 Solution

For the infinite system with outward drift, total probability remains one, but probability at a fixed distance from the root eventually decreases as the walker moves away. The question concerns this local decay, not a loss of total probability.

The independent dimensions of the continuum coefficients are $[D]=L^2/t$ and $[C]=L/t$. The drift-diffusion decay time and its associated length are, up to numerical factors:Equivalently, equating the drift distance $|C|t$ to the diffusive width of order $\sqrt{Dt}$ gives this crossover scale. Rescaling $x=(D/|C|)y$ and $t=(D/C^2)s$ removes the dimensional coefficients from the equation.

With the conventional coefficients used in the requested continuum derivation,where the last expression suppresses the numerical factor as requested. The elementary drifted Gaussian distribution on the full line hasAt fixed $x$, its exponential decay factor is $e^{-C^{2}t/(4D)}$, so an exponential e-folding time would conventionally be called $4D/C^2$, consistent with the same dimensional scale. The reflecting half-line changes the spatial form and algebraic prefactor. The substitution $P=e^{-Cx/(2D)}\psi$ gives ${\partial }_t\psi =D{\partial }_x^2\psi -(C^2/4D)\psi$, displaying the same exponential scale for outward drift.

For $z=2$, $C=0$ and no finite drift-induced exponential decay time exists: the probability near the root decays only algebraically. For the finite-bias, deterministic-step Gaussian envelope discussed in (b), the corresponding dimensional expression is instead $D_{\mathrm{fixed}}/C^2=2\tau (z-1)/(z-2{)}^2$. These continuum estimates are not exact return-probability decay rates for the lattice walk at arbitrary $z$; in particular, a diffusion approximation need not describe far-tail return events accurately at strong bias.