It is shown that the initial value problem (IVP) of the generalized KdV equation @tu + @x(a(u)) + @ 3 xu = 0; u(x; 0) = (x) is well posed in the classical Sobolev space H(R) with s > 3=4, which thus establishes a nonlinear map K from H(R) to C([ T; T ];H(R)). Then it is proved that (i) if a = a(x) is a C function on R to R, then K is in nitely many times Frechet di erentiable; (ii) if a = a(x) ...

probability 1. (Note that we do not assume any relation of dependence or indepenaence between X(r) and X(s) for m $ n.) 1 See M. Frechet, Recherches theoriques modernes, Vol. 1, Paris, 1937, p. 27. 2 See H. Cram&r, Mathematical methods of statistics, Princeton, 1946, p. 93. Compare F. P. Cantelli, Considerazioni sulla legge uniforme dei grandi numeri ecc., Giornale dell 'Istituto Italiano degli...