Homogenization of an advection equation with locally stationary random coefficients

2018/09/06 by Chojecki, Tymoteusz, Komorowski, Tomasz
#FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1809.02099

Abstract

In the paper we consider the solution of an advection equation with rapidly changing coefficients ∂_t u_\eps+(1/\eps)V(t\eps^-2,x/\eps)⋅∇_x u_\eps=0 for t0 is some small parameter and the drift term (V(t,x))_{(t,x)∈ \bbR^1+d} is assumed to be a d-dimensional, vector valued random field with incompressible spatial realizations. We prove that when the field is Gaussian, locally stationary, quasi-periodic in the x variable and strongly mixing in time the solutions u_\eps(t,x) converge in law, as \eps→0, to u₀(x(T;t,x)), where (x(s;t,x))_{s≥ t} is a diffusion satisfying x(t;t,x)=x. The averages of u_\eps(T,x) converge then to the solution of the corresponding Kolmogorov backward equation.

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