# Understanding two definitions of diffusion processes

1. From Wikipedia:

A diffusion process is a Markov process with continuous sample paths for which the Kolmogorov forward equation is the Fokker-Planck equation.

I was wondering if "for which the Kolmogorov forward equation is the Fokker-Planck equation" is redundant and therefore can be removed, because if I am correct, the Fokker-Planck equation is the same thing as the Kolmogorov forward equation.

2. From Lamperti's Stochastic Processes:

The transition function of a Brownian motion or Wiener process satisfy $$p_t(x, [x-\epsilon, x+\epsilon]) = 1-o(t)$$ for every $\epsilon >0$ and every $x \in \mathbb{R}$. This means that the Wiener process never stands still, but is in constant motion. The condition, however, implies that the process changes state not by jumps but by continuous motion. A Markov process with this property is called a diffusion.

Suppose a stochastic process (Markovian or not necessarily Markovian) satisfies $$p_t(x, [x-\epsilon, x+\epsilon]) = 1-o(t)$$ for every $\epsilon >0$ and every $x \in \mathbb{R}$.

• Why does it never stand still, but is in constant motion? If a process takes a constant value $x$ over all times and outcomes, it still satisfies the condition, doesn't it?

• The condition, however, implies that the process changes state not by jumps but by continuous motion. Does it mean the condition implies the process is continuous a.s.?

3. Is Lamperti's definition stronger than Wikipedia's definition?

Thanks and regards!

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