introduction.md

Introduction

In previous presentations, we [AR, AvM] talked about stochastic differential equations (middle box in {numref}table-desc-levels). Examples:

:::{margin} :class: sticky

Brownian motion ~ $\dot{X}(t) = ξ(t)$ ~ $X(t) = \mathcal{N}(0, t)$ ::: Brownian motion ~ $\dot{X}(t) = ξ(t)$ ~ $X(t) = \mathcal{N}(0, t)$

Drift-diffusion processes ~ $\dot{X}(t) = f(t,X) + ξ(t)$ ~ $dX(t) = f(t,X)dt + g(t,X)dW$

:width: 500px
:name: table-desc-levels

(Risken, Table 1.1)

We also talked about the fact that the “white noise” $ξ$ in a Langevin equation is ill-defined.

::::{margin} :::{note} All of these conventions are defined in terms of finite intervals. ::: :::: Itô convention ~ $$ΔX(t_i) = f(t_i,X(t_i))Δt + g(t_i,X(t_i))ΔW(t_i)\bigr)\hphantom{+ g(t_{i+1},X(t_{i+1}))}$$

Stratonovich convention ~ $$ΔX(t_i) = f(t_i,X(t_i))Δt + \frac{g(t_i,X(t_i)) + g(t_{i+1},X(t_{i+1}))}{2}ΔW(t_i)$$

Anticipatory (or Hänggi-Klimontovich) convention ~ $$ΔX(t_i) = f(t_i,X(t_i))Δt + g(t_{i+1},X(t_{i+1}))ΔW(t_i)\hphantom{+ g(t_{i},X)}$$

This ambiguity of convention arises because the infinitesimal limit of white noise is mathematically well-defined (within a given convention), but non physical.

A full solution to a Langevin equation typically takes the form of a probability density function (PDF), which, if the system is Markovian, can be written as $p(t, X)$. Now, this PDF is physical, so a differential equation for $p(t, X)$ should not suffer from ambiguity. The Fokker-Planck equation is such a differential equation:

$$ \frac{\partial p(t, x)}{\partial t} = - \sum_i \frac{\partial}{\partial x_i} \left[D_i^{(1)}(x) , p(t,x)\right] + \sum_{i,j} \frac{\partial^2}{\partial x_i \partial x_j} \left[D_{ij}^{(2)}(x) , p(t, x) \right] $$ (eq-FPE-intro)

:name: fig-density-3-times

The kind of problem we want to solve: given an initial probability distribution, how does it evolve over time ?
Often, but not always, the initial condition is a Dirac δ.
(Risken, Fig. 2.2)

A stochastic process is a generalization of a random variable. Intuitively, we assign to each $t \in \mathbb{R}$ a random variable (as suggested by {numref}fig-density-3-times). More precisely, to any countable set of times, the process associates a joint distribution. So if $x \in \mathbb{R^N}$,

• $p(t_1, x_1)$ is the PDF of a random variable on $\mathbb{R}^N$;
• $p(t_2, x_2, t_1, x_1)$ is the PDF of a random variable on $\mathbb{R}^{2N}$;
• $p(t_3, x_3, t_2, x_2, t_1, x_1)$ is the PDF of a random variable on $\mathbb{R}^{3N}$;
• etc.

One obtains a lower dimensional distribution by marginalising over certain time points:

$$p(t_3, x_3, t_1, x_1) = \int p(t_3, x_3, t_2, x_2, t_1, x_1) dx_2$$

:::{margin} Markov process: $p(t_3, x_3 \mid t_2, x_2, t_1, x_1) = p(t_3, x_3 \mid t_2, x_2)$ ::: For a Markov process, this becomes the Chapman-Kolmogorov equation:

$$p(t_3, x_3 \mid t_1, x_1) = \int p(t_3, x_3 \mid t_2, x_2) p(t_2, x_2 \mid t_1, x_1)$$