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An infinite system of point particles placed in $\mathds{R}^d$ is studied.Its constituents perform random jumps with mutual repulsion described by atranslation-invariant jump kernel and interaction potential, respectively. Thepure states of the system are locally finite subsets of $\mathds{R}^d$, whichcan also be interpreted as locally finite Radon measures. The set of all suchmeasures $\Gamma$ is equipped with the vague topology and the correspondingBorel $\sigma$-field. For a special class $\mathcal{P}_{\rm exp}$ of(sub-Poissonian) probability measures on $\Gamma$, we prove the existence of aunique family $\{P_{t,\mu}: t\geq 0, \ \mu \in \mathcal{P}_{\rm exp}\}$ ofprobability measures on the space of cadlag paths with values in $\Gamma$ thatsolves a restricted initial-value martingale problem for the mentioned system.Thereby, a Markov process with cadlag paths is specified which describes thestochastic dynamics of this particle system.

A Markov process for an infinite interacting particle system in the continuum