On semilinear stochastic partial differential equations on the real line. (English) Zbl 0887.60070

Manthey, Ralf

Stochastics Stochastics Rep. 57, No. 3-4, 213-234 (1996).

The author constructs by the help of the comparison method a solution of the Cauchy problem to a parabolic stochastic partial differential equation of the form \[ {\partial\over\partial t} u(t,x)= (\Delta u)(t,x)+ f(t,x,u(t,x))+ \sigma(t,x,u(t,x))\dot W(t,x), \] \(t\in(0,T]\), \(x\in\mathbb{R}\), \(u(0,x)= v(x)\), \(x\in\mathbb{R}\), where \(\Delta\) denotes the Laplace operator and \(\dot W\) stands for the space-time Gaussian noise. The “drift” is assumed to be one-sided linearly bounded, continuous and of at most polynomial growth while the “diffusion” is Lipschitzian and linearly bounded.

Reviewer: B.G.Pachpatte (Aurangabad)

MSC:

60H15

Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

Cauchy problem; parabolic stochastic partial differential equation; Laplace operator