A nonlinear parabolic equation with noise
The authors consider the semilinear stochastic parabolic partial differential equation (PDE, in short) with multiplicative white noise \[ \partial_t \varphi+ \sum^d_{k=1} \partial_k\bigl(f(t,x, \varphi(t,x) \bigr)= \nu\sum^d_{k=1} \partial^2_{x_k} \varphi+\sigma (t)\varphi(t,x)\dot W_t, \quad (t,x) \in[0,T] \times\mathbb{R}^n, \] where the stochastic integral in this equation is defined in Skorokhod's sense. The approach is based on Hida's white noise analysis, the probability space is composed of the tempered Schwartz distributions; since the space of Hida test functions turns out to be not rich enough, the authors' analysis has to use the dual pair of spaces of smooth and generalized random variables studied by \textit{J. Potthoff} and \textit{M. Timpel} [Potential Anal. 4, No. 6, 637-654 (1995; Zbl 0839.46035)]. They prove existence and uniqueness of the solution by splitting \(\varphi\) into the Wick product of the solution of a pathwise random PDE and a normalized exponential of a Wiener integral. \textit{G. Da Prato} and \textit{J. Zabczyk} [``Stochastic equations in infinite dimensions'' (1992; Zbl 0761.60052)] applied a similar method to linear stochastic parabolic PDEs. The reader interested in white noise methods applied to PDEs can consult earlier works by \textit{J. Potthoff} [in: Stochastic partial differential equations and their applications. Lect. Notes Control Inf. Sci. 176, 238-251 (1992; Zbl 0783.60056)], \textit{F. E. Benth} [Stochastics Stochastics Rep. 51, No. 3/4, 195-216 (1994; Zbl 0851.60061)] and \textit{H. Holden}, \textit{B. Øksendal}, \textit{J. Ubøe} and \textit{T. Zhang} [``Stochastic partial differential equations. A modeling, white noise functional approach'' (1996; Zbl 0860.60045)].
semilinear stochastic partial differential equations
scientific article; zbMATH DE number 1507484