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

Let ${\mathbf{e}}_i$ 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

Multiplying the master equation by $a_i$, summing over every state, and shifting indices in each gain term gives, for $2\le i\le m-1$,At the absorbing left edge and reflecting right edge the corresponding equations are

A centered nearest-neighbour discretization of drift--diffusion is obtained fromFor sufficiently small $h$ these rates are nonnegative. Taylor expansion of the mean equations givesAbsorption into the target to the left of the first compartment giveswhile the absence of outward jumps at the right endpoint gives the zero-flux boundary condition

The Fokker-Planck equation corresponds to the Itô diffusionabsorbed at $0$ and reflected at $L$.

For $v(x)=x$, an Euler--Maruyama proposal isIf $Y\le 0$, kill the path. If $X_n>0$ and $Y>0$, an endpoint-only test can miss a crossing. Conditional on the endpoints, the local Brownian bridge crossing probability isKill the path with this probability; otherwise impose reflection at $L$ by replacing an overshoot $Y>L$ with $2L-Y$ and set $X_{n+1}=Y$. Repeated reflection handles very rare multiple overshoots, and the approximation converges as $\Delta t\to 0$.

Under the independence closure, the mean rightward flux across the bond $(i,i+1)$ isThe mean equation is the discrete conservation law ${\dot{M}}_i=J_{i-1/2}-J_{i+1/2}$. Substitution of the rates from part c and Taylor expansion show that the exclusion factors cancel from the symmetric diffusive contribution but remain in the biased contribution:Therefore the mean-field asymmetric simple exclusion process limit isThe factor $1-c$ is the probability that the destination site is vacant.