Dynamical Theories of Brownian Motion (Mathematical Notes by Edward Nelson

By Edward Nelson

These notes are according to a process lectures given by way of Professor Nelson at Princeton in the course of the spring time period of 1966. the topic of Brownian movement has lengthy been of curiosity in mathematical likelihood. In those lectures, Professor Nelson strains the historical past of prior paintings in Brownian movement, either the mathematical thought, and the traditional phenomenon with its actual interpretations. He maintains via contemporary dynamical theories of Brownian movement, and concludes with a dialogue of the relevance of those theories to quantum box concept and quantum statistical mechanics.

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T is stationary. VI - Proof: first we get formula (13) by noticing that the law of A ® A under (u, v) ~ Csu + C~v is worth A according to [9],[10]. Then following [10] we are back to show that for any S > 0 and T > 0, M ",t = r(e-(S-s)A e-(T-t)B)F(Hs,t ) is a martingale on [0, S] x [0, T] with respect to the 3's,t. It suffices to do so when F = k f = e(f-lfI 2 / Z ) with fEE'. 4)), which is linear in w so that its bracket is worth INs,tI Z by [9] and proposition 9. DENIS FEYEL AND ARNAUD DE LA PRADELLE 354 IV.

According to the Hahn-Banach theorem and the Lindelof property of E (cf. [4], n067), g is the upper enveloppe of a sequence of linear continuous forms J; E E'. Let v = JJe- sA - tB p(ds,dt) = «1 + A)(I + B»-1/2 The operator V-I is linearly extendable as a 'x-measurable operator of E according to [8]. Let us put hi = fi 0 V-I, one get a sequence of elements hi E H', whose the upper enveloppe is denoted q. One has gdO < +00 so that q is 'x-measurable and satisfies J qd'x = J gdB < +00. One has for every i J 2hi(CsX) = hi(C"x + C~y) + hi(Csx - C~y) for every (x,y) E E x E, where C's = J1- C; ('x-mesurable linear extension).

Let (~, A) a couple of centered gaussian measures on E, then there exists a unique centered gaussian measure 8 on C(ll4, E) such that the Wt coordinate process is brownian motion starting from ~ with law A. It means that for any couple (f, g) of continuous linear forms on E. In this formula At is the convolution power. This comes from [9],[10]. In particular, Wt (8) = ~ * At. We recall that 8 is carried by a Lusin sub-space (of K/7-type) The Cameron-Martin subspace satisfying 1l(~, A) = tth n c C(ll4,E).

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