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 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
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.
exists as uniform limit
{\displaystyle D(A)=\left\{f\in C_{0}(E):\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}}{\text{ exists as uniform limit}}\right\},}
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.
{\displaystyle Af(x)=-c(x)f(x)+l(x)\cdot \nabla f(x)+{\frac {1}{2}}{\text{div}}Q(x)\nabla f(x)+\int _{\mathbb {R} ^{d}\setminus {\{0\}}}\left(f(x+y)-f(x)-\nabla f(x)\cdot y\chi (|y|)\right)N(x,dy),}
is a Lévy triplet for fixed
The generator of Lévy semigroup is of the form
{\displaystyle Af(x)=l\cdot \nabla f(x)+{\frac {1}{2}}{\text{div}}Q\nabla f(x)+\int _{\mathbb {R} ^{d}\setminus {\{0\}}}\left(f(x+y)-f(x)-\nabla f(x)\cdot y\chi (|y|)\right)\nu (dy)}
denotes the Fourier transform.
So the generator of a Lévy process (or semigroup) is a Fourier multiplier operator with symbol
be a Lévy process with symbol
be locally Lipschitz and bounded.
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.
with a Brownian motion driving noise.
are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol
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] For finite-state continuous time Markov chains the generator may be expressed as a transition rate matrix.
The general n-dimensional diffusion process
The following are commonly used special cases for the general n-dimensional diffusion process.