In this work we construct general additive processes on the nuclear spaces, and prove Khintchin's formula and Paul Levy's decomposition for these processes. As applications, we construct some Ornstein-Uhlenbeck processes with jumps and solve some (stochastic) partial differential equations obtained from the transformations of these processes by a random diffeomorphism corresponding to a finite dimensional diffusion process.
"Additive Processes on Nuclear Spaces." Ann. Probab. 12 (3) 858 - 868, August, 1984. https://doi.org/10.1214/aop/1176993234