If the deterministic skeleton converges to zero and small jumps vanish, a symmetric pure-jump SDDE is weakly irreducible to zero.
Ishikawa, Support theorem for jump process of canonical type
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Weak irreducibility of stochastic delay differential equation driven by pure jump noise
If the deterministic skeleton converges to zero and small jumps vanish, a symmetric pure-jump SDDE is weakly irreducible to zero.