Dissipativity in Mean Square of Stochastic Reaction Diffusion Systems

It is by means of Lyapunov method that stochastic ordinary differential equations and stochastic functional differential equations have been studied intensively. However, for stochastic reaction diffusion equations, this useful technique seems to find no way out on account of the empty of its own Ito´s formula. To get over this difficulty, we will regard the integral of the considered trajectory with respect to spatial variables as the solution of the corresponding stochastic ordinary differential equations, via employing Ito´s formula under integral operator instead of directly applying Ito´s formula to Lyapunov functions in the case of stochastic ordinary differential equations, to aim at establishing the theory of dissipativity for Ito stochastic reaction diffusion systems. Some sufficient conditions for dissipativity and uniform dissipativity in mean square are given and this paper ends up with an example illustrating the obtained results.