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By Desloge E.A.

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But then the boundaries can be closed. To derive the tangent approximation we put the following conditions on the mixing measure: F has a density fIel with the properties (1) a) there exists a monotone increasing function certain positive constants K and Vo h:~++[',oo) and 46 J (1+Kv)'h(Kv)cjl(v)dv < 00 Vo such that b) The ratio f(6(1tE))/f(6) (lI) for all f(6(1+a))~f(6)h(a) 1 uniformly for all 6 as E + 2-8 ) < sup F(O,y)/(f(y)e Y O

2) na(t) is defined as the solution x=na(t) of the implicit equation f(x,t)=a. 10) together yield that the boundaries for t=l satisfy the equations s (2. 14) x ; , t s a Brownian Bridge with endpoints (8,0) and (yo,so) , is mapped into a Brownian motion with drift e starting at (Yo ,~) and running up to So So infinity. 14) and vice versal. The reader is invited to check some of the examples of both sections on their correspondence. These observations raise the further leading question: how are the methods themselves related?

7) below), that the first exit density at the tangent is bigger than that at the curve at time t. In this section we study the asymptotic behaviour of Palt) when a + co for boundaries ~a with strong curvature. 3) Aa (t) 1/1 (t) Palt) = ~ ~(~ ) (1+0(1» holds uniformlyon intervals (O,t 1 ] when a+oo • This statement is studied from different perspectives, at first from an analytic and further below from a probabilistic point of view. 3) has the advantage that it is easy to calculate. Its quality of approximation for finite situations is discussed in the next section.

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Classical Mechanics by Desloge E.A.

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