Random Dynamical Systems pp 49-107 | Cite as

# Generation

## Summary

In this foundational chapter we will associate (infinitesimal) generators with all reasonably regular RDS.

The task is easy for discrete time where all RDS have obvious generators, their time-one maps (Sect. 2.1). Products of i.i.d. random mappings and their associated Markov chains are studied in more detail in Subsect. 2.1.3.

There are two basically different answers:dynamical system = exp(vector field).

- (1)
In Sect. 2.2 we obtain the following one-to-one correspondence: Every w-wise random differential equation of the form \({\dot x_t} = f({\theta _t}\omega ,{x_t})\) generates an RDS which is absolutely continuous with respect to time t (see Theorem 2.2.1 for the local case and Theorem 2.2.2 for the global case). Conversely, every RDS

*φ*for which*t*↦*φ*(*t*,*ω*)*x*is absolutely continuous is a solution of such a random differential equation (Theorem 2.2.13). All we need for the proof is a new look at classical results for ordinary differential equations. - (2)
It seems surprising at first sight that there is a large and very important class of RDS for which

*t*↦*φ*(*t*,*ω*)*x*is, though continuous, not of bounded variation, and yet have generators. The key extra property is that*t*↦*φ*(*t*,*ω*)*x*is a semimartingale. The generators are stochastic differential equations (Sect. 2.3).

semimartingale cocycle = exp(semimartingale helix).

The proof relies crucially on our Perfection Theorem 1.3.2.

Criteria for when a classical (i.e. Brownian motion driven) stochastic differential equation generates a continuous or *C* ^{ k } RDS are given in Subsect. 2.3.6.

Subsect. 2.3.7 is devoted to the detailed treatment of a one-dimensional example.

In the final Subsect. 2.3.9 we sketch the route (historically and systematically) that lead from Markov processes via classical stochastic differential equations to random dynamical systems. As a climax, we characterize those invariant measures which are related to the solutions of the Fokker-Planck equation (Theorem 2.3.45).

Our hope is that this chapter can serve as a reference text for the generation problem.

## Keywords

Invariant Measure Local Characteristic Continuous Time Stochastic Differential Equation Random Mapping## Preview

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