Continuous Stochastic Dynamics Research Articles

Stability analysis of stochastic differential systems with state‐dependent delay subject to average‐delay impulses

In this paper, we study the stochastic weak local exponential stability of stochastic differential systems with state‐dependent delay (SDD) subject to average‐delay impulses by using Lyapunov–Krasovskii functional and stochastic analytical skill. Unlike time‐dependent delay discussed in the previous literature, we introduce SDD in impulsive stochastic differential systems, where SDD is a stochastic variable associated with system state, consequently leading to an indeterminate delay boundary. Our findings reveal that when destabilizing delayed impulses satisfying specific conditions are generated, the stability of stochastic differential systems with destabilizing delayed impulses and stable continuous stochastic dynamics can still be maintained. Additionally, if stable delayed impulses satisfy certain conditions, stability of the systems under consideration can also be achieved, irrespective of the stable status on the continuous stochastic dynamics. Finally, two examples are given to demonstrate the validity of theoretical results.

Practical exponential stability of impulsive stochastic functional differential systems with distributed-delay dependent impulses

This paper develops new practical stability criteria for impulsive stochastic functional differential systems with distributed-delay dependent impulses by using the Lyapunov–Razumikhin approach and some inequality techniques. In the given systems, the state variables on the impulses are concerned with a history time period, which is very appropriate for modelling some practical problems. Moreover, different from the existing practical stabilization results for the systems with unstable continuous stochastic dynamics and stabilizing impulsive effects, we take the systems with stable continuous stochastic dynamics and destabilizing impulsive effects into account. It shows that under the impulsive perturbations, the practical exponential stability of the stochastic functional differential systems can remain unchanged when the destabilizing distributed-delay dependent impulses satisfy some conditions on the frequency and amplitude of the impulses. In other words, it reveals that how to control the impulsive perturbations such that the corresponding stochastic functional differential systems still maintain practically exponentially stable. Finally, an example with its numerical simulation is offered to demonstrate the efficiency of the theoretical findings.

Practical exponential stability of hybrid impulsive stochastic functional differential systems with delayed impulses

SummaryThis paper is concerned with the problem of practical exponential stability for hybrid impulsive stochastic functional differential systems with delayed impulses, which comprise three classes of systems: the systems with unstable continuous stochastic dynamics and stable discrete dynamics, the systems with stable continuous stochastic dynamics and unstable discrete dynamics, and the systems where both the continuous stochastic dynamics and the discrete dynamics are stable. By using the Lyapunov‐Razumikhin approach, several new sufficient conditions for the practical exponential stability are established for each class of systems. It shows that the stabilizing and destabilizing delayed impulses that satisfy some conditions on their frequency and amplitude can stabilize the systems with unstable continuous stochastic dynamics in the practical exponential stability sense and ensure the practical exponential stability of the systems with stable continuous stochastic dynamics, respectively. Conversely, if the continuous stochastic dynamics are practically exponentially stable and the delayed impulses are stabilizing, then the systems can be practically exponentially stable regardless of the restrictions on the delayed impulses frequency and amplitude. Finally, two numerical examples are presented to illustrate the efficiency of the results.

Stochastic entropy production for continuous measurements of an open quantum system

We investigate the total stochastic entropy production of a two-level bosonic open quantum system under protocols of time dependent coupling to a harmonic environment. These processes are intended to represent the measurement of a system observable, and consequent selection of an eigenstate, whilst the system is also subjected to thermalising environmental noise. The entropy production depends on the evolution of the system variables and their probability density function, and is expressed through system and environmental contributions. The continuous stochastic dynamics of the open system is based on the Markovian approximation to the exact, noise-averaged stochastic Liouville-von Neumann equation, unravelled through the addition of stochastic environmental disturbance mimicking a measuring device. Under the thermalising influence of time independent coupling to the environment, the mean rate of entropy production vanishes asymptotically, indicating equilibrium. In contrast, a positive mean production of entropy as the system responds to time dependent coupling characterises the irreversibility of quantum measurement, and a comparison of its production for two coupling protocols, representing connection to and disconnection from the external measuring device, satisfies a detailed fluctuation theorem.

Housekeeping entropy in continuous stochastic dynamics with odd-parity variables

We investigate the decomposition of the total entropy production in continuous stochastic dynamics when there are odd-parity variables that change their signs under time reversal. The first component of the entropy production, which satisfies the fluctuation theorem, is associated with the usual excess heat that appears during transitions between stationary states. The remaining housekeeping part of the entropy production can be further split into two parts. We show that this decomposition can be achieved in infinitely many ways characterized by a single parameter σ. For an arbitrary value of σ, one of the two parts contributing to the housekeeping entropy production satisfies the fluctuation theorem. We show that for a range of σ values this part can be associated with the breakage of the detailed balance in the steady state, and can be regarded as a continuous version of the corresponding entropy production that has been obtained previously for discrete state variables. The other part of the housekeeping entropy does not satisfy the fluctuation theorem and is related to the parity asymmetry of the stationary state distribution. We discuss our results in connection with the difference between continuous and discrete variable cases especially in the conditions for the detailed balance and the parity symmetry of the stationary state distribution.

Fisher information metric for the Langevin equation and least informative models of continuous stochastic dynamics

The evaluation of the Fisher information matrix for the probability density of trajectories generated by the over-damped Langevin dynamics at equilibrium is presented. The framework we developed is general and applicable to any arbitrary potential of mean force where the parameter set is now the full space dependent function. Leveraging an innovative Hermitian form of the corresponding Fokker-Planck equation allows for an eigenbasis decomposition of the time propagation probability density. This formulation motivates the use of the square root of the equilibrium probability density as the basis for evaluating the Fisher information of trajectories with the essential advantage that the Fisher information matrix in the specified parameter space is constant. This outcome greatly eases the calculation of information content in the parameter space via a line integral. In the continuum limit, a simple analytical form can be derived to explicitly reveal the physical origin of the information content in equilibrium trajectories. This methodology also allows deduction of least informative dynamics models from known or available observables that are either dynamical or static in nature. The minimum information optimization of dynamics is performed for a set of different constraints to illustrate the generality of the proposed methodology.

Entropy production in full phase space for continuous stochastic dynamics

Total entropy production and its three constituent components are described both as fluctuating trajectory-dependent quantities and as averaged contributions in the context of the continuous Markovian dynamics, described by stochastic differential equations with multiplicative noise, of systems with both odd and even coordinates with respect to time reversal, such as dynamics in full phase space. Two of these constituent quantities obey integral fluctuation theorems and are thus rigorously positive in the mean due to Jensen's inequality. The third, however, is not and furthermore cannot be uniquely associated with irreversibility arising from relaxation, nor with the breakage of detailed balance brought about by nonequilibrium constraints. The properties of the various contributions to total entropy production are explored through the consideration of two examples: steady-state heat conduction due to a temperature gradient, and transitions between stationary states of drift diffusion on a ring, both in the context of the full phase space dynamics of a single Brownian particle.

Computational Methods for Verification of Stochastic Hybrid Systems

Stochastic hybrid system (SHS) models can be used to analyze and design complex embedded systems that operate in the presence of uncertainty and variability. Verification of reachability properties for such systems is a critical problem. Developing sound computational methods for verification is challenging because of the interaction between the discrete and the continuous stochastic dynamics. In this paper, we propose a probabilistic method for verification of SHSs based on discrete approximations focusing on reachability and safety problems. We show that reachability and safety can be characterized as a viscosity solution of a system of coupled Hamilton-Jacobi-Bellman equations. We present a numerical algorithm for computing the solution based on discrete approximations that are derived using finite-difference methods. An advantage of the method is that the solution converges to the one for the original system as the discretization becomes finer. We also prove that the algorithm is polynomial in the number of states of the discrete approximation. Finally, we illustrate the approach with two benchmarks: a navigation and a room heater example, which have been proposed for hybrid system verification.