By Nigel J. Cutland

This increased model of the 1997 eu Mathematical Society Lectures given by way of the writer in Helsinki, starts with a self-contained advent to nonstandard research (NSA) and the development of Loeb Measures, that are wealthy measures chanced on in 1975 by way of Peter Loeb, utilizing suggestions from NSA. next chapters caricature quite a number fresh functions of Loeb measures as a result of writer and his collaborators, in such diversified fields as (stochastic) fluid mechanics, stochastic calculus of adaptations ("Malliavin" calculus) and the mathematical finance idea. The exposition is designed for a normal viewers, and no prior wisdom of both NSA or many of the fields of purposes is assumed.

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Note that to describe each measure µt it is suﬃcient to characterize the integrals θ(u)dµt (u). 19) H for a suﬃciently broad class of test functions θ. 18)) H = (θ (u), F (u))H dµt (u) H (by deﬁnition of µt again). 18: t θ(u)dµt (u) − H θ(u)dµ0 (u) = (θ (u), F (t, u))H dµs (u)ds. 18). 18) have the uniqueness property – otherwise the function S does not exist. However, S does not occur in the Foias equation, so as an abstract equation for the time evolution of a family of measures it makes sense even when the underlying equation does not have a unique solution.

A measure attractor for statistical solutions will be a subset of the set M1 (H) of Borel probabilities on H, viewed as a subset of the space of Borel measures M(H) on H (which we equip with the topology of weak convergence). The paper [15], on which the present section is based, establishes results on existence of measure attractors for Navier–Stokes equations that are more general than those obtained by Schmalfuß. 7 Measure Attractors for Stochastic Navier–Stokes Equations 51 Here are brief details.

18) have the uniqueness property – otherwise the function S does not exist. However, S does not occur in the Foias equation, so as an abstract equation for the time evolution of a family of measures it makes sense even when the underlying equation does not have a unique solution. This is the crucial point that was observed by Foias. Thus, in the case of the Navier–Stokes equations in 3-dimensions, since it is not known whether there is a unique solution, the above derivation does not make sense; however, the end result – the corresponding Foias equation – does makes sense and it is possible ﬁnd statistical solutions.