Feller process
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In mathematics, a Feller process is a particular kind of stochastic process.
[edit] Definitions
Let <math>X</math> be some locally compact topological space with a countable base. Let <math>C_{c} (X; \mathbb{R})</math> denote the space of all continuous functions <math>f : X \to \mathbb{R}</math> with compact support with the uniform norm.
A Feller semigroup on <math>C_{c} (X; \mathbb{R})</math> is a collection <math>\{ T_{t} \}_{t \geq 0}</math> of positive linear maps from <math>C_{c} (X; \mathbb{R})</math> to itself such that
- <math>T_{0} = \mathrm{id} : C_{c} (X; \mathbb{R}) \to C_{c} (X; \mathbb{R})</math> and <math>\| T_{t} \| \leq 1</math> for all <math>t \geq 0</math>;
- the semigroup property: <math>T_{t + s} = T_{t} \circ T_{s}</math> for all <math>s, t \geq 0</math>;
- <math>\lim_{t \to 0} \| T_{t} f - f \| = 0</math> for every <math>f \in C_{c} (X; \mathbb{R}).</math>
A Feller transition function is a probability transition function associated with a Feller semigroup.
A Feller process is a Markov process with a Feller transition function.
