Continuous Stochastic Processes

Abstract

In this chapter we define notions of stochastic continuity and differentiability and describe Lindeberg’s condition for continuity of stochastic Markovian trajectories. We also show that the Fokker–Planck equation describes a continuous stochastic process. Finally, we derive the stationary solutions of the Fokker–Planck equation and define potential function of dynamics.

Problems

4.1

Second statistical moment from Fokker–Planck equation

Suppose that the drift and diffusion coefficients in the Fokker–Planck equation are given by: $D^{(1)}(x) = -\gamma x$ and $D^{(2)}(x) = \alpha + \beta x^2$, where $\alpha , \beta , \gamma > 0$.

(a) Derive the stationary solution of Fokker–Planck equation with vanishing current.

(b) Show that second statistical moment $\langle x^2 \rangle$ diverges for $\beta \ge \gamma$.

4.2

The Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process is stationary at steady-state, and its correlation function is given by,

$$ f(\tau ) = \frac{1}{2\tau _c} \exp (-|\tau |/\tau _c) $$

where $\tau _c>0$ is a constant (correlation time scale).

Using (4.3) and (4.8), check the mean-square continuity and differentiability of the Ornstein-Uhlenbeck process for finite and vanishing (limit) $\tau _c$.