Wave front sets for ultradistribution solutions of linear partial differential operators with coefficients in non‐quasianalytic classes

We prove the following inclusion where WF_* denotes the non‐quasianalytic Beurling or Roumieu wave front set, Ω is an open subset of $\mathbb {R}^n$, P is a linear partial differential operator with suitable ultradifferentiable coefficients, and Σ is the characteristic set of P. The proof relies on some techniques developed in the study of pseudodifferential operators in the Beurling setting. Some applications are also investigated.     

Exponential Moments of Solutions for Nonlinear Equations with Catalytic Noise and Large Deviation

Nonlinear equation with catalytic noise is considered. We discuss the existence of catalytic superprocess associated with the equation and derive the exponential moment formula. Moreover, we prove the large deviation principle for catalytic superprocesses.