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 $\sup\_x \Phi'(x)$ occurring around $x = 1$.


Any help in attacking the problem in higher dimensions and with non-simple probability transitions would be appreciated, though I suspect a thorough answer exists in the literature. Pointers to the relevant resources are welcome (even preferred!).


This is a migrant copy of the following [MSE post](https://math.stackexchange.com/questions/1362995/gradient-of-probability-distribution).



Gradient of probability distribution
The estimate you seek is a consequence of a strong local CLT, valid for aperiodic walks (at least on $Z^d$). See for example chapter 2 in Lawler-Limic. (Theorem 2.1.1 should be enough.)


