Stochastic Analysis on Groups of Diffeomorphisms
Everywhere in this chapter we use the notions and notation introduced in Chapter 5. We describe a stochastic differential equation of a special sort on those groups of diffeomorphisms that arise in the applications to viscous hydrodynamics described in Section 16.4 below (see, e.g., Gliklikh (1989, 1994, 2003)). This class of equations is characterized by the fact that they involve finite-dimensional Wiener processes. It should be pointed out that a theory of stochastic differential equations on infinite-dimensional manifolds involving infinite-dimensional Wiener processes exists (see, e.g., Belopolskaya and Dalecky (1989), Brzezniak and Elworthy (2000), Elworthy (1982)) and is used in viscous hydrodynamics (see, e.g., Cipriano and Cruzeiro (2007)). However, the description of this theory requires a complicated functional-analytic machinery that is not included in our exposition. For simplicity of presentation, we restrict ourselves to the finite-dimensional version of the theory since, in applications, the theories yield very similar results.
KeywordsStochastic Differential Equation Wiener Process Stochastic Analysis Local Connector Quadratic Operator
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