Hypothesis Testing For The Parameter In One Class Of Nonlinear Stochastic Partial Diffenrential Equations

Hypothesis testing problem H₀:θ=θ₀,H₁:θ=θ₁ is studied for the parameter 0 in the following stochastic partial differential equation defined on（0,π）dU（t,x）+θ（-Δ）U（t,x）dt=sin U（t,x）dt+σdW（t,x）,x∈（0,π）,U（t,0）=U（t,π）=0,U（0,x）=u₀（x）where Δ=（?）²x,W（t,x）is a H=L²（0,π）-valued Q-Wiener process and σ is the noise intensity.We first derive the finite dimensianl Log-Likelihood Ration of θ from N-dimensional approximation on H.For a pre-specified significance level α,we con-struct the rejection region （?） with respect to time T and the rejection region （?） with respectto N.And,we further prove the rejection region is asymptotic,that is,（?） and lim （?） with N and T fixed respectively.