# probabilistic interpretation of a finite difference scheme

Consider the heat equation :

$\frac{\partial p}{ \partial t} = \frac{1}{2} \frac{\partial^2 p}{\partial y^2} \quad \mbox{in} \quad \mathbb{R}\times (0,\infty), \quad p(y,0) = \delta_0(y).$

It is well known that it is the Fokker-Planck equation of a standard Wiener process $y_t$ in $\mathbb{R}$ and $p(y,t) = \frac{1}{\sqrt{2 \pi t}} \exp(-\frac{y^2}{2t})$ is the probability that the particle is at position $y$ at time $t$.

In discrete setting, if we consider a two-dimensional rectangular lattice: comprising the sites

$\{ (j\Delta y, n\Delta t), j \in \mathbb{Z}, n \in \mathbb{N}\}$

for a given spacing $\Delta y>0$ and time duration $\Delta t >0$, then it is well known that the following finite difference scheme discretizes the heat equation:

$\frac{p_j^{n+1}-p_j^n}{ \Delta t} = \frac{D}{2} \frac{p_{j+1}^n - 2 p_{j}^n + p_{j-1}^n}{(\Delta y)^2} \quad \mbox{in} \quad \mathbb{Z}\times \mathbb{N}, \quad p^0(y) = \delta_0(y), \quad D := \frac{(\Delta y)^2}{ \Delta t}$

and it can be interpreted as the Fokker-Planck equation of a random walker satisfying $y^{n+1} = y^n + \Delta y \xi^n$ with $y^0 =0$ and $\{ \xi^n, n \geq 0 \}$ is a sequence of i.i.d satisfying $\mathbb{P} (\xi = \pm 1) = \frac{1}{2}$. Here $p_j^n$ is the probability that the random walker is at position $j \Delta y$ at time $n \Delta t$.

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My question concerns the identification of the random walker for the simplest Langevin equation.

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Here, we consider the simplest Langevin process $(\int_0^t y_s d s, y_t)$ and $p(x,y,t)$ ( which can be easily computed) : the probability that the particle is at position $x,y$ at time $t$. Here, the Fokker-Planck equation is:

$\frac{\partial p}{ \partial t} = \frac{1}{2} \frac{\partial^2 p}{\partial y^2} - y \frac{\partial p}{\partial x} \quad \mbox{in} \quad \mathbb{R}^2 \times (0,\infty), \quad p(x,y,0) = \delta_0(x,y).$

Then, similarly to what was done before, if we consider a three-dimensional rectangular lattice: comprising the sites

$\{ (i\Delta x, j \Delta y, n\Delta t), (i,j) \in \mathbb{Z}^2, n \in \mathbb{N}\}$

for given spacings $\Delta y, \Delta x>0$ and time duration $\Delta t >0$, then it is well known that the following finite difference scheme discretizes equation above:

$\frac{p_{i,j}^{n+1}-p_{i,j}^n}{ \Delta t} = \frac{D}{2} \frac{p_{i,j+1}^n - 2 p_{i,j}^n + p_{i,j-1}^n}{(\Delta y)^2} - j \Delta y \frac{p_{i,j}^n - p_{i-1,j}^n}{\Delta x} \mathbf{1}_{(j < 0)} - j \Delta y \frac{p_{i+1,j}^n - p_{i,j}^n}{\Delta x} \mathbf{1}_{(j > 0)} \quad \mbox{in} \quad \mathbb{Z}^2 \times \mathbb{N}, \quad p^0(x,y) = \delta_0(x,y).$

question Is it possible to identify an underlying random walker?

I would like to interpret the equation above as a Fokker-Planck in a discrete context. thanks.