Stochastic differential equations of jump type and Lévy processes in diffeomorphisms group

Abstract

Consider a stochastic differential equation with respect to Brownian motions.It is a know n fact that the solution defines a stochastic flow of homeomorphisms, p ro v id e d th a t coefficients of the equation are Lipschitz c o n tin u o u s. S e e e. g.B is m u t [2 ], K u n ita [6 ].H e n c e the solution defines a B row nian motion in homeomorphisms group G .Recently, Brownian motion in homeomorphisms group itself is studied a lot.S e e H arris [4].L e Jan [9], B axendale [1], K unita [7] and Le Jan-W atanabe [ 1 0 ] .T hey characterize it by th e infinitesimal mean and covariance, called local characteristics.T he purpose of this paper is to study the related problem for jump processes.T he stochastic differential (integral) equations considered in th is p a p e r is b a se d o n th e L é v y p ro c e ss Xt -= X ,(x ), x E R d (process with independent increments, continuous in probability), which takes values in the vector space o f continuous m a p s (o r c o n tin u o u s v e c to r fields).W e c a l l i t a C-valued Lévy process.In Section 1, w e study system atically the C-valued Lévy p ro c e ss.W e introduce the sy ste m (a, b , y , U ) which characterizes th e law of C-valued process and discuss the existence problem of C-valued Lévy process associated with a given characte r is tic s .The Lévy process w ith values in the vector space of Cui-maps is also considered.

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