Stochastic differential equations random forces

The stochastic differential equation and the second moment of the random force are insufficient to determine which calculus is to be preferred. The two calculus correspond to different physical models [11,12]. It is beyond the scope of the present article to describe the difference in details. We only note that the Ito calculus consider r t) to be a function of the edge of the interval while the Stratonovich calculus takes an average value. Hence, in the Ito calculus using a discrete representation rf t) becomes r] tn) i]n — y n — A i) -I- j At. Developing the determinant of the Jacobian -... 

A further complication that has been much studied in the literature is that of multiplicative noise in which the random force in stochastic differential equations like Eq. (1) is modified by a modulating term, i.e.,... 

The random force is not a prescribed function of the time, but a random function of the time. The Langevin equation is consequently a stochastic differential equation. In order to solve such an equation, the statistical properties of the random force must be specified. Of course, the Langevin equation can be solved formally to give... 

In the foregoing demonstration, we had limited ourselves to include only the kinematic aspects of bubble motion. A dynamic model including force balances on bubble motion would have called for adding the bubble velocity also as a particle state variable. Such a model could also have been considered allowing for bubble velocity to be a random process satisfying a stochastic differential equation of the type (2.11.14). The basic objective of this example has been to demonstrate applications in which particle state can be a random process. The next and the last example in this chapter considers a similar application, but with a distinction that can help address an entirely different class of problems. 

Consider relative motion by Brownian motion as well as by gravitational settling. We neglect interparticle force and correlated random motion, although their inclusion in the manner in which it was done in Section 3.3.5.3 is straightforward. In fact, the issue of interest in Section 3.3.5.3 does not differ from that in this section. The relative motion of particles of volume x as viewed from a particle of volume x is described by the stochastic differential equation... 

Equations [18] and [20] are stochastic differential equations because the force iR is taken from a random distribution. The stochastic nature means that one must produce many independent trajectories that are averaged together, producing the time evolution of an ensemble-averaged property. 

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... 

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