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In this paper we introduce a new locally convex space of distributions, as a generalization of the space from [12], in which we have the product of any distributions as a series expansion. Then we discuss higher powers of the complex white noise on the space consisting of distributions without any renormalization. We also extend the Lévy and Voltera Laplacians to operators on a locally convex space taking the completion of the set of all distribution-coefficient polynomials on distributions with respect to some topology, and give an infinite dimensional Brownian motion generated by the Lévy Laplacian with a divergent part as a distribution. Based on those results, we obtain white noise distribution-valued stochastic differential equations, for the delta distribution centered at the infinite dimensional Brownian motion and also a sum of delta distributions centered at one dimensional Brownian motions.

The product of distributions and white noise distribution-valued stochastic differential equations