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Summary. Let a be a non-isolated point of a topological space E. Suppose we are given standard processes X 0 and b X 0 on E0 = E \ {a} in weak duality with respect to a σ-finite measure m on E0 which are of no killings inside E0 but approachable to a. We first show that their extensions X and bX to E admitting no sojourn at a and keeping the weak duality are uniquely determined by the approaching probabilities of X 0 , b X 0 and m up to a non-negative constant δ0 representing the killing rate of X at a. We then construct, starting from X 0 , such X by piecing together returning excursions around a and a possible non-returning excursion including the instant killing. This extends a recent result by M. Fukushima and H. Tanaka [16] which treats the case where X 0 , X are m-symmetric diffusions and X admits no sojourn nor killing at a. Typical examples of jump type symmetric Markov processes and non-symmetric diffusions on Euclidean domains are given at the end of the paper.

Time change is one of the most basic and very useful transformations for Markov processes. The time changed process can also be regarded as the trace of the original process on the support of the Revuz measure used in the time change. In this paper we give a complete characterization of time changed processes of an arbitrary symmetric Markov process, in terms of the Beurling-Deny decomposition of their associated Dirichlet forms and of Feller measures of the process. In particular, we determine the jumping and killing measure (or, equivalently, the Lévy system) for the time-changed process. We further discuss when the trace Dirichlet form for the time changed process can be characterized as the space of finite Douglas integrals defined by Feller measures. Finally, we give a probabilistic characterization of Feller measures in terms of the excursions of the base process.