# Generation

• Ludwig Arnold
Part of the Springer Monographs in Mathematics book series (SMM)

## 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.

In the continuous time case we ask for the stochastic equivalent of the deterministic one-to-one correspondence between dynamical systems and vector fields, symbolically written as

dynamical system = exp(vector field).

There are two basically different answers:
1. (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 tφ(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. (2)

It seems surprising at first sight that there is a large and very important class of RDS for which tφ(t, ω)x is, though continuous, not of bounded variation, and yet have generators. The key extra property is that tφ(t, ω)x is a semimartingale. The generators are stochastic differential equations (Sect. 2.3).

As we are again aiming at a general one-to-one correspondence between those RDS and their generators we have to use Kunita’s [224] general concepts, extended to two-sided time. Our main results are: Every stochastic differential equation driven by a semimartingale helix (additive cocycle) generates a semimartingale RDS (Theorem 2.3.26 for the global case, Theorem 2.3.29 for the local case). Conversely, every semimartingale RDS is generated by a stochastic differential equation driven by a semimartingale helix (Theorem 2.3.30). This one-to-one relation can be succinctly written as

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

Filtration Manifold Covariance Assure Sine