This is from "Ross, Introduction to Probability Models (10th ed.)", p. 180, exercise 45.

Exercise instructions

How should one tackle this formally?

Simulation (with Gaussian stepping) suggests the traveler stays at his starting position on average. I am less sure about the standard deviation of the ending position. With small values of $\beta$, it appears the deviation grows somewhat as square root of the number of steps, or perhaps plain linearly. With $\beta$ large, simulation results "blow out" indeterminately.

With $\beta = 0.01$, I started at $x_0 = 100$ for the plot below, and did $10 \, 000$ simulations for every $n$ between 5 and 100 with step 5. Large dots are for mean, from which lines extend upward and downward for the values of the deviation.

Simulation results



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