A Local Limit Theorem for Nonhomogeneous Random Walk on a Lattice

Abstract

In this paper, we study the walk of a particle on the $\nu$-dimensional lattice ${\bf Z}^\nu ,\nu = 1,2,3$, of which the one-step transition probabilities ${\bf Pr} (y \to x)$ 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 ${\bf Pr} (x_t = x|x_0 = y)$ as $t \to \infty $ is studied, $x,y \in {\bf 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 \[ \frac{{{\text{const}}}}{{\sqrt t }}\left( {{\text{sign}}\,\exp \left\{ { - \frac{{{\text{const}}}} {t}(|x| + |y|)^2 } \right\} + o\left( {\frac{1}{{|x|}}} \right)} \right), \] 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 $\mathcal{T}$).