Stability in Probability of Partial Variables for Stochastic Reaction Diffusion Systems

Stochastic ordinary differential equations and stochastic functional differential equations have recently been studied intensively by means of Lyapunov function. However, it is a pity for stochastic reaction diffusion equations that this useful technique seems to find no way out due to 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 investigating stability in probability of partial variables for Ito stochastic reaction diffusion equations. Some sufficient conditions for stability and uniform stability in probability of partial variables are given and this paper is ended up with an example illustrating the obtained results.