ERGODIC PROPERTIES OF CONTINUOUS PARAMETER ADDITIVE PROCESSES

Ergodic properties of Markov processes

In these notes we discuss Markov processes, in particular stochastic differential equations (SDE) and develop some tools to analyze their long-time behavior. There are several ways to analyze such properties, and our point of view will be to use systematically Liapunov functions which allow a nice characterization of the ergodic properties. In this we follow, at least in spirit, the excellent b...

Ergodic Properties of Markov Processes

Markov processes describe the time-evolution of random systems that do not have any memory. Let us demonstrate what we mean by this with the following example. Consider a switch that has two states: on and off. At the beginning of the experiment, the switch is on. Every minute after that, we throw a dice. If the dice shows 6, we flip the switch, otherwise we leave it as it is. The state of the ...

Ergodic properties of Markov processes

Warning. If F and G are σ-algebras, H = F ∪G is not necessarily a σ-algebra unless of course G ⊂ F or vice versa. Indeed, even if {Fn} is a countable collection of σ-algebras satisfying Fn ⊂ Fn+1 for each n = 1, 2, . . ., the union ⋃∞ n=1Fn is not necessarily a σ-algebra. For example, let Ω = N and let Fn be the smallest σ-algebra containing {1}, {2}, . . . , {n}. (This is the power set of {1, ...

Ergodic Properties of Max–Infinitely Divisible Processes

We prove that a stationary max–infinitely divisible process is mixing (ergodic) iff its dependence function converges to 0 (is Cesaro summable to 0). These criteria are applied to some classes of max–infinitely divisible processes.

Probability, Random Processes, and Ergodic Properties