Weak convergence of Dirichlet processes (Q1392625)

A Dirichlet form is considered which is a generalization on an arbitrary metric space of the bilinear functional \[ {\mathcal E}(u,v)={1\over 2}\int_{R^d}\sum_{ij}a_{ij}\partial_iu\partial_jvdx+\int_{R^d}\sum_ib_i\partial _iu\cdot vdx+\int_{R^d}Cuv dx, \] where \(d\geq 1\), \(u,v\) are some differentiable functions, \(a_{ij},b_i,C\) are diffusion coefficients. There corresponds some Markov-Hunt process to this form. Two theorems about weak convergence of a sequence of Markov processes are proved in terms of associated Dirichlet forms. In the first theorem the main condition is convergence of the sequence of square forms \(({\mathcal E}^n(u,u))^\infty_1\), and in the second one the main condition is strong convergence of the sequence of semigroups \((T^n_t)^\infty_1\) for all \(t>0\). Examples of applications of these theorems to special Dirichlet forms are mentioned, in particular to the above form associated to a Markov process of a diffusion type.