Stochastic evolution equations with Wick-polynomial nonlinearities

Abstract

We study nonlinear stochastic partial differential equations with Wick-analytic type nonlinearities set in the framework of white noise analysis. These equations include the stochastic Fisher-KPP equations, stochastic Allen-Cahn, stochastic Newell-Whitehead-Segel, and stochastic Fujita-Gelfand equations. By implementing the theory of C 0 −semigroups and evolution systems into the chaos expansion theory in infinite dimensional spaces, we prove existence and uniqueness of solutions for this class of stochastic partial differential equations.

Wick-type stochastic parabolic equations with random potentials

“…[1,2,3,6]. Stochastic evolution equations with multiplicative noise are studied in [7] and stochastic evolution problems with polynomial nonlinearities are studied in [8].…”

Solutions of Hyperbolic Stochastic PDEs on Bounded and Unbounded Domains

“…Having all these highly random terms will lead to singular solutions that do not allow to use ordinary multiplication, but require its renormalization also known as the Wick product. The Wick product is known to represent the highest order stochastic approximation of the ordinary product [32], and has been used in many models together with the Wiener chaos expansion method [18,19,26,27,29,30,[40][41][42]47].…”

Stochastic parabolic equations with singular potentials

“…Semilinear heat equation which is driven by a space-time Gaussian white noise in suitable algebras of generalized functions is considered in [38]. Stochastic evolution problems with polynomial nonlinearities were studied in [25]. We aim to consider more general classes of problems, i.e., to study stochastic parabolic evolution equations with singular space depending potentials, where the operator L is not necessarily the Laplace operator.…”

“…We aim to consider more general classes of problems, i.e., to study stochastic parabolic evolution equations with singular space depending potentials, where the operator L is not necessarily the Laplace operator. For the analysis of (1.2) we propose a new method which combines the chaos expansion method [17,20,25,42,44] and the very weak solution concept [3,4,5,9,10,27,39,40]. The chaos expansion method is based on constructing the solution to the initial stochastic PDE (SPDE) as a Fourier series in terms of a Hilbert space basis of orthogonal stochastic polynomials, with unknown coefficients being elements in an appropriate space of deterministic functions.…”