Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 215 / 3 / a

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- Solution a

Solution

Choose a shortest path

x = x_{0}, \dots, x_{m} = y

. Glue the prescribed one-step edge couplings successively. The triangle inequality for the Wasserstein transportation metric gives

ρ_{K} (P (x, \cdot), P (y, \cdot)) \leq \sum_{j = 1}^{m} e^{- α} ρ (x_{j - 1}, x_{j}) = e^{- α} ρ (x, y) .

Now couple initial states

(U, V)

optimally for

μ, ν

and conditionally use these one-step couplings. Taking expectations and then the infimum gives

ρ_{K} (μ P, ν P) \leq e^{- α} ρ_{K} (μ, ν) .

ν = π

yields

ρ_{K} (P^{t} (x, \cdot), π) \leq e^{- α t} diam (V)

. Since this metric dominates total variation, it is at most

ε

once

t \geq \frac{1}{α} (lo g diam (V) + lo g (1/ ε)) .

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