ABSTRACT
Smooth deterministic dynamical systems φ(t) and differential equations x ˙ = f ( x ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351075053/29beae47-70ce-4fb5-959b-f0d782e1229b/content/eq566.tif"/> (generating φ through their solution flows) are basically the same class of objects. Here we investigate the situation in the random case: When is a random dynamical system φ(t,ω) generated by some sort of random differential equation x ˙ = f ( x , t , ω ) https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351075053/29beae47-70ce-4fb5-959b-f0d782e1229b/content/eq567.tif"/> , and which random differential equation generates a random dynamical system through its solution?
The situation will turn out to be roughly as follows: If φ(t, ω)x is differentiable with respect to time t then there exists a pathwise generator that is a pathwise random differential equation, and conversely. However, even if φ(t, ω)x is not differentiable with respect to t (and not even absolutely continuous), but is a semimartingale, it is generated by a stochastic differential equation, driven by a semimartingale with stationary increments, and conversely.
This contribution collects in a systematic way the existence, uniqueness and regularity statements for the various situations.
