**Julian Newman**

**Department of Mathematics, Imperial College London**

**Measurable stochastic dynamics and convergence to "statistical equilibrium"**

We consider the simultaneous evolution of all initial conditions in a given phase space, under the consecutive application of a sequence of i.i.d. randomly selected measurable self-maps of the phase space. One way to analyze this statistically is to fix an "initial distribution" (i.e. a probability measure on the phase space), and look at how this distribution evolves under the sequence of random maps. This gives us a measure-valued Markov process, and it turns out that in a "nice" class of cases this Markov process has a well-defined "limiting distribution" called a "statistical equilibrium". In the case that this statistical equilibrium is supported on the set of Dirac masses on the phase space (which turns out to occur remarkably often), we have that "the trajectories of almost all initial conditions move towards each other"; but what is remarkable is that we can say this without first defining a distance function on the phase space!

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