Feller process

Remarks

Example: Brownian motion on $\R$ is a Feller process. Here $T_t f(x) = \int f(y) \frac{1}{\sqrt{2\pi t}} e^{-(y - x)^2/(2t)}\, dy$; this is in $C_0(\R)$ when $f$ is, it preserves $0 \le f \le 1$, and it tends to $f$ uniformly because $f$ is uniformly continuous. Non-example: on $E = \R$, let the process started at $x$ jump at once to $0$ if $x > 0$ and to $1$ if $x \le 0$, then stay there. Then $T_t f = f(0)$ on $(0, \infty)$ and $T_t f = f(1)$ on $(-\infty, 0]$, which is not continuous in $x$, so $T_t$ does not map $C_0(E)$ into itself. Clause 1 allows $T_t 1 < 1$, so the process may be killed; one then adds a cemetery point to $E$. Every Feller process has a modification with right-continuous paths with left limits, and that modification is strong Markov.