  We consider the Korteweg--de Vries equation with white noise initial data,
posed on the whole real line, and prove the almost sure existence of solutions.
Moreover, we show that the solutions obey the group property and follow a white
noise law at all times, past or future.
  As an offshoot of our methods, we also obtain a new proof of the existence of
solutions and the invariance of white noise measure in the torus setting.
