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Journal articles on the topic 'Non-markovian stochastic Schrodinger equations'

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Author: Grafiati

Published: 4 June 2021

Last updated: 20 February 2023

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1

Vanchurin, Vitaly. "Towards a Theory of Quantum Gravity from Neural Networks." Entropy 24, no. 1 (December 21, 2021): 7. http://dx.doi.org/10.3390/e24010007.

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Neural network is a dynamical system described by two different types of degrees of freedom: fast-changing non-trainable variables (e.g., state of neurons) and slow-changing trainable variables (e.g., weights and biases). We show that the non-equilibrium dynamics of trainable variables can be described by the Madelung equations, if the number of neurons is fixed, and by the Schrodinger equation, if the learning system is capable of adjusting its own parameters such as the number of neurons, step size and mini-batch size. We argue that the Lorentz symmetries and curved space-time can emerge from the interplay between stochastic entropy production and entropy destruction due to learning. We show that the non-equilibrium dynamics of non-trainable variables can be described by the geodesic equation (in the emergent space-time) for localized states of neurons, and by the Einstein equations (with cosmological constant) for the entire network. We conclude that the quantum description of trainable variables and the gravitational description of non-trainable variables are dual in the sense that they provide alternative macroscopic descriptions of the same learning system, defined microscopically as a neural network.

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2

Costanza, G. "Non-Markovian stochastic evolution equations." Physica A: Statistical Mechanics and its Applications 402 (May 2014): 224–35. http://dx.doi.org/10.1016/j.physa.2014.01.038.

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3

Adamian, G. G., N. V. Antonenko, Z. Kanokov, and V. V. Sargsyan. "Quantum Non-Markovian Stochastic Equations." Theoretical and Mathematical Physics 145, no. 1 (October 2005): 1443–56. http://dx.doi.org/10.1007/s11232-005-0170-2.

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4

Semina, I., V. Semin, F. Petruccione, and A. Barchielli. "Stochastic Schrödinger Equations for Markovian and non-Markovian Cases." Open Systems & Information Dynamics 21, no. 01n02 (March 12, 2014): 1440008. http://dx.doi.org/10.1142/s1230161214400083.

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Firstly, the Markovian stochastic Schrödinger equations are presented, together with their connections with the theory of measurements in continuous time. Moreover, the stochastic evolution equations are translated into a simulation algorithm, which is illustrated by two concrete examples — the damped harmonic oscillator and a two-level atom with homodyne photodetection. We then consider how to introduce memory effects in the stochastic Schrödinger equation via coloured noise. Specifically, the approach by using the Ornstein-Uhlenbeck process is illustrated and a simulation for the non-Markovian process proposed. Finally, an analytical approximation technique is tested with the help of the stochastic simulation in a model of a dissipative qubit.

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5

Li, Xiantao. "Markovian embedding procedures for non-Markovian stochastic Schrödinger equations." Physics Letters A 387 (January 2021): 127036. http://dx.doi.org/10.1016/j.physleta.2020.127036.

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6

Tilloy, Antoine. "Time-local unraveling of non-Markovian stochastic Schrödinger equations." Quantum 1 (September 19, 2017): 29. http://dx.doi.org/10.22331/q-2017-09-19-29.

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Non-Markovian stochastic Schrödinger equations (NMSSE) are important tools in quantum mechanics, from the theory of open systems to foundations. Yet, in general, they are but formal objects: their solution can be computed numerically only in some specific cases or perturbatively. This article is focused on the NMSSE themselves rather than on the open-system evolution they unravel and aims at making them less abstract. Namely, we propose to write the stochastic realizations of linear NMSSE as averages over the solutions of an auxiliary equation with an additional random field. Our method yields a non-perturbative numerical simulation algorithm for generic linear NMSSE that can be made arbitrarily accurate for reasonably short times. For isotropic complex noises, the method extends from linear to non-linear NMSSE and allows to sample the solutions of norm-preserving NMSSE directly.

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7

Farias, R. L. S., Rudnei O. Ramos, and L. A. da Silva. "Numerical solutions for non-Markovian stochastic equations of motion." Computer Physics Communications 180, no. 4 (April 2009): 574–79. http://dx.doi.org/10.1016/j.cpc.2008.12.005.

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8

Gough, John E., Matthew R. James, and Hendra I. Nurdin. "Single photon quantum filtering using non-Markovian embeddings." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, no. 1979 (November 28, 2012): 5408–21. http://dx.doi.org/10.1098/rsta.2011.0524.

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We determine quantum master and filter equations for continuous measurement of systems coupled to input fields in certain non-classical continuous-mode states, specifically single photon states. The quantum filters are shown to be derivable from an embedding into a larger non-Markovian system, and are given by a system of coupled stochastic differential equations.

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9

Breuer, Heinz-Peter, Bernd Kappler, and Francesco Petruccione. "Stochastic wave-function method for non-Markovian quantum master equations." Physical Review A 59, no. 2 (February 1, 1999): 1633–43. http://dx.doi.org/10.1103/physreva.59.1633.

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10

Sezer, Ali Devin, Thomas Kruse, and Alexandre Popier. "Backward stochastic differential equations with non-Markovian singular terminal values." Stochastics and Dynamics 19, no. 02 (March 27, 2019): 1950006. http://dx.doi.org/10.1142/s0219493719500060.

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We solve a class of BSDE with a power function [Formula: see text], [Formula: see text], driving its drift and with the terminal boundary condition [Formula: see text] (for which [Formula: see text] is assumed) or [Formula: see text], where [Formula: see text] is the ball in the path space [Formula: see text] of the underlying Brownian motion centered at the constant function [Formula: see text] and radius [Formula: see text]. The solution involves the derivation and solution of a related heat equation in which [Formula: see text] serves as a reaction term and which is accompanied by singular and discontinuous Dirichlet boundary conditions. Although the solution of the heat equation is discontinuous at the corners of the domain, the BSDE has continuous sample paths with the prescribed terminal value.

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11

Hong, Yin. "Non-Markovian Forward-Backward Stochastic Differential Equations with Discontinuous Coefficients." Applied Mathematics 11, no. 04 (2020): 328–43. http://dx.doi.org/10.4236/am.2020.114024.

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12

Costanza, G. "A theorem allowing the derivation of deterministic evolution equations from stochastic evolution equations. III The Markovian–non-Markovian mix." Physica A: Statistical Mechanics and its Applications 391, no. 6 (March 2012): 2167–81. http://dx.doi.org/10.1016/j.physa.2011.11.055.

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13

Yin, Hong. "Forward–backward stochastic partial differential equations with non-monotonic coefficients." Stochastics and Dynamics 16, no. 06 (November 6, 2016): 1650025. http://dx.doi.org/10.1142/s0219493716500258.

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In this paper we study the solvability of a class of fully-coupled forward–backward stochastic partial differential equations (FBSPDEs) with non-monotonic coefficients. These FBSPDEs cannot be put into the framework of stochastic evolution equations in general, and the usual decoupling methods for the Markovian forward–backward SDEs are difficult to apply. We prove the well-posedness of such FBSPDEs by using the method of continuation. Contrary to the common belief, we show that the usual monotonicity assumption can be removed by a change of the diffusion term.

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14

CAO, GUILAN, KAI HE, and XICHENG ZHANG. "SUCCESSIVE APPROXIMATIONS OF INFINITE DIMENSIONAL SDES WITH JUMP." Stochastics and Dynamics 05, no. 04 (December 2005): 609–19. http://dx.doi.org/10.1142/s0219493705001584.

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In this paper, we study the existence and uniqueness of solutions to non-Markovian stochastic differential equations with jump and non-Lipschitz coefficients in infinite dimensional spaces by successive approximation.

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15

Mamis, K. I., G. A. Athanassoulis, and Z. G. Kapelonis. "A systematic path to non-Markovian dynamics: new response probability density function evolution equations under Gaussian coloured noise excitation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 475, no. 2226 (June 2019): 20180837. http://dx.doi.org/10.1098/rspa.2018.0837.

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Determining evolution equations governing the probability density function (pdf) of non-Markovian responses to random differential equations (RDEs) excited by coloured noise, is an important issue arising in various problems of stochastic dynamics, advanced statistical physics and uncertainty quantification of macroscopic systems. In the present work, such equations are derived for a scalar, nonlinear RDE under additive coloured Gaussian noise excitation, through the stochastic Liouville equation. The latter is an exact, yet non-closed equation, involving averages over the time history of the non-Markovian response. This non-locality is treated by applying an extension of the Novikov–Furutsu theorem and a novel approximation, employing a stochastic Volterra–Taylor functional expansion around instantaneous response moments, leading to efficient, closed, approximate equations for the response pdf. These equations retain a tractable amount of non-locality and nonlinearity, and they are valid in both the transient and long-time regimes for any correlation function of the excitation. Also, they include as special cases various existing relevant models, and generalize Hänggi's ansatz in a rational way. Numerical results for a bistable nonlinear RDE confirm the accuracy and the efficiency of the new equations. Extension to the multidimensional case (systems of RDEs) is feasible, yet laborious.

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16

Frank, T. D. "Strongly Nonlinear Stochastic Processes in Physics and the Life Sciences." ISRN Mathematical Physics 2013 (March 28, 2013): 1–28. http://dx.doi.org/10.1155/2013/149169.

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Strongly nonlinear stochastic processes can be found in many applications in physics and the life sciences. In particular, in physics, strongly nonlinear stochastic processes play an important role in understanding nonlinear Markov diffusion processes and have frequently been used to describe order-disorder phase transitions of equilibrium and nonequilibrium systems. However, diffusion processes represent only one class of strongly nonlinear stochastic processes out of four fundamental classes of time-discrete and time-continuous processes evolving on discrete and continuous state spaces. Moreover, strongly nonlinear stochastic processes appear both as Markov and non-Markovian processes. In this paper the full spectrum of strongly nonlinear stochastic processes is presented. Not only are processes presented that are defined by nonlinear diffusion and nonlinear Fokker-Planck equations but also processes are discussed that are defined by nonlinear Markov chains, nonlinear master equations, and strongly nonlinear stochastic iterative maps. Markovian as well as non-Markovian processes are considered. Applications range from classical fields of physics such as astrophysics, accelerator physics, order-disorder phase transitions of liquids, material physics of porous media, quantum mechanical descriptions, and synchronization phenomena in equilibrium and nonequilibrium systems to problems in mathematics, engineering sciences, biology, psychology, social sciences, finance, and economics.

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17

Pandey, Devashish, Enrique Colomés, Guillermo Albareda, and Xavier Oriols. "Stochastic Schrödinger Equations and Conditional States: A General Non-Markovian Quantum Electron Transport Simulator for THz Electronics." Entropy 21, no. 12 (November 25, 2019): 1148. http://dx.doi.org/10.3390/e21121148.

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A prominent tool to study the dynamics of open quantum systems is the reduced density matrix. Yet, approaching open quantum systems by means of state vectors has well known computational advantages. In this respect, the physical meaning of the so-called conditional states in Markovian and non-Markovian scenarios has been a topic of recent debate in the construction of stochastic Schrödinger equations. We shed light on this discussion by acknowledging the Bohmian conditional wavefunction (linked to the corresponding Bohmian trajectory) as the proper mathematical object to represent, in terms of state vectors, an arbitrary subset of degrees of freedom. As an example of the practical utility of these states, we present a time-dependent quantum Monte Carlo algorithm to describe electron transport in open quantum systems under general (Markovian or non-Markovian) conditions. By making the most of trajectory-based and wavefunction methods, the resulting simulation technique extends to the quantum regime, the computational capabilities that the Monte Carlo solution of the Boltzmann transport equation offers for semi-classical electron devices.

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18

Kitagawa, Akihiro, and Atsushi Takeuchi. "Asymptotic Behavior of Densities for Stochastic Functional Differential Equations." International Journal of Stochastic Analysis 2013 (February 28, 2013): 1–17. http://dx.doi.org/10.1155/2013/537023.

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Consider stochastic functional differential equations depending on whole past histories in a finite time interval, which determine non-Markovian processes. Under the uniformly elliptic condition on the coefficients of the diffusion terms, the solution admits a smooth density with respect to the Lebesgue measure. In the present paper, we will study the large deviations for the family of the solution process and the asymptotic behaviors of the density. The Malliavin calculus plays a crucial role in our argument.

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19

Martyr, Randall, John Moriarty, and Magnus Perninge. "Discrete-time risk-aware optimal switching with non-adapted costs." Advances in Applied Probability 54, no. 2 (June 2022): 625–55. http://dx.doi.org/10.1017/apr.2021.44.

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AbstractWe solve non-Markovian optimal switching problems in discrete time on an infinite horizon, when the decision-maker is risk-aware and the filtration is general, and establish existence and uniqueness of solutions for the associated reflected backward stochastic difference equations. An example application to hydropower planning is provided.

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20

Gwiżdż, Piotr. "Applications of Stochastic Semigroups to Queueing Models." Annales Mathematicae Silesianae 33, no. 1 (September 1, 2019): 121–42. http://dx.doi.org/10.2478/amsil-2018-0007.

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AbstractNon-markovian queueing systems can be extended to piecewise-deterministic Markov processes by appending supplementary variables to the system. Then their analysis leads to an infinite system of partial differential equations with an infinite number of variables and non-local boundary conditions. We show how one can study such systems by using the theory of stochastic semigroups.

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21

Farias, R. L. S., R. O. Ramos, and L. A. da Silva. "Nonlinear effects in the dynamics governed by non-Markovian stochastic Langevin-like equations." Journal of Physics: Conference Series 246 (September 1, 2010): 012029. http://dx.doi.org/10.1088/1742-6596/246/1/012029.

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22

Fleming, C. H., and B. L. Hu. "Non-Markovian dynamics of open quantum systems: Stochastic equations and their perturbative solutions." Annals of Physics 327, no. 4 (April 2012): 1238–76. http://dx.doi.org/10.1016/j.aop.2011.12.006.

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23

Leão, Dorival, Alberto Ohashi, and Francesco Russo. "Discrete-type approximations for non-Markovian optimal stopping problems: Part I." Journal of Applied Probability 56, no. 4 (December 2019): 981–1005. http://dx.doi.org/10.1017/jpr.2019.57.

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AbstractWe present a discrete-type approximation scheme to solve continuous-time optimal stopping problems based on fully non-Markovian continuous processes adapted to the Brownian motion filtration. The approximations satisfy suitable variational inequalities which allow us to construct $\varepsilon$ -optimal stopping times and optimal values in full generality. Explicit rates of convergence are presented for optimal values based on reward functionals of path-dependent stochastic differential equations driven by fractional Brownian motion. In particular, the methodology allows us to design concrete Monte Carlo schemes for non-Markovian optimal stopping time problems as demonstrated in the companion paper by Bezerra et al.

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24

Costanza, G. "A theorem allowing to derive deterministic evolution equations from stochastic evolution equations, II: The non-Markovian extension." Physica A: Statistical Mechanics and its Applications 390, no. 12 (June 2011): 2267–75. http://dx.doi.org/10.1016/j.physa.2011.02.046.

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25

Buckdahn, R. "A Regularity Condition for Non-Markovian Solutions of Stochastic Differential Equations in the Plane." Mathematische Nachrichten 149, no. 1 (1990): 125–32. http://dx.doi.org/10.1002/mana.19901490109.

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26

Wouters, Jeroen, Stamen Iankov Dolaptchiev, Valerio Lucarini, and Ulrich Achatz. "Parameterization of stochastic multiscale triads." Nonlinear Processes in Geophysics 23, no. 6 (November 28, 2016): 435–45. http://dx.doi.org/10.5194/npg-23-435-2016.

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Abstract. We discuss applications of a recently developed method for model reduction based on linear response theory of weakly coupled dynamical systems. We apply the weak coupling method to simple stochastic differential equations with slow and fast degrees of freedom. The weak coupling model reduction method results in general in a non-Markovian system; we therefore discuss the Markovianization of the system to allow for straightforward numerical integration. We compare the applied method to the equations obtained through homogenization in the limit of large timescale separation between slow and fast degrees of freedom. We numerically compare the ensemble spread from a fixed initial condition, correlation functions and exit times from a domain. The weak coupling method gives more accurate results in all test cases, albeit with a higher numerical cost.

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27

Leimkuhler, B., C. Matthews, and M. V. Tretyakov. "On the long-time integration of stochastic gradient systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 470, no. 2170 (October 8, 2014): 20140120. http://dx.doi.org/10.1098/rspa.2014.0120.

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This article addresses the weak convergence of numerical methods for Brownian dynamics. Typical analyses of numerical methods for stochastic differential equations focus on properties such as the weak order which estimates the asymptotic (stepsize h → 0 ) convergence behaviour of the error of finite-time averages. Recently, it has been demonstrated, by study of Fokker–Planck operators, that a non-Markovian numerical method generates approximations in the long-time limit with higher accuracy order (second order) than would be expected from its weak convergence analysis (finite-time averages are first-order accurate). In this article, we describe the transition from the transient to the steady-state regime of this numerical method by estimating the time-dependency of the coefficients in an asymptotic expansion for the weak error, demonstrating that the convergence to second order is exponentially rapid in time. Moreover, we provide numerical tests of the theory, including comparisons of the efficiencies of the Euler–Maruyama method, the popular second-order Heun method, and the non-Markovian method.

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28

Cruces, Diego. "Review on Stochastic Approach to Inflation." Universe 8, no. 6 (June 17, 2022): 334. http://dx.doi.org/10.3390/universe8060334.

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We present a review on the state-of-the-art of the mathematical framework known as stochastic inflation, paying special attention to its derivation, and giving references for the readers interested in results coming from the application of the stochastic framework to different inflationary scenarios, especially to those of interest for primordial black hole formation. During the derivation of the stochastic formalism, we will emphasise two aspects in particular: the difference between the separate universe approach and the true long wavelength limit of scalar inhomogeneities and the generically non-Markovian nature of the noises that appear in the stochastic equations.

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29

Kadetov, Aleksandr, Alexey Kipriyanov, and Aleksandr Doktorov. "Kinetic Equations for Local Concentrations of Reactants in Spatially Inhomogeneous Solutions." Siberian Journal of Physics 4, no. 3 (October 1, 2009): 78–87. http://dx.doi.org/10.54362/1818-7919-2009-4-3-78-87.

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Spatially inhomogeneous dilute solutions of reactants placed in a continual solvent (Waite approach) are examined. Based on the familiar non-Markovian kinetic equations of the Integral Encounter Theory, the Markovian kinetic equations were first obtained that allowed for the force action of reactants on each other due to binary encounters in solution during reactant macroscopic displacement. It has been shown that the value of “collisional integral” of reactants determined by their stochastic motion is small as compared to the “collisional integral” with the solvent molecules in the frame of the binary approach to the description of the solvent evolution. Nevertheless, taking account of the encounters of reactants with each other may turn out to be of primary importance, since it results in new physical effects.

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30

Dieu, Nguyen Thanh. "Some Results on Almost Sure Stability of Non-Autonomous Stochastic Differential Equations with Markovian Switching." Vietnam Journal of Mathematics 44, no. 4 (January 8, 2016): 665–77. http://dx.doi.org/10.1007/s10013-015-0181-8.

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31

Deng, Qiqi, and Tianshou Zhou. "Memory-Induced Bifurcation and Oscillations in the Chemical Brusselator Model." International Journal of Bifurcation and Chaos 30, no. 10 (August 2020): 2050151. http://dx.doi.org/10.1142/s0218127420501515.

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Previous studies assumed that the reaction processes in the chemical Brusselator model are memoryless or Markovian. However, as long as a reactant interacts with its environment, the reaction kinetics cannot be described as a memoryless process. This raises a question: how do we predict the behavior of the chemical Brusselator system with molecular memory characterized by nonexponential waiting-time distributions? Here, a novel technique is developed to address this question. This technique converts a non-Markovian question to a Markovian one by introducing effective transition rates that explicitly decode the memory effect. Based on this conversion, it is analytically shown that molecular memory can induce bifurcations and oscillations. Moreover, a set of sufficient conditions are derived, which can guarantee that the system of the rate equations for the Markovian reaction system generates oscillations via memory index-induced bifurcation. In turn, these conditions can guarantee that the original non-Markovian reaction system generates stochastic oscillations. Numerical simulation verifies the theoretical prediction. The overall analysis indicates that molecular memory is not a negligible factor affecting a chemical system’s behavior.

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32

Barchielli, Alberto. "Quantum Stochastic Equations for an Opto-Mechanical Oscillator with Radiation Pressure Interaction and Non-Markovian Effects." Reports on Mathematical Physics 77, no. 3 (June 2016): 315–33. http://dx.doi.org/10.1016/s0034-4877(16)30033-7.

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33

Mao, Xuerong, Chenggui Yuan, and G. Yin. "Approximations of Euler–Maruyama type for stochastic differential equations with Markovian switching, under non-Lipschitz conditions." Journal of Computational and Applied Mathematics 205, no. 2 (August 2007): 936–48. http://dx.doi.org/10.1016/j.cam.2006.01.052.

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34

de Vega, Inés, Daniel Alonso, Pierre Gaspard, and Walter T. Strunz. "Non-Markovian stochastic Schrödinger equations in different temperature regimes: A study of the spin-boson model." Journal of Chemical Physics 122, no. 12 (March 22, 2005): 124106. http://dx.doi.org/10.1063/1.1867377.

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35

Röst, G., Z. Vizi, and I. Z. Kiss. "Pairwise approximation for SIR -type network epidemics with non-Markovian recovery." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 474, no. 2210 (February 2018): 20170695. http://dx.doi.org/10.1098/rspa.2017.0695.

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We present the generalized mean-field and pairwise models for non-Markovian epidemics on networks with arbitrary recovery time distributions. First we consider a hyperbolic partial differential equation (PDE) system, where the population of infective nodes and links are structured by age since infection. We show that the PDE system can be reduced to a system of integro-differential equations, which is analysed analytically and numerically. We investigate the asymptotic behaviour of the generalized model and provide an implicit analytical expression involving the final epidemic size and pairwise reproduction number. As an illustration of the applicability of the general model, we recover known results for the exponentially distributed and fixed recovery time cases. For gamma- and uniformly distributed infectious periods, new pairwise models are derived. Theoretical findings are confirmed by comparing results from the new pairwise model and explicit stochastic network simulation. A major benefit of the generalized pairwise model lies in approximating the time evolution of the epidemic.

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36

Janssen, A., and J. Segers. "Markov Tail Chains." Journal of Applied Probability 51, no. 4 (December 2014): 1133–53. http://dx.doi.org/10.1239/jap/1421763332.

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The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions inRd. We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

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37

Janssen, A., and J. Segers. "Markov Tail Chains." Journal of Applied Probability 51, no. 04 (December 2014): 1133–53. http://dx.doi.org/10.1017/s0001867800012027.

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The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in R d . We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

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38

Janssen, A., and J. Segers. "Markov Tail Chains." Journal of Applied Probability 51, no. 04 (December 2014): 1133–53. http://dx.doi.org/10.1017/s002190020001202x.

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The extremes of a univariate Markov chain with regularly varying stationary marginal distribution and asymptotically linear behavior are known to exhibit a multiplicative random walk structure called the tail chain. In this paper we extend this fact to Markov chains with multivariate regularly varying marginal distributions in R d . We analyze both the forward and the backward tail process and show that they mutually determine each other through a kind of adjoint relation. In a broader setting, we will show that even for non-Markovian underlying processes a Markovian forward tail chain always implies that the backward tail chain is also Markovian. We analyze the resulting class of limiting processes in detail. Applications of the theory yield the asymptotic distribution of both the past and the future of univariate and multivariate stochastic difference equations conditioned on an extreme event.

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39

Mikhailov, V. A., and N. V. Troshkin. "Non-Markovian decoherence of a two-level system in a Lorentzian bosonic reservoir and a stochastic environment with finite correlation time." Computer Optics 45, no. 3 (June 2021): 372–81. http://dx.doi.org/10.18287/2412-6179-co-776.

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In this paper we investigate non-Markovian evolution of a two-level system (qubit) in a bosonic bath under inﬂuence of an external classical ﬂuctuating environment. The interaction with the bath has the Lorentzian spectral density, and the ﬂuctuating environment (stochastic ﬁeld) is represented by a set of Ornstein-Uhlenbeck processes. Each of the subenvironments of the composite environment is able to induce non-Markovian dynamics of the two-level system. By means of the numerically exact method of hierarchical equations of motion, we study steady states of the two-level system, evolution of the reduced density matrix and the equilibrium emission spectra in dependence on the frequency cutoffs and the coupling strengths of the subenvironments. Additionally, we investigate the impact of the rotating wave approximation (RWA) for the interaction with the bath on accuracy of the results.

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40

Wu, Shu Jin, and Bin Zhou. "Existence and uniqueness of stochastic differential equations with random impulses and Markovian switching under non-lipschitz conditions." Acta Mathematica Sinica, English Series 27, no. 3 (February 4, 2011): 519–36. http://dx.doi.org/10.1007/s10114-011-9753-z.

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41

Fujii, Masaaki, and Akihiko Takahashi. "Anticipated backward SDEs with jumps and quadratic-exponential growth drivers." Stochastics and Dynamics 19, no. 03 (May 30, 2019): 1950020. http://dx.doi.org/10.1142/s0219493719500205.

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In this paper, we study a class of Anticipated Backward Stochastic Differential Equations (ABSDE) with jumps. The solution of the ABSDE is a triple [Formula: see text] where [Formula: see text] is a semimartingale, and [Formula: see text] are the diffusion and jump coefficients. We allow the driver of the ABSDE to have linear growth on the uniform norm of [Formula: see text]’s future paths, as well as quadratic and exponential growth on the spot values of [Formula: see text], respectively. The existence of the unique solution is proved for Markovian and non-Markovian settings with different structural assumptions on the driver. In the former case, some regularities on [Formula: see text] with respect to the forward process are also obtained.

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Callaham, J. L., J. C. Loiseau, G. Rigas, and S. L. Brunton. "Nonlinear stochastic modelling with Langevin regression." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2250 (June 2021): 20210092. http://dx.doi.org/10.1098/rspa.2021.0092.

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Many physical systems characterized by nonlinear multiscale interactions can be modelled by treating unresolved degrees of freedom as random fluctuations. However, even when the microscopic governing equations and qualitative macroscopic behaviour are known, it is often difficult to derive a stochastic model that is consistent with observations. This is especially true for systems such as turbulence where the perturbations do not behave like Gaussian white noise, introducing non-Markovian behaviour to the dynamics. We address these challenges with a framework for identifying interpretable stochastic nonlinear dynamics from experimental data, using forward and adjoint Fokker–Planck equations to enforce statistical consistency. If the form of the Langevin equation is unknown, a simple sparsifying procedure can provide an appropriate functional form. We demonstrate that this method can learn stochastic models in two artificial examples: recovering a nonlinear Langevin equation forced by coloured noise and approximating the second-order dynamics of a particle in a double-well potential with the corresponding first-order bifurcation normal form. Finally, we apply Langevin regression to experimental measurements of a turbulent bluff body wake and show that the statistical behaviour of the centre of pressure can be described by the dynamics of the corresponding laminar flow driven by nonlinear state-dependent noise.

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Kazakov, V. A. "Statistical analysis of linear dynamic systems driven by the binary Markovian noise on the basis of the kinetic equations for non-Markovian stochastic processes." Signal Processing 76, no. 2 (July 1999): 167–80. http://dx.doi.org/10.1016/s0165-1684(99)00006-7.

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Pei, Bin, and Yong Xu. "Mild solutions of local non-Lipschitz neutral stochastic functional evolution equations driven by jumps modulated by Markovian switching." Stochastic Analysis and Applications 35, no. 3 (December 16, 2016): 391–408. http://dx.doi.org/10.1080/07362994.2016.1257945.

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Jung, Gerhard. "Non-Markovian systems out of equilibrium: exact results for two routes of coarse graining." Journal of Physics: Condensed Matter 34, no. 20 (March 10, 2022): 204004. http://dx.doi.org/10.1088/1361-648x/ac56a7.

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Abstract Generalized Langevin equations (GLEs) can be systematically derived via dimensional reduction from high-dimensional microscopic systems. For linear models the derivation can either be based on projection operator techniques such as the Mori–Zwanzig (MZ) formalism or by ‘integrating out’ the bath degrees of freedom. Based on exact analytical results we show that both routes can lead to fundamentally different GLEs and that the origin of these differences is based inherently on the non-equilibrium nature of the microscopic stochastic model. The most important conceptional difference between the two routes is that the MZ result intrinsically fulfills the generalized second fluctuation–dissipation theorem while the integration result can lead to its violation. We supplement our theoretical findings with numerical and simulation results for two popular non-equilibrium systems: time-delayed feedback control and the active Ornstein–Uhlenbeck process.

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Vacchini, Bassano. "General structure of quantum collisional models." International Journal of Quantum Information 12, no. 02 (March 2014): 1461011. http://dx.doi.org/10.1142/s0219749914610115.

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We point to the connection between a recently introduced class of non-Markovian master equations and the general structure of quantum collisional models. The basic construction relies on three basic ingredients: a collection of time dependent completely positive maps, a completely positive trace preserving transformation and a waiting time distribution characterizing a renewal process. The relationship between this construction and a Lindblad dynamics is clarified by expressing the solution of a Lindblad master equation in terms of demixtures over different stochastic trajectories for the statistical operator weighted by suitable probabilities on the trajectory space.

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Morozov, A. N., and A. V. Skripkin. "Description of evaporation of a spherical liquid drop by a non-Markovian random process based on integral stochastic equations." Russian Physics Journal 53, no. 11 (April 2011): 1167–78. http://dx.doi.org/10.1007/s11182-011-9546-y.

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Moix, Jeremy M., and Jianshu Cao. "A hybrid stochastic hierarchy equations of motion approach to treat the low temperature dynamics of non-Markovian open quantum systems." Journal of Chemical Physics 139, no. 13 (October 7, 2013): 134106. http://dx.doi.org/10.1063/1.4822043.

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Venturi, D., T. P. Sapsis, H. Cho, and G. E. Karniadakis. "A computable evolution equation for the joint response-excitation probability density function of stochastic dynamical systems." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 468, no. 2139 (November 16, 2011): 759–83. http://dx.doi.org/10.1098/rspa.2011.0186.

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By using functional integral methods, we determine a computable evolution equation for the joint response-excitation probability density function of a stochastic dynamical system driven by coloured noise. This equation can be represented in terms of a superimposition of differential constraints, i.e. partial differential equations involving unusual limit partial derivatives, the first one of which was originally proposed by Sapsis & Athanassoulis. A connection with the classical response approach is established in the general case of random noise with arbitrary correlation time, yielding a fully consistent new theory for non-Markovian systems. We also address the question of computability of the joint response-excitation probability density function as a solution to a boundary value problem involving only one differential constraint. By means of a simple analytical example, it is shown that, in general, such a problem is undetermined, in the sense that it admits an infinite number of solutions. This issue can be overcome by completing the system with additional relations yielding a closure problem, which is similar to the one arising in the standard response theory. Numerical verification of the equations for the joint response-excitation density is obtained for a tumour cell growth model under immune response.

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Rao, Ruofeng, Xiongrui Wang, Shouming Zhong, and Zhilin Pu. "LMI Approach to Exponential Stability and Almost Sure Exponential Stability for Stochastic Fuzzy Markovian-Jumping Cohen-Grossberg Neural Networks with Nonlinearp-Laplace Diffusion." Journal of Applied Mathematics 2013 (2013): 1–21. http://dx.doi.org/10.1155/2013/396903.

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The robust exponential stability of delayed fuzzy Markovian-jumping Cohen-Grossberg neural networks (CGNNs) with nonlinearp-Laplace diffusion is studied. Fuzzy mathematical model brings a great difficulty in setting up LMI criteria for the stability, and stochastic functional differential equations model with nonlinear diffusion makes it harder. To study the stability of fuzzy CGNNs with diffusion, we have to construct a Lyapunov-Krasovskii functional in non-matrix form. But stochastic mathematical formulae are always described in matrix forms. By way of some variational methods inW1,p(Ω),Itôformula, Dynkin formula, the semi-martingale convergence theorem, Schur Complement Theorem, and LMI technique, the LMI-based criteria on the robust exponential stability and almost sure exponential robust stability are finally obtained, the feasibility of which can efficiently be computed and confirmed by computer MatLab LMI toolbox. It is worth mentioning that even corollaries of the main results of this paper improve some recent related existing results. Moreover, some numerical examples are presented to illustrate the effectiveness and less conservatism of the proposed method due to the significant improvement in the allowable upper bounds of time delays.

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