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 [cases{displaystyle {frac{partial u}{partial t}}(t,x)={frac{1}{2}}Delta u(t,x)+V(t,x)u(t,x),cr u(0,x)=u_0(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 [lim_{Rtoinfty}(log R)^{-2/3}logmax_{|x|le R}u(t,x)={frac{3}{4}}root 3of {frac{2t}{3}}qquad a.s.] is obtained for the parabolic Anderson model $partial_tu={frac{1}{2}}partial_{xx}^2u+dot{W}u$ with the $(1+1)$-white noise $dot{W}(t,x)$. In addition, some links between time and space asymptotics for the parabolic Anderson equation are also pursued.