The following are classical diffusion operators on open intervals. Prove that all of them generate ergodic semigroups.
  1. $M=(-1,1)$, $a(x)=1-x^2$, $b(x)=2x$ for some $\a,\b > -1$. Compute the density $\r$. The Jacobi polynomials $P_n^{(\a,\b)}$ are eigen functions of $H$ for the eigenvalues $\l_n=n(n+\a+\b+1)$ and the form a complete orthogonal set. For $\a=\b$ these polynomials are called ultraspherical or Gegenbauer polynomials.
  2. $M=\R$, $a(x)=1$, $b(x)=2x$. Compute the density $\r$. The Hermite polynomials $H_n$ are eigen functions of $H$ for the eigenvalues $\l_n=2n$ and the form a complete orthogonal set.
  3. $M=\R^+$, $a(x)=x$, $b(x)=x-\a-1$ for some $\a > -1$. Compute the density $\r$. The Laguerre polynomials $L_n^\a$ are eigen functions of $H$ for the eigenvalues $\l_n=n$ and the form a complete orthogonal set.
2. $\r^\prime+2\r x=0$ implies that $\r(x)=e^{-x^2}/\sqrt{\pi}$. As $HH_n=2nH_n$ it follows that $P_tH_n=e^{-2nt}H_n$ and thus for $f=\sum c_nH_n$ the equation $P_tf=f$ implies by orthogonality of the Hermite polynomials: $c_n=0$ for all $n\neq0$, i.e. $f$ is constant.