Symmetric diffusions on manifolds with boundary are studied. Symmetric diffusions are nicer to work with than non-symmetric diffusions because (1) it is easier to tell if an equilibrium density exists, and (2) it is easier to find the equilibrium density when it does exist. If an equilibrium density exists, a symmetric diffusion is time reversible. On the line essentially all diffusions are symmetric.
Using symmetric diffusions, it is shown that a large family of densities can be realized as equilibrium densities of time-reversible diffusions. Some examples are given.
