Starting  with a transient irreducible diffusion process $X^0$ on a locally compact separable metric space $(D, d)$ (for example, absorbing Brownian motion in a snowflake domain), one can construct a canonical symmetric reflected diffusion process $\bar X$ on a completion $D^*$ of $(D, d)$ through the theory of  reflected Dirichlet spaces. The boundary trace process $\check X$ of $X$ on the boundary $\partial D:=D^*\setminus D$ is the reflected diffusion process $\bar X$ time-changed by a smooth measure $\nu$ having full quasi-support on $\partial D$. The Dirichlet form of the trace process $\check X$ is called the trace Dirichlet form. In this talk, I will address the following two fundamental questions:

1) How to characterize the boundary trace Dirichlet space in a concrete way?

2) How does the boundary trace process behave? 

Based on a joint work with Shiping Cao.