Given a random walk on a lattice $L$ (not necessarily centered - we allow $E[X_i] \neq 0$ for the i.i.d. increments $X_i$), let $p_t(x)$ denote the probability measure of state $x \in L$ after $t$ time increments. I am interested in the time evolution of the maximum difference of neighboring states' measures:

$$\max_{x \in L, ~ x \sim y} |p_t(x) - p_t(y)|.$$

Here $x \sim y$ indicates sites $x$ and $y$ are adjacent, or that

$P_x(y) = P($a walker started at $x$ transitions to $y$ in a single step$) > 0$.

I've explored various texts and online notes, but they don't seem to cover questions of the pointwise nature of these distributions; typically a distribution's weak limit is found via CLT or saddle-point approximation and it's left at that.

As a first effort, it's straightforward to look at the simple random walk on $\mathbb{Z}$ (where the increments are $+1$ or $-1$ with equal probability) and use Stirling's approximation to confirm that

$$\max_{k \in \mathbb{Z}} |p_t(k+1) - p_t(k)| \approx C/t$$

for some constant $C$ (the $\approx$ indicating omission of lower order terms). One also finds that this occurs around $\lfloor k\rfloor = \sqrt{t}$, just as one would expect considering the limiting gaussian distribution, $\Phi(x) = (2\pi)^{-1/2} e^{-x^2/2}$, which has $\sup_x \Phi'(x)$ occuring around $x = 1$.

Any help in attacking the problem in higher dimensions and with non-simple probability transitions would be appreciated. I suspect a thorough answer exists in the literature; pointers to the relevant resources could also form an answer.