The renormalized projection operator technique for nonlinear stochastic equations. III

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Abstract
The solution of the nonlinear stochastic equation L (x,t,ω) ψ (x,t,ω) =g (x,t)+f [ψ (x,t,ω)] is found via the renormalized projection operator technique and is approximated to be 〈ψ (x,t) 〉=F Fdx′dt′ 〈Gp(x,t ‖x′,t′) 〉 {ψint(x′,0)+g (x′,t′) } −F Fdx′dt′〈P (x′,t′) 〉 = o e ing−brace)=F Fdx″dt″〈Gp(x,t ‖x′,t ′) G0(x′,t′‖x″,t″〉 [ψint(x″,0)+g (x″,t″)]}+ F Fdx′dt′〈Gp(x,t ‖x′,t′〉 〈f [ψH(x′,t′)]〉. The terms 〈Gp(x,t ‖x′,t′) 〉 and 〈Gp(x,t ‖x′,t′) G0(x′,t′‖x″,t″) 〉 are the stochastic one‐ and two‐point Green’s functions. Also three conditions are shown that the projection operator must have in order to insure convergence.

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