• Young measures
• statistical equilibrium
• random dynamical systems

In climate studies we are faced with variuos input of different time scales, those inputs can be external or internal

Those are the reason why we focus on a subset of systems, which they generate a Non autonomous dynamical systems, as they can be seen as a system in which
we have two exteranl inputs, one slow-variyng and another fast-varying.

This can be seen as the dichotomy between Climate and Weather,indeed by weather we mean the set of weather conditions on a time scale of a few hours,
while climate is the set of the same conditions on a much larger time scale.
The goal of this thesis is to provide a mathematical background in which to
give a definition of Climate and Weather supported by properties enjoyed by the above systems.

Since the systems currently governing weather forecasting (general circulation model, GCM for short) are very complicated,
we will study a non-autonomous SDE with some initial datum a generic $t_0$ time, in fact the study
will be primarily focus on the product in the pullback sense (as we observe a process that goes on for centuries)
of various objects, namely sets that attract dynamics or their statistical equivalent at the level of measures that are produced by the
system asymptotically.

In Chapter 1, we review the theory of attractors for nonautonomous dynamical system, this theory is very well know in literature,
we are intrested in showing results regarding the existence and uniqueness of a global attractor for the system, i.e., which definitely attracts the orbits of the system.

In Chapter 2, we address the study of a concept of special interest produced by dynamics, namely statistical equilibrium.
Given a family of measure that are invariant for markov semigroup associated to~\eqref{eq:1} the \textit{statistical equilibrium} (if exists) is a measure given by the weak limit of
of their pullback via the propagator.
We will dwell on what conditions are necessary or sufficient, in order
for such a measure to be obtained as a weak limit of any measure with relevant physical significance, for example as a pullback of
Lebesgue measure.\\

Finally, Chapter 3 is devoted to the study of the relationship enjoyed by climate and weather, in an attempt to give meaning to the fact that climate, in its recognized definition,
is nothing more than the time average of weather over a period of time, like 30 years.

We will show how it is possible through the introduction of Young's measures, and a scaling parameter to
relate the theory of attractors to that of invariant measures, to give provide a rigorous theorem on the convergence
of temporal averages.