Spatial asymptotics for the parabolic Anderson models with generalized time–space Gaussian noise

Partially motivated by the recent papers of Conus, Joseph and Khoshnevisan [Ann. Probab. 41 (2013) 2225–2260] and Conus et al. [Probab. Theory Related Fields 156 (2013) 483–533], this work is concerned with the precise spatial asymptotic behavior for the parabolic Anderson equation {∂u∂t(t,x)=12Δu(t,x)+V(t,x)u(t,x),u(0,x)=u0(x), where the homogeneous generalized Gaussian noise V(t,x) is, among other forms, white or fractional white in time and space. Associated with the Cole–Hopf solution to the KPZ equation, in particular, the precise asymptotic form limR→∞(logR)−2/3logmax|x|≤Ru(t,x)=342t3−−−√3a.s. is obtained for the parabolic Anderson model ∂tu=12∂2xxu+W˙u with the (1+1)-white noise W˙(t,x). In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.