By M. Göckeler, T. Schücker
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Additional resources for Differential Geometry, Gauge Theories, and Gravity
If d~ t function is i n d e p e n d e n t of b(x,t) dw t is independent of Ws,S < t , but we cannot expect this p r o p e r t y Indeed, ~t ' then in the situation discusses above, ~ (d~tl~t) = ~ (d~t) , and the w o u l d have to be spacially constant. We can, however, 35 assume that where the particle moves to between pends only of the position of the particle of the rest of the history of the particle. and the future of the particle present, its past and future property. 27) order still equal to a model in the past but given the otherwise.
We are qoin~ of the previous ~ = ~. 6): Let ~ be a measure on the filtration under ~ , ~t(~), t £ [O,T] measure dx negative function such that space (~,~t,~) such that is a Wiener process with Lebesgue's as initial distribution, and let O(x,O) S p(x,O)dx = I [S~ b+(~s,S)d~s be a non- If I t - ~ ~o I b+(~s,S)l 2 ds] Qt = p (x,O) e is a martingale, dP = QTd~ then under the probability P defined by the process t Wt = ~t - I b+(~s'S)ds o is also a Wiener process with the same diffusion coefficient ~t and with initial distribution Now, let us suppose that p(x,O)dx ~ = C(O,T) as .
P(Xn_ 1 , t n _ ] , X n , t n) P ( X o , t o ) d X o for ..... dx n t o < t I < t 2 ... 73) p and the density and p . ~3). In of the t r a n s i t i o n p r o b a b i l i t y p does not w h i c h in fact depends on some specified ini- tial condition. If we are i n t e r e s t e d in the b e h a v i o u r of the r a n d o m trajectories t ~ Wt of the B r o w n i a n particle, we need to k n o w m o r e than the transi- tion function (and c o r r e l a t i o n functions). We have to prove the exis- tence of a p r o b a b i l i t y m e a s u r e on the space of trajectories c o n s i s t e n t w i t h the B r o w n i a n t r a n s i t i o n function in the following sense: finite c o l l e c t i o n of m e a s u r a b l e 0 < t I < t 2 ...
Differential Geometry, Gauge Theories, and Gravity by M. Göckeler, T. Schücker