A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the fol...A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the following conditions for someα∈(0,1]:||∫_(0)^(t)e−(t−s)^(A)dW(s)||L^(2)(Ω;L^(2)(O))≤C^(t^(α/2))and||∫_(0)^(t)e−(t−s)^(A)dW(s)||_B^(∞)L^(2)(Ω:H^(α)(O))≤C..The conditions above are shown to be satisfied by both trace-class noises(withα=1)and one-dimensional space-time white noises(withα=1/2).The latter would fail to satisfy the conditions withα=1/2 if the stochastic Besov norm||·||B∞L^(2)(Ω;˙H^(α)(O))is replaced by the classical Sobolev norm||·||L^(2)(Ω;˙Hα(O)),and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation.In this paper,the convergence of a modified exponential Euler method,with a spectral method for spatial discretization,is proved to have orderαin both the time and space for possibly nonsmooth initial data in L^(4)(Ω;˙H^(β)(O))withβ>−1,by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t=0.展开更多
基金supported by National Natural Science Foundation of China(Grant Nos.12071020,12131005 and U2230402)the Research Grants Council of Hong Kong(Grant No.Poly U15300519)an Internal Grant of The Hong Kong Polytechnic University(Grant No.P0038843,Work Programme:ZVX7)。
文摘A class of stochastic Besov spaces BpL^(2)(Ω;˙H^(α)(O)),1≤p≤∞andα∈[−2,2],is introduced to characterize the regularity of the noise in the semilinear stochastic heat equation du−Δudt=f(u)dt+dW(t),under the following conditions for someα∈(0,1]:||∫_(0)^(t)e−(t−s)^(A)dW(s)||L^(2)(Ω;L^(2)(O))≤C^(t^(α/2))and||∫_(0)^(t)e−(t−s)^(A)dW(s)||_B^(∞)L^(2)(Ω:H^(α)(O))≤C..The conditions above are shown to be satisfied by both trace-class noises(withα=1)and one-dimensional space-time white noises(withα=1/2).The latter would fail to satisfy the conditions withα=1/2 if the stochastic Besov norm||·||B∞L^(2)(Ω;˙H^(α)(O))is replaced by the classical Sobolev norm||·||L^(2)(Ω;˙Hα(O)),and this often causes reduction of the convergence order in the numerical analysis of the semilinear stochastic heat equation.In this paper,the convergence of a modified exponential Euler method,with a spectral method for spatial discretization,is proved to have orderαin both the time and space for possibly nonsmooth initial data in L^(4)(Ω;˙H^(β)(O))withβ>−1,by utilizing the real interpolation properties of the stochastic Besov spaces and a class of locally refined stepsizes to resolve the singularity of the solution at t=0.
