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Alpha-cut-complete Boolean algebras by Hager A.W.

By Hager A.W.

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316, Serie I, p. 287-292 (1993) 30 [AM1] S. M. Ma, Additive functionals, nowhere Radon and Kato class smooth measure associated with Dirichlet forms, Osaka Math J. 29, 247-265 (1992) [AM2] S. M. Ma, Perturbation of Dirichlet forms - lower semiboundedness, closability and form cores, J. Funet. Anal. 99,332-356 (1991) [AMR1] S. M. Ma, M. Rockner, Regularization of Dirichlet spaces and applications, CRAS, Ser. I, Paris 314, 859-864 (1992) [AMR2] S. M. Ma, M. Rockner, Non symmetric Dirichlet forms and Markov processes on general state space, CRAS, Ser.

One then shows the regularization can be removed. e. lot ( C;2 ) (x)+c; u;(x) = Gt*uo(x)--1 . dsG~_s* (U;)2 - -rK(O) 2 0 2v lot G~_s*dBs (x) k 0 (24) Notice that the term r K(O) actually disappears due to the integration of G/. Since Ot( x) is not differentiable Ute x) can be defined as a distribution valued process Ut(f) = 2v J dx f'(x) logOt(x) (25) In the limit k ~ 00 the non linear term in (24) becomes then ill defined, the limiting equation may be meaningless. A reasonable expectation is that Ute x) can be interpreted as a solution of the limit equation in a weak probabilistic sense.

For the non conservative case we here outline how an existence theorem for the Cauchy problem for equation (1) can be obtained, a detailed proof being in [BCJ]. This theorem gives an explicit expression of a strong solution (in the sense of stochastic differential equation) as a continuous process (in time) with values in the class of continuous functions (in space). A uniqueness result has been recently proven when the Burgers equation is considered in a bounded domain with Dirichlet boundary conditions [DDT].

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