Small perturbation of stochastic parabolic equations: A power series analysis
The author investigates a parameter-depending SPDE of the form \[ \begin{aligned} du^\varepsilon(t,x)=&\left(\sum_{i,j}a^{ij,\varepsilon} (t,x)D_iD_ju^\varepsilon +f^\varepsilon(t,x,Du^\varepsilon,u^\varepsilon)\right) dt\\ &+\sum_k\left(\sum_i\sigma^{ik,\varepsilon}(t,x)D_iu^\varepsilon+ g^{k,\varepsilon}(t,x,u^\varepsilon)\right) dw_k(t),\end{aligned} \] for \(t>0\), \(x\in {\mathbb R}^d\), and \(u^\varepsilon(0,x)=\varphi^\varepsilon(x)\). Here, \(w_k\) are independent standard Wiener processes (their number may be infinite), \(D_i=\partial/\partial x_i\), \(i,j=1,\dots,d\), and all coefficients can depend on \(\omega\). The main result is that, under certain conditions, \(u^\varepsilon=\sum_{m=0}^n\varepsilon^mu^{(m)}+ o(\varepsilon^n)\), where \(u^{(m)}\) are defined recursively as solutions of linear equations.
stochastic partial differential equation
scientific article; zbMATH DE number 1836674