Long time behaviour for a diffusion process associated with a porous medium equation (Q1894031)

For given \(\alpha > 1\), let \(\{X(t)\}\) be a diffusion process such that the density \(u\) of the distribution \(P(X(t) \in dx)\) is a solution of a porous medium equation \(u_t = \begin{smallmatrix} {1 \over 2} \end{smallmatrix} \Delta u^\alpha\) and the process \(\{X(t)\}\) is a solution of the following S.D.E. \[ dX(t) = u \bigl( t,X(t) \bigr)^{(\alpha - 1)/2} dB(t). \] The process \(\{X(t) \}\) is called a diffusion process associated with a porous medium equation. Put \[ K(t) = \int^t_0 u \bigl( s, X(s) \bigr)^{\alpha - 1} ds, \] then the limit distribution of \(K(t)^{- 1/2} X(t)\) (as \(t \to \infty)\) is the normal distribution \(N(0,1)\), but the limit distribution of \(\overline K(t)^{-1/2} X(t)\) (as \(t \to \infty)\) is a beta distribution, where \(\overline K(t) = E[K(t)]\).