Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 356 / 2 / a

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- Solution a

Solution

Let

e_{i}

be the

i

th coordinate vector and take

p

to be zero whenever any argument is negative. Right jumps

i \to i + 1

, left jumps

i \to i - 1

, and absorption from compartment

1

give the chemical master equation

\partial_{t} p (a, t) = i = 1 \sum m - 1 k_{i}^{+} [(a_{i} + 1) p (a + e_{i} - e_{i + 1}, t) - a_{i} p (a, t)] + i = 2 \sum m k_{i}^{-} [(a_{i} + 1) p (a - e_{i - 1} + e_{i}, t) - a_{i} p (a, t)] + k_{1}^{-} [(a_{1} + 1) p (a + e_{1}, t) - a_{1} p (a, t)] .

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