= Solution
Let $p_j^n$ be the probability of occupying site $ja$ after $n$ steps, with $p_j^0=\mathbf1_{\{j=0\}}$ and $\alpha+\beta=1$. The <master equation> is
$$
\boxed{p_j^{n+1}=\alpha p_{j-1}^n+\beta p_{j+1}^n.}
$$
Write $p_j^n\approx a p(ja,n\tau)$ so that the limiting $p$ is a <probability density>. A <Taylor expansion> of this <master equation> gives
$$
\tau p_t=(\beta-\alpha)a p_x+\frac{(\alpha+\beta)a^2}{2}p_{xx}+O(a^3,\tau^2).
$$
The weakly biased <diffusion limit of a weakly biased random walk> keeps $a^2/\tau$ and $(\alpha-\beta)a/\tau$ finite as $a,\tau\to0$, so $\alpha-\beta=O(a)$. The limiting <advection-diffusion equation> is
$$
\boxed{p_t+Vp_x=Dp_{xx},\qquad V=\frac{a(\alpha-\beta)}\tau,\quad D=\frac{a^2(\alpha+\beta)}{2\tau}=\frac{a^2}{2\tau}.}
$$
Its initial condition is the <Dirac delta function> at the origin. At finite spacing, the exact variance gained per step is $a^2[1-(\alpha-\beta)^2]=4a^2\alpha\beta$. Thus a finite-step <central limit theorem> gives the variance-matched coefficient $2a^2\alpha\beta/\tau$; its difference from the displayed $D$ vanishes in the stated weak-bias <diffusion limit of a weakly biased random walk>. Keeping a fixed nonzero bias while taking diffusive space-time scaling would instead make $V$ diverge.
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