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^{[1]} 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 *L*^{2} Hermitian adjoint is used in evolution equations such as the Fokker–Planck equation (which describes the evolution of the probability density functions of the process).

## Definition

### General case

For a Feller process with Feller semigroup and state space we define the generator^{[2]} by

- ,

Where 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 but it is always closed and densely defined. If is a valued and contains the test functions (compactly supported smooth functions) then^{[3]}

where is for fixed a Lévy triplet.

### Lévy processes

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 .

### Stochastic differential equations driven by Lévy processes

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

## Generators of some common processes

- For finite-state continuous time Markov chains the generator may be expressed as a transition rate matrix
- Standard Brownian motion on , which satisfies the stochastic differential equation , has generator , where denotes the Laplace operator.
- The two-dimensional process satisfying:

- where is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator:

- The Ornstein–Uhlenbeck process on , which satisfies the stochastic differential equation , has generator:

- Similarly, the graph of the Ornstein–Uhlenbeck process has generator:

- A geometric Brownian motion on , which satisfies the stochastic differential equation , has generator:

## See also

## References

- Calin, Ovidiu (2015).
*An Informal Introduction to Stochastic Calculus with Applications*. Singapore: World Scientific Publishing. p. 315. ISBN 978-981-4678-93-3. (See Chapter 9) - Øksendal, Bernt K. (2003).
*Stochastic Differential Equations: An Introduction with Applications*(Sixth ed.). Berlin: Springer. doi:10.1007/978-3-642-14394-6. ISBN 3-540-04758-1. (See Section 7.3)

**^**Böttcher, Björn; Schilling, René; Wang, Jian (2013).*Lévy Matters III: Lévy-Type Processes: Construction, Approximation and Sample Path Properties*. Lévy Matters. Springer International Publishing. ISBN 978-3-319-02683-1.**^**Böttcher, Björn; Schilling, René; Wang, Jian (2013).*Lévy Matters III: Lévy-Type Processes: Construction, Approximation and Sample Path Properties*. Lévy Matters. Springer International Publishing. ISBN 978-3-319-02683-1.**^**Böttcher, Björn; Schilling, René; Wang, Jian (2013).*Lévy Matters III: Lévy-Type Processes: Construction, Approximation and Sample Path Properties*. Lévy Matters. Springer International Publishing. ISBN 978-3-319-02683-1.