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Feller process
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Feller process
In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process.
Let X be a locally compact Hausdorff space with a countable base. Let C0(X) denote the space of all real-valued continuous functions on X that vanish at infinity, equipped with the sup-norm ||f ||. From analysis, we know that C0(X) with the sup norm is a Banach space.
A Feller semigroup on C0(X) is a collection {Tt}t ≥ 0 of positive linear maps from C0(X) to itself such that
Warning: This terminology is not uniform across the literature. In particular, the assumption that Tt maps C0(X) into itself is replaced by some authors by the condition that it maps Cb(X), the space of bounded continuous functions, into itself. The reason for this is twofold: first, it allows including processes that enter "from infinity" in finite time. Second, it is more suitable to the treatment of spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.
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.
Feller processes (or transition semigroups) can be described by their infinitesimal generator. A function f in C0 is said to be in the domain of the generator if the uniform limit
exists. The operator A is the generator of Tt, and the space of functions on which it is defined is written as DA.
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Feller process
In probability theory relating to stochastic processes, a Feller process is a particular kind of Markov process.
Let X be a locally compact Hausdorff space with a countable base. Let C0(X) denote the space of all real-valued continuous functions on X that vanish at infinity, equipped with the sup-norm ||f ||. From analysis, we know that C0(X) with the sup norm is a Banach space.
A Feller semigroup on C0(X) is a collection {Tt}t ≥ 0 of positive linear maps from C0(X) to itself such that
Warning: This terminology is not uniform across the literature. In particular, the assumption that Tt maps C0(X) into itself is replaced by some authors by the condition that it maps Cb(X), the space of bounded continuous functions, into itself. The reason for this is twofold: first, it allows including processes that enter "from infinity" in finite time. Second, it is more suitable to the treatment of spaces that are not locally compact and for which the notion of "vanishing at infinity" makes no sense.
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.
Feller processes (or transition semigroups) can be described by their infinitesimal generator. A function f in C0 is said to be in the domain of the generator if the uniform limit
exists. The operator A is the generator of Tt, and the space of functions on which it is defined is written as DA.