  The Laplace transforms of the transition probability density and distribution
functions for the Ornstein-Uhlenbeck process contain the product of two
parabolic cylinder functions, namely D_{v}(x)D_{v}(y) and D_{v}(x)D_{v-1}(y),
respectively. The inverse transforms of these products have as yet not been
documented. However, the transition density and distribution functions can be
obtained by alternatively applying Doob's transform to the Kolmogorov equation
and casting the problem in terms of Brownian motion. Linking the resulting
transition density and distribution functions to their Laplace transforms then
specifies the inverse transforms to the aforementioned products of parabolic
cylinder functions. These two results, the recurrence relation of the parabolic
cylinder function and the properties of the Laplace transform then enable the
calculation of inverse transforms also for countless other combinations in the
orders of the parabolic cylinder functions such as D_{v}(x)D_{v-2}(y),
D_{v+1}(x)D_{v-1}(y) and D_{v}(x)D_{v-3}(y).
