Irreducibility of SPDEs driven by pure jump noise
A talk in the SPDEs, optimal control and mean field games series by
Tusheng Zhang from Manchester
| Abstract: | The irreducibility is fundamental for the study of ergodicity of stochastic dynamical
systems. In the literature, there are very few results on the irreducibility
of stochastic partial differential equations (SPDEs) and stochastic differential
equations (SDEs) driven by pure jump noise. The existing methods on this topic
are basically along the same lines as that for the Gaussian cases. They rely
on that the driving noises are additive type and more or less in the class of
stable processes. The use of such methods to deal with the case of other types
of additive pure jump noises appears to be unclear, let alone the case of multiplicative
noises. In this paper, we develop a new, effective method to obtain the irreducibility
of SPDEs and SDEs driven by multiplicative pure jump noise. The conditions placed
on the coefficients and the driving noise are very mild, and in some sense they
are necessary and sufficient. This leads to not only significantly improving
the results in the literature, but also to new irreducibility results of a much
larger class of equations driven by pure jump noise with much weaker requirements
than those treatable by the known methods. As a result, new applications apply
to SPDEs with locally monotone coefficients, SPDEs/SDEs with singular coefficients,
Euler equations etc. |