Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 215 / 1 / a

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- Solution a

Solution

The laziness of

P = (Q + I) /2

makes all its eigenvalues nonnegative. Write

1 = λ_{1} > λ_{2} \geq \dots \geq λ_{∣ S ∣} \geq 0.

The relaxation time is

t_{rel} = \frac{1}{1 - λ _{2}} .

If

(f_{i})

is an orthonormal eigenbasis of

L^{2} (π)

with

f_{1} = 1

, the spectral decomposition is

\frac{P ^{t} ( x , y )}{π ( y )} = \sum_{i = 1}^{∣ S ∣} λ_{i}^{t} f_{i} (x) f_{i} (y) .

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