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The irreducibility is fundamental for the study of ergodicity of stochasticdynamical systems. In the literature, there are very few results on theirreducibility of stochastic partial differential equations (SPDEs) andstochastic differential equations (SDEs) driven by pure jump noise. Theexisting methods on this topic are basically along the same lines as that forthe Gaussian case. They heavily rely on the fact that the driving noises areadditive type and more or less in the class of stable processes. The use ofsuch methods to deal with the case of other types of additive pure jump noisesappears to be unclear, let alone the case of multiplicative noises.  In this paper, we develop a new, effective method to obtain theirreducibility of SPDEs and SDEs driven by multiplicative pure jump noise. Theconditions placed on the coefficients and the driving noise are very mild, andin some sense they are necessary and sufficient. This leads to not onlysignificantly improving all of the results in the literature, but also to newirreducibility results of a much larger class of equations driven by pure jumpnoise with much weaker requirements than those treatable by the known methods.As a result, we are able to apply the main results to SPDEs with locallymonotone coefficients, SPDEs/SDEs with singular coefficients, nonlinear $\rmSchr\ddot{o}dinger$ equations, Euler equations etc. We emphasize that under oursetting the driving noises could be compound Poisson processes, even allowed tobe infinite dimensional. It is somehow surprising.

Irreducibility of SPDEs driven by pure jump noise