Differential Equation for brownian bridge?

Question

For the brownian motion, we know that probability density of the particle's position at time $ t $, $ \rho(x,t) $ satisfies the diffusion equation pde: $ \partial_t \rho = d \; \partial_x^2 \rho $. Is there some similar statement for the brownian bridge, something of the form: the probability density of the particle's position $\rho $ satisfies $ L \rho = 0 $ for some differential operator $L $ with the boundary conditions $ \rho(x,0) = \rho(x,1) = \delta(x) $?