Nuclear space-valued stochastic differential equations driven by Poisson random measures.

摘要

The thesis is devoted primarily to the study of stochastic differential equations on duals of nuclear spaces driven by Poisson random measures. The existence of a weak solution is obtained by the Galerkin method and the uniqueness is established by implementing the Yamada-Watanabe argument in the present setup.;When the magnitudes of the driving terms are small enough and the Poisson streams occur frequently enough, it is proved that the stochastic differential equations mentioned above can be approximated by diffusion equations.;Finally, we consider a system of interacting stochastic differential equations driven by Poisson random measures. Let ;The above problems are motivated by applications to neurophysiology, in particular, to the fluctuation of voltage potentials of spatially distributed neurons and to the study of asymptotic behavior of large systems of interacting neurons.