  We introduce a stochastic process and functional that should describe the
semigroup generated by the stochastic Bessel operator. Recently Gorin and
Shkolnikov showed that the largest eigenvalues for certain random matrix
ensembles with soft edge behavior can be understood by analyzing large powers
of tridiagonal matrices, which converge to operators in the stochastic Airy
semigroup. In this article we make some progress towards realizing Gorin and
Shkolnikov's program at the random matrix hard edge. We analyze large powers of
a suitable tridiagonal matrix model (a slight modification of the
$\beta$-Laguerre ensemble). For finite $n$ we represent the matrix powers using
Feynman-Kac type formulas, which identifies a sequence of stochastic processes
$X^n$ and functionals $\Phi_n$. We show that $\Phi_n(X^n)$ converges in
probability to the limiting functional $\Phi(X)$ for our proposed stochastic
Bessel semigroup. We also discuss how the semigroup method may be used to
understand transitions from a hard edge to a soft edge in the $\beta$-Laguerre
models.

Probability