  Using the theory of Dirichlet forms we construct a large class of continuous
semimartingales on an open domain $E \subset \mathbb{R}^d$, which are governed
by rank-based, in addition to name-based, characteristics. Using the results of
Baur et al. [Potential Analysis, 38(4):1233-1258,2013] we obtain a strong
Feller property for this class of diffusions. As a consequence we are able to
establish the nonexistence of triple collisions and obtain a simplified formula
for the dynamics of its rank process. We also establish conditions under which
the process is ergodic. Our main motivation is Stochastic Portfolio Theory
(SPT), where rank-based diffusions of this type are used to model financial
markets. We show that three main classes of models studied in SPT -- Atlas
models, generalized volatility-stabilized models and polynomial models -- are
special cases of our framework.

Probability;Mathematical Finance