Stochastic difference equation in time

A. Stochastic Difference in Time Definition A Stochastic Model for a Trajectory Weights of Trajectories and Sampling Procedures Mean Field Approach, Fast Equilibration, and Molecular Labeling Stochastic Difference Equation in Length Fractal Refinement of Trajectories Parameterized by Length... 

The stochastic difference equation in length is conceptually similar to the stochastic difference in time. We therefore do not repeat all of the arguments and... 

Elber, R., Ghosh, A., Cardenas, A. Stochastic difference equation as a tool to compute long time dynamics, in Nielaba, R, Mareschal, M., Ciccotti, G., editors. Bridging the Time Scale Gap. Berlin Springer Verlag 2002. 

We begin with a simple example of a particle performing a discrete-time random walk (DTRW) in one dimension. Assume that it is initially at point 0. The random walk can be defined by the stochastic difference equation for the particle position... 

It may be useful to point out a few topics that go beyond a first course in control. With certain processes, we cannot take data continuously, but rather in certain selected slow intervals (c.f. titration in freshmen chemistry). These are called sampled-data systems. With computers, the analysis evolves into a new area of its own—discrete-time or digital control systems. Here, differential equations and Laplace transform do not work anymore. The mathematical techniques to handle discrete-time systems are difference equations and z-transform. Furthermore, there are multivariable and state space control, which we will encounter a brief introduction. Beyond the introductory level are optimal control, nonlinear control, adaptive control, stochastic control, and fuzzy logic control. Do not lose the perspective that control is an immense field. Classical control appears insignificant, but we have to start some where and onward we crawl. 

The different theoretical models for analyzing particle deposition kinetics from suspensions can be classified as either deterministic or stochastic. The deterministic methods are based on the formulation and solution of the equations arising from the application of Newton s second law to a particle whose trajectory is followed in time, until it makes contact with the collector or leaves the system. In the stochastic methods, forces are freed of their classic duty of determining directly the motion of particles and instead the probability of finding a particle in a certain place at a certain time is determined. A more detailed classification scheme can be found in an overview article [72]. 

Signal Modification. The decomposition approach has been applied successfully to speech and music modification [Serra, 1989] where modification is performed differently on the two deterministic/stochastic components. Consider, for example, time-scale modification. With the deterministic component, the modification is performed as with the baseline system using Equation (9.27), sustained (i.e., steady ) sine waves are compressed or stretched. For the aharmonic component, the white noise input lingers over longer or shorter time intervals and is matched to impulse responses (per frame) that vary slower or faster in time. 

Stochastic Models for the Disturbances The type of stochastic process disturbances N-t occurring in practice can usually be modelled quite conveniently by statistical time series models (Box and Jenkins (k)). These models are once again simple linear difference equation models in which the input is a sequence of uncorrelated random Normal deviates (a. ) (a white noise sequence)... 

It is important to emphasize, however, that our model is different from the Langevin equation, which is a stochastic differential equation. Our model has no noise in the limit of small time steps in which the numerical errors approach zero. The noise we introduce is numerical. Once we filter the rapid oscillations, it is impossible for us to recover the tme trajectory using only the low-frequency modes. The noise in the SDE approach is introduced when we approximate a differential equation by a finite difference formula and filter out high-frequency motions. 

Equation (8.54) is a stochastic equation of motion similarto Eq. (8.13). However, we see an important difference Eq. (8.54) is an integro-differential equation in which the term yx of Eq. (8.13) is replaced by the integral /J drZ t — r)x(r). At the same time the relationship between the random force R t) and the damping, Eq. (8.20), is now replaced by (8.59). Equation (8.54) is in fact the non-Markovian generalization of Eq. (8.13), where the effect of the thermal environment on the system is not instantaneous but characterized by a memory—at time t it depends on the past interactions between them. These past interactions are important during a memory time, given by the lifetime of the memory kernel Z t). The Markovian limit is obtained when this kernel is instantaneous... 

We have already noted the difference between the Langevin description of stochastic processes in terms of the stochastic variables, and the master or Fokker-Planck equations that focus on their probabilities. Still, these descriptions are equivalent to each other when applied to the same process and variables. It should be possible to extract information on the dynamics of stochastic variables from the time evolution of their probabihty distribution, for example, the Fokker-Planck equation. Here we show that this is indeed so by addressing the passage time distribution associated with a given stochastic process. In particular we will see (problem 14.3) that the first moment of this distribution, the mean first passage time, is very useful for calculating rates. 

We now know that there are processes, which are not stochastic, whose output mimics stochastic behavior. This phenomenon is now called chaos. Chaos is a jargon word that means that a system has certain mathematical properties. It should not be confused with its nontechnical homonym that means confusion or disorder. A chaotic system can be described by a set of nonlinear difference or differential equations that have a small number of independent variables. Because these equations can be integrated in time, the future values of the variables are completely determined by their past values. 

---