Consider a symmetric strongly local regular Dirichlet form on a locally compact separable metric measure space \$({\cal X}, m)\$ whose heat kernel admits sub-Gaussian heat kernel estimate. Let \$D\$ be a uniform subdomain of \${\cal X}\$. There is a conservative reflected diffusion process \$X\$ on \$\overline D\$. The trace Dirichlet form on the boundary of \$D\$ is the Dirichlet form of the diffusion process \$X\$ time-changed by a smooth measure \$\mu\$ with full quasi-support on \$\partial D\$. In this talk, we give Besov space type characterization of the domain of the trace Dirichlet forms for any good smooth measure \$\mu\$ on the boundary \$\partial D\$. We investigate properties of the harmonic measure of \$X\$ on the boundary \$\partial D\$. In particular, we provide a condition equivalent to the doubling property of the harmonic measure. We then present characterization and estimates of the jump kernel of the trace Dirichlet form under the doubling condition of the harmonic measure on \$\partial D\$.
This is based on a joint work with Shiping Cao.