Homogenization of random parabolic operator with large potential (Q2769664)

This paper deals with the homogenization problem for random parabolic operators of the form: NEWLINE\[NEWLINE u_t(t,x)-\text{div}\Big[a\Big(\frac x{\varepsilon}, \xi_{t/\varepsilon^\alpha}\Big)\nabla u(t,x)\Big] -\varepsilon^{-\beta}c\Big(\frac x{\varepsilon}, \xi_{t/\varepsilon^\alpha}\Big)u(t,x), \quad x\in \mathbb R^n, NEWLINE\]NEWLINE where \(\varepsilon\) is a small positive parameter, \(\alpha,\beta>0\), the coefficients \(a^{ij}(z,y)\) and \(c(z,y)\) are periodic in the first argument, and \(\xi_t\) is a diffusion process with values in \(\mathbb R^d\). The parameter \(\alpha\) represents the ratio between space and time microscopic length scales. The authors show that for \(\beta=\alpha/2\wedge 1\) under the natural regularity assumptions, the homogenization results hold, while the structure of the limit problem depends crucially on whether \(\alpha=2\), or \(\alpha<2\) or \(\alpha>2\). If \(\alpha>2\), the family of solutions of the corresponding Cauchy problem converges in probability in a proper functional space to the solution of the Cauchy problem for parabolic operators with constant nonrandom coefficients. If \(\alpha\leq 2\), the family of measures generated by the solutions, converges weakly to the unique solution of the limit martingale problem which involves the one-dimensional Brownian motion. The formulae for the coefficients of the limit problem are different in the cases \(\alpha=2\) and \(\alpha<2\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].