Inflation in multidimensional quantum cosmology

We extend to multidimensional cosmology Vilenkin's prescription of tunneling from nothing for the quantum origin of the observable Universe. Our model consists of a (D+4)-dimensional spacetime of topology RXL(3)XL(D), with a scalar field (''chaotic inflaton'') for the matter component. Einstein gravity and Casimir compactification are assumed. The resulting minisuperspace is three-dimensional. Patchwise we find an approximate analytic solution of the Wheeler-DeWitt equation through which we discuss the tunneling picture and the probability of nucleation of the classical Universe with compactifying extra dimensions. Our conclusion is that the most likely initial conditions, although they do not lead to the compactification of the internal space, still yield (power-law) inflation for the outer space. The scenario is physically acceptable because the inner space growth is limited to similar to 10(11) in 100 e-foldings of inflation, starting from the Planck scale.