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Concept
Infinitesimal generator (stochastic processes)
Summary
In mathematics — specifically, in stochastic analysis — the infinitesimal generator of a Feller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is a Fourier multiplier operator that encodes a great deal of information about the process.
The generator is used in evolution equations such as the Kolmogorov backward equation, which describes the evolution of statistics of the process; its L2 Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation, also known as Kolmogorov forward equation, which describes the evolution of the probability density functions of the process.
The Kolmogorov forward equation in the notation is just , where is the probability density function, and is the adjoint of the infinitesimal generator of the underlying stochastic process. The Klein–Kramers equation is a special case of that.
For a Feller process with Feller semigroup and state space we define the generator by
for any .
Here denotes the Banach space of continuous functions on vanishing at infinity, equipped with the supremum norm, and . In general, it is not easy to describe the domain of the Feller generator. However, the Feller generator is always closed and densely defined. If is -valued and contains the test functions (compactly supported smooth functions) then
where , and is a Lévy triplet for fixed .
The generator of Lévy semigroup is of the form
where is positive semidefinite and is a Lévy measure satisfying
and for some with is bounded. If we define
for then the generator can be written as
where denotes the Fourier transform. So the generator of a Lévy process (or semigroup) is a Fourier multiplier operator with symbol .
Let be a Lévy process with symbol (see above). Let be locally Lipschitz and bounded. The solution of the SDE exists for each deterministic initial condition and yields a Feller process with symbol
Note that in general, the solution of an SDE driven by a Feller process which is not Lévy might fail to be Feller or even Markovian.
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