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We consider the Cauchy problem for a generalized KdV equation\begin{eqnarray*}  u_{t}+\partial_{x}^{3}u+u^{7}u_{x}=0,  \end{eqnarray*} with random data on \R. Kenig, Ponce, Vega(Comm. Pure Appl.Math.46(1993), 527-620)proved that the problem is globally well-posed inH^{s}(\R)$ with s> s_{crit}=\frac{3}{14}, which is the scaling criticalregularity indices. Birnir, Kenig, Ponce, Svanstedt, Vega(J. London Math. Soc.53 (1996), 551-559.) proved that the problem is ill-posed in the sense that thetime of existence T and the continuous dependence cannot be expressed in termsof the size of the data in the H^{\frac{3}{14}}-norm. In this present paper, weprove that almost sure local in time well-posedness holds in H^{s}(\R) withs>\frac{17}{112}, whose lower bound is below \frac{3}{14}.  The key ingredients are the Wiener randomization of the initial data andprobabilistic Strichartz estimates together with some important embeddingTheorems.

Random data Cauchy problem for a generalized KdV equation in the supercritical case