order are obtained.We show that stochastic paths in a complex extension of the original phase-space allow an implementation of such higher-order derivative terms.The resulting stochastic process is equivalent to the original partial differential equation in the sense of having equivalent analytic moments.However, the correspondence has unusual properties.Only the analytic moments are convergent, while non-analytic moments such as the complex moduli are non-convergent.These results unify previous approaches that transform higher-derivative equations into probabilistic stochastic equations.Larger ensembles are required as time-steps are reduced, giving these equations unusual convergence properties.This type of process is relevant to the question of how to obtain stochastic quantum simulations using the Wigner representation.

DOI
10.1088/1751-8113/47/33/335001

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Stochastic processes that correspond to equations of arbitrary differential order are obtained.We show that stochastic paths in a complex extension of the original phase-space allow an implementation of such higher-order derivative terms.The resulting stochastic process is equivalent to the original partial differential equation in the sense of having equivalent analytic moments.However, the correspondence has unusual properties.Only the analytic moments are convergent, while non-analytic moments such as the complex moduli are non-convergent.These results unify previous approaches that transform higher-derivative equations into probabilistic stochastic equations.Larger ensembles are required as time-steps are reduced, giving these equations unusual convergence properties.This type of process is relevant to the question of how to obtain stochastic quantum simulations using the Wigner representation.
General higher-order stochastic processes that correspond to any diffusion-type tensor of higher than second order are obtained. The relationship of multivariate higher-order stochastic differential equations with tensor decomposition theory and tensor rank is explained. Techniques for generating the requisite complex higher-order noise are proved to exist either using polar coordinates and $\ensuremath{\gamma}$ distributions, or from products of Gaussian variates. This method is shown to allow the calculation of the dynamics of the Wigner function, after it is extended to a complex phase space. The results are illustrated physically through dynamical calculations of the positive Wigner distribution for three-mode parametric downconversion, widely used in quantum optics. The approach eliminates paradoxes arising from truncation of the higher derivative terms in Wigner function time evolution. Anomalous results of negative populations and vacuum scattering found in truncated Wigner quantum simulations in quantum optics and Bose-Einstein condensate dynamics are shown not to occur with this type of stochastic theory.
The classical stochastic differential equation given as dX(t,ω)=a(t,X) dt+b(t,X) dQ(t),X(t0)=X0,t∈[t0,T],dX(t, \omega) = a(t, X) \, dt + b(t, X) \, dQ(t), \quad X(t_0) = X_0, \quad t \in [t_0, T],dX(t,ω)=a(t,X)dt+b(t,X)dQ(t),X(t0​)=X0​,t∈[t0​,T], where the driving process Q(t) Q(t) Q(t) is a martingale, and a a a and b b b are adapted functionals has several numerical schemes developed for it, where each exhibits distinct stability and convergence properties depending on the nature of the driving process and the functional space of the solution. A generalization of this equation is the quantum stochastic differential equation introduced by Hudson and Parthasarathy, expressed as dX(t)=E(t,X(t)) dΛπ(t)+F(t,X(t)) dAf†(t)+G(t,X(t)) dAg(t)+H(t,X(t)) dt,dX(t) = E(t, X(t)) \, d\Lambda_\pi(t) + F(t, X(t)) \, dA_f^\dagger(t) + G(t, X(t)) \, dA_g(t) + H(t, X(t)) \, dt,dX(t)=E(t,X(t))dΛπ​(t)+F(t,X(t))dAf†​(t)+G(t,X(t))dAg​(t)+H(t,X(t))dt, where E,F,G, E, F, G, E,F,G, and H H H are stochastic operator-valued functions defined on a Hilbert space. In 1992, Ekhaquere reformulated this equation in an equivalent form involving the non-classical differential equation stated as; ddt⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0)=X0,t∈[t0,T],\frac{d}{dt} \langle \eta, X(t) \xi \rangle = P(t, X(t)) \langle \eta, \xi \rangle, \quad X(t_0) = X_0, \quad t \in [t_0, T],dtd​⟨η,X(t)ξ⟩=P(t,X(t))⟨η,ξ⟩,X(t0​)=X0​,t∈[t0​,T], which represents an ordinary differential equation of non-classical type. The stochastic process X(t) X(t) X(t) acts as a densely defined linear operator on a tensor product of two Hilbert spaces, one of which is the Boson Fock space. The vectors η \eta η and ξ \xi ξ belong to a dense subset of this tensor product Hilbert space, and the mapping (η,ξ)↦P(t,X)(η,ξ)(\eta, \xi) \mapsto P(t, X)(\eta, \xi)(η,ξ)↦P(t,X)(η,ξ) defines a sesquilinear form for each fixed pair (t,X)(t, X)(t,X). We analyze the numerical approximation of the QSDE using its equivalent inner-product form. The numerical investigation employs three principal methods: the Lagrangian Quadrature Method, the Linear Multistep Scheme, and the Newton–Cotes Method. Each method is carefully designed to ensure the consistency, stability, and convergence of the approximate solution X(t)X(t)X(t) toward the exact stochastic process. These schemes are implemented using Python programming language, which provides a flexible and efficient computational environment for testing the proposed algorithms. Our computational results show that the developed schemes generalize Taylor-type approximations of Itô processes under weak convergence criteria. The proposed algorithms exhibit numerical stability and yield results that compare favorably with analytical benchmarks. Furthermore, the implementation avoids direct approximation of the driving quantum processes, thereby preserving the natural adaptation of the schemes to quantum dynamical models.
Previously [2004, AIP Conf. Proc., 750, 361], we considered a stochastic mechanics in the form of a stochastic differential equation mdxdt = μ(t) where μ(t) is a stochastic process defined by the set of Bohmian momentum time histories from an ensemble of particles. In this paper, we consider another stochastic differential equation d2xdt2 = 1m[−dV(x)dx + ζ(t)] where the stochastic force ζ(t) is a stochastic process defined by the set of quantum force time histories from an ensemble of particles. We show that, if the stochastic process is a purely random process characterized by an n‐th order joint probability density in the form of products of delta functions, then the stochastic mechanics with the stochastic force ζ(t) is equivalent to quantum mechanics in the sense that the former yields the same position probability density as the latter. However, for a particular non‐purely random process, we show that the stochastic mechanics with the stochastic force ζ(t) is not equivalent to quantum mechanics. Therefore, this stochastic mechanics is also generally not equivalent to quantum mechanics.
White noise theory allows to formulate quantum white noises explicitly as elemental quantum stochastic processes. A traditional quantum stochastic differential equation of Itô type is brought into a normal-ordered white noise differential equation driven by lower powers of quantum white noises. The class of normal-ordered white noise differential equations covers quantum stochastic differential equations with highly singular noises such as higher powers or higher order derivatives of quantum white noises, which are far beyond the traditional Itô theory. For a general normal-ordered white noise differential equation unique existence of a solution is proved in the sense of white noise distribution. Its regularity properties are investigated by means of weighted Fock spaces interpolating spaces of white noise distributions and associated characterization theorems for S-transform and for operator symbols.
Phase-space representations based on coherent states (PPP,QQQ, Wigner) have been successful in the creation of stochastic differential equations (SDEs) for the efficient stochastic simulation of high-dimensional quantum systems. However, many problems using these techniques remain intractable over long integration times. We present a number-phase Wigner representation that can be unraveled into SDEs. We demonstrate convergence to the correct solution for an anharmonic oscillator with small dampening for significantly longer than other phase-space representations. This process requires an effective sampling of a nonclassical probability distribution. We describe and demonstrate a method of achieving this sampling using stochastic weights.
We present a survey of the relations between infinite dimensional integrals, both of the probabilistic type (e.g. Wiener path integrals) and of oscillatory type (e.g. Feynman path integrals).Besides their mutual relations (analogies and differences) we also discuss their relations with certain types of partial differential equations (parabolic resp. hyperbolic), describing time evolution with or without stochastic terms.The connection of these worlds of deterministic and stochastic evolutions with the world of quantum phenomena is also briefly illustrated. The survey spans a bridge from basic concepts and methods in these areas to recent developments concerning their relations.