Potential Theory and Degenerate Partial Differential by Marco Biroli

By Marco Biroli

Contemporary years have witnessed an more and more shut courting growing to be among capability idea, likelihood and degenerate partial differential operators. the idea of Dirichlet (Markovian) kinds on an summary finite or infinite-dimensional area is usual to all 3 disciplines. it is a interesting and demanding topic, significant to a few of the contributions to the convention on `Potential conception and Degenerate Partial Differential Operators', held in Parma, Italy, February 1994.

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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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