This article is about infinitesimal generator for general stochastic processes. For generators for the special case of finite-state continuous time Markov chains, see transition rate matrix.
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[1] by
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[1]
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 .
Stochastic differential equations driven by Lévy processesedit
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.
As a simple example consider with a Brownian motion driving noise. If we assume are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol
The mean first passage time satisfies . This can be used to calculate, for example, the time it takes for a Brownian motion particle in a box to hit the boundary of the box, or the time it takes for a Brownian motion particle in a potential well to escape the well. Under certain assumptions, the escape time satisfies the Arrhenius equation.[2]
Calin, Ovidiu (2015). An Informal Introduction to Stochastic Calculus with Applications. Singapore: World Scientific Publishing. p. 315. ISBN978-981-4678-93-3. (See Chapter 9)