Pinned diffusions and Markov bridges

In this article we consider a family of real-valued diffusion processes on the time interval $[0,1]$ indexed by their prescribed initial value $x \in $\mathbb{R}$$andanotherpointinspace,$y \in $\mathbb{R}$$. We first present an easy-to-check condition on their drift and diffusion coefficients ensuring that the diffusion is pinned in $y$ at time $t=1$. Our main result then concerns the following question: can this family of pinned diffusions be obtained as the bridges either of a Gaussian Markov process or of an It^o diffusion? We eventually illustrate our precise answer with several examples.