Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 201 / 2 / a / v / Solution

Almost surely, Brownian motion has a unique minimizer on every interval with rational endpoints, by rescaling part (iv). Every local minimum is the minimum on some rational interval contained in a witnessing neighbourhood. Each rational interval contributes at most one point, and there are countably many such intervals. The set of local minima is therefore countable.