I'm reading Stochastic Processes and Applications by Pavliotis, in Ch 2.5 Diffusion Processes and the Forward and Backward Kolmogorov Equations, it defines diffusion process as:
A Markov process $X_t$ in $\mathbb R$ with transition function $P(\Gamma, t | x,s)$ is called a diffusion process if the following conditions are satisfied.
- (Continuity). For every $x$ and every $\varepsilon>0$, $$\int_{|x-y|>\varepsilon} P(dy, t | x,s) = o(t-s)$$ uniformly over $s<t$.
- (Definition of drift coefficient). There exists a function $b(x,s)$ such that for every $x$ and every $\varepsilon>0$, $$\int_{|y-x|\le\varepsilon} (y-x) P(dy, t | x,s) = b(x,s)(t-s) + o(t-s)$$ uniformly over $s<t$.
- (Definition of diffusion coefficient). There exists a function $\Sigma(x,s)$ such that for every $x$ and every $\varepsilon>0$, $$\int_{|y-x|\le\varepsilon} (y-x)^2 P(dy, t | x,s) = \Sigma(x,s)(t-s) + o(t-s)$$ uniformly over $s<t$.
I don't quite get the meaning of the uniformly here. For example, in the first condition of Continuity, what does the uniformly mean?
My understanding of uniformly is more on calculus, regarding to a family of functions. For example, in Uniform convergence the definition is
Suppose $E$ is a set and ${(f_n)}_{n\in \mathbb N}$ is a sequence of real-valued functions on it. We say the sequence $({f_n})_{n\in \mathbb N}$ is uniformly convergent on $E$ with limit $f:E\to \mathbb R$ if for every $\epsilon > 0$, there exists a natural number $N$ such that for all $n\geq N$ and $x \in E$, $ |f_{n}(x)-f(x)|<\epsilon$.
But in the above diffusion process definition there is no family of functions, so what does the uniformly mean?