Feller process

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In mathematics, a Feller process is a particular kind of stochastic process.

[edit] Definitions

Let $X$ be some locally compact topological space with a countable base. Let $C_{c} (X; \mathbb{R})$ denote the space of all continuous functions $f : X \to \mathbb{R}$ with compact support with the uniform norm.

A Feller semigroup on $C_{c} (X; \mathbb{R})$ is a collection $\{ T_{t} \}_{t \geq 0}$ of positive linear maps from $C_{c} (X; \mathbb{R})$ to itself such that

$T_{0} = \mathrm{id} : C_{c} (X; \mathbb{R}) \to C_{c} (X; \mathbb{R})$ and $\| T_{t} \| \leq 1$ for all $t \geq 0$ ;
the semigroup property: $T_{t + s} = T_{t} \circ T_{s}$ for all $s, t \geq 0$ ;
$\lim_{t \to 0} \| T_{t} f - f \| = 0$ for every $f \in C_{c} (X; \mathbb{R}).$