Weak time discretization for slow-fast stochastic reaction-diffusion equations (Q2056199)

In this article, the authors construct a time discretization for the slow-fast stochastic partial differential equations (SPDEs) on the interval \(D = [0,1]\) \[ \partial_t u^{\varepsilon}(t,x) = \partial_{xx} u^{\varepsilon}(t,x) + f(u^{\varepsilon}(t,x),v^{\varepsilon}(t,x)) \] \[ \partial_t v^{\varepsilon}(t,x) = \frac{1}{\varepsilon} \big[ \partial_{xx} v^{\varepsilon}(t,x) + g(u^{\varepsilon}(t,x),v^{\varepsilon}(t,x)) \big] + \frac{1}{\sqrt{\varepsilon}} \partial_t W(t) \] \[ u^{\varepsilon}(0,x) = u_0(x), \,\,\,\,\,\, v^{\varepsilon}(0,x) = v_0(x), \,\,\,\,\,\, x \in D, \] \[ u^{\varepsilon}(t,0) = u^{\varepsilon}(t,1) = 0, \,\,\,\,\,\, v^{\varepsilon}(t,0) = v^{\varepsilon}(t,1) = 0, \,\,\,\,\,\, t > 0, \] where \(f(u,v) = u - u^3 + uv\), the function \(g\) is a Lipschitz nonlinearity, and \(\{ W(t) : t \geq 0 \}\) is an \(L^2(D)\)-valued Wiener process. For this purpose, the authors show that the SPDEs above can be written in abstract form as \[ d u^{\varepsilon}(t) = [ A u^{\varepsilon}(t) + f(u^{\varepsilon}(t),v^{\varepsilon}(t)) ] dt \] \[ d v^{\varepsilon}(t) = \frac{1}{\varepsilon} \big[ A v^{\varepsilon}(t) + g(u^{\varepsilon}(t),v^{\varepsilon}(t)) \big] dt + \frac{1}{\sqrt{\varepsilon}} dW(t) \] \[ u^{\varepsilon}(0) = u_0, \,\,\,\,\,\, v^{\varepsilon}(0) = v_0, \] where \(A = \Delta\) on \(D\) with Dirichlet boundary conditions. Then the idea is to consider the averaging equation \[ d \bar{u}(t) = [A \bar{u}(t) + \bar{f}(\bar{u}(t))] dt. \] Here the function \(\bar{f}\) is given by \[ \bar{f}(u) = \int_{L^2(D)} f(u,v) \mu^u(dv), \] where \(\mu^u\) denotes the unique invariant measure for the fast component with frozen slow component \(u\). Afterwards, they introduce the implicit Euler scheme \[ u_{n+1} = S_{\Delta t} u_n + \Delta t S_{\Delta t} \bar{f}(u_n), \] where \(S_{\Delta t} = (I - A \Delta t)^{-1}\). In their main result (Theorem 3.1) the authors provide an estimate for the weak error \[ | \mathbb{E} \phi(u^{\varepsilon}(T)) - \mathbb{E} \phi(u_n) | \] for any \(T > 0\) and \(\phi \in C_b^3\). The authors also present a result about the discretization approximation of the solution of the averaging equation (see Theorem 4.5) and a result about asymptotic expansions (see Theorem 5.1).