Homogenization of parabolic equations with large time-dependent random potential (Q468729)

The equations NEWLINE\[NEWLINE\begin{aligned} \partial_tu_\varepsilon(t,x) & =2^{-1}\Delta u_\varepsilon(t,x)+i\varepsilon^{-((\alpha/2)\lor 1)}V(\varepsilon^{-\alpha}t,\varepsilon^{-1}x)u_\varepsilon(t,x),\\ \partial_tu_0(t,x) & =2^{-1}\Delta u_0(t,x)-\rho(\alpha)u_0(t,x),\\ \partial_tU_\varepsilon(t,x) & =2^{-1}\Delta U_\varepsilon(t,x)+i\varepsilon^{-1/2}V(\varepsilon^{-1}t,x)U_\varepsilon(t,x),\\ \partial_tU_0(t,x) & =2^{-1}\Delta U_0(t,x)+i\dot W(t,x)\circ U_0(t,x)\end{aligned} NEWLINE\]NEWLINE are considered on \(\mathbb R^d\) with \(d\geq 3\), with the initial condition \(u_\varepsilon(0,x)=u_0(0,x)=U_\varepsilon(0,x)=U_0(0,x)=f(x)\) for some \(f\in C_b(\mathbb R^d)\) and with \(\alpha>0\). Here, \(V(t,x,\omega)=\mathbb V(\tau_{(t,x)}\omega))\) is a mean-zero, time-dependent, stationary random potential, where \(\mathbb V\) is a centred square integrable random variable and \(\{\tau_{(t,x)}:t\in\mathbb R,\,x\in\mathbb R^d\}\) is a group of measure-preserving, ergodic transformations on a probability space \((\Omega,\mathcal F,\mathbb P)\) modelling the random medium. The term \(\rho(\alpha)\) is defined explicitly in the paper in terms of the covariance of \(V\) and its definition differs significantly if \(\alpha\in[0,2)\), \(\alpha=2\) or \(\alpha\in(2,\infty)\). Here, \(\dot W(t,x)\) is a Gaussian noise white in time and with a specific spatially homogeneous covariance in space related to the covariance of \(V\). The potential \(V\) is further assumed to be uniformly bounded and there is an additional hypothesis that there exists \(M>0\) such that \(\sigma(V(t,x):(t,x)\in S_1)\) and \(\sigma(V(t,x):(t,x)\in S_2)\) are independent whenever \(S_1\) and \(S_2\) are non-empty subsets of \(\mathbb R^{1+d}\) with the Euclidean distance \(\text{dist}(S_1,S_2)\geq M\). It is proved that \(u_\varepsilon(t,x)\to u_0(t,x)\) in probability and \(U_\varepsilon(t,x)\to U_0(t,x)\) in law, as \(\varepsilon\to 0\).