A local limit theorem for nonhomogeneous random walk on a lattice

In this paper, we study the walk of a particle on the $\nu$-dimensional lattice $\mathbf{Z}^\nu$, $\nu=1,2,3$, of which the one-step transition probabilities $\mathbf{Pr}(y\to x)$ differ from those of the homogeneous symmetric walk only in a finite neighborhood of the point $x=0$. For such a walk, the main term (having the order $O(1/t^{\nu/2})$) of the asymptotics of the probability $\mathbf{Pr}(x_t=x\mid x_0=y)$ as $t\to\infty$ is studied, $x,y\in\mathbf{Z}^\nu$, $x_t$ being the position of the particle at time $t$. It turns out that, for $\nu=2,3$, this main term of the asymptotics differs from the corresponding term of the asymptotics for the homogeneous walk (which has a usual Gaussian form) by a quantity of the order $O(t^{t-\nu/2}(|y|+1)^{-(\nu-1)/2})$. Thus the correction to the Gaussian term is comparable with it only in a finite neighborhood of the origin. In the case $\nu=1$, this correction has the form
$$ \frac{\mathrm{const}}{\sqrt t}\biggl(\operatorname{sign}y\exp\biggl\{-\frac{\mathrm{const}}t(|x|+|y|)^2\biggr\}+O\biggl(\frac 1{|y|}\biggr)\biggr), $$
i.e., remains of the same order as the Gaussian term on distances $|y|\sim\sqrt t$. The proof of these results is obtained by a detailed study of the structure of the resolvent $(\mathcal{T}-zE)^{-1}$ of the stochastic operator $\mathcal{T}$ of our model for $z$ lying in a small neighborhood of the point $z=1$ (the right boundary of the continuous spectrum of $\mathcal{T}$).