Mild Solution To Parabolic Anderson Model In Gaussian And Renormalized Poisson Potential

Stochastic Partial Differential Equations have been widely used in fluid mechanics,quantum field theory financial mathematics,biology,and random control.The parabolic Anderson model studied in this paper is a special case of stochastic partial differential equations.The random potential of the parabolic Anderson model is composed of Gaussian potential and renormalized Poisson potential.When the random potential is Holder continuous,you can use the Feynman-Kac formula to get the unique solution.However,the random potential does not have Holder continuity,so we look for a mild solution similar to the Feynman-Kac solution.In the first chapter,We give the construction process of the characteristic function of infinitely separable measures and the necessary and sufficient conditions to judge whether a function is integrable with respect to infinitely separable measures,The latter is the main theorem for deriving that the random potential is less than infinityIn the second chapter,first we propose our improved random potential V(x)=?Rd(y-x)[?(dy)-dy+W(dy)].here is composed by The renormalized Poisson potential and Gaussian potential.Then,a sufficie-nt and necessary condition for a function about ?(dy)-dy+W(dy) integrability and some basic properties of such integration are given.The last part,the integrability of the Brownian motion and the square of the Brownian motion under the renormalized Poisson potential and the Gaussian potential are given.Finally,two forms of parabolic Anderson model are given:a random potential is-V(x),and the other random po-tential is V(x)?(t),where ?(t)is time white noise,and their mild solut-ion is given.