Solution to Partial Differential Equations: Methods and Applications (Robert McOwen) Section 5.4

1. Suppose that the probability that a Brownian motion particle will move to the left a distance $\epsilon$ is $p$ and to the right is $q$, where $p+q=1$. If $p\neq q$, find the diffusion equation that the probability density satisfies. The constant $p-q$ is called the drift constant; can you explain why?
Solution
Assume that in a small increment of time $\Delta t$, the probability that a Brownian motion particle will move to the left a distance $\epsilon$ is $p$ and to the right is $q$. Thus,
$u(x,t+\Delta t)=p\cdot u(x-\epsilon,t)+q\cdot u(x+\epsilon,t)$.
Using the Taylor expansion: $u(x\pm\epsilon,t)\approx u(x,t)\pm u_x(x,t)\epsilon+\frac12 u_{xx}(x,t)\epsilon^2+O(\epsilon^3)$, we obtain
$\begin{aligned}u(x,t+\Delta t)&\approx p\cdot\left(u(x,t)+u_x(x,t)\epsilon+u_{xx}(x,t)\epsilon^2+O(\epsilon^3)\right)+q\cdot\left(u(x,t)-u_x(x,t)\epsilon+u_{xx}(x,t)\epsilon^2+O(\epsilon^3)\right)\\&=u(x,t)+(p-q)u_x(x,t)\epsilon+\frac12u_{xx}(x,t)\epsilon^2+O(\epsilon^3),\end{aligned}$
which implies
$\displaystyle\frac{u(x,t+\Delta t)-u(x,t)}{\Delta t}\approx\frac{\epsilon^2}{2\Delta t}u_{xx}(x,t)+(p-q)u_x(x,t)\frac{\epsilon}{\Delta t}+O\left(\frac{\epsilon^3}{\Delta t}\right)$.
Suppose that $\displaystyle\frac{\epsilon^2}{2\Delta t}$ tends to $\gamma$. Moreover, we assume that $(p-q)/\Delta t$ tends to a finite value $\beta$. Therefore, we derive
$\displaystyle u_t=\frac\gamma2u_{xx}+\beta u_x$.
Here $p-q\approx\beta\Delta t$ represents the tendency of the limiting continuou motion so we call it the drift constant, i.e., if $p-q>0$, the movement is towards the left; if $p-q<0$, the movement is towards the right.