Inmathematics — specifically, instochastic analysis — theinfinitesimal generator of aFeller process (i.e. a continuous-time Markov process satisfying certain regularity conditions) is aFourier multiplier operator^[1] that encodes a great deal of information about the process.

The generator is used in evolution equations such as theKolmogorov backward equation, which describes the evolution of statistics of the process; itsL²Hermitian adjoint is used in evolution equations such as theFokker–Planck equation, also known as Kolmogorov forward equation, which describes the evolution of theprobability density functions of the process.

The Kolmogorov forward equation in the notation is just${\displaystyle {\mathrm{\partial }}_t\rho ={\mathcal{A}}^{\ast }\rho }$, where${\displaystyle \rho }$ is the probability density function, and${\displaystyle {\mathcal{A}}^{\ast }}$ is the adjoint of the infinitesimal generator of the underlying stochastic process. TheKlein–Kramers equation is a special case of that.

For aFeller process${\displaystyle (X_t{)}_{t\ge 0}}$ with Feller semigroup${\displaystyle T=(T_t{)}_{t\ge 0}}$ and state space${\displaystyle E}$ the generator${\displaystyle (A,D(A))}$ is defined as^[1]

Here${\displaystyle C_0(E)}$ denotes theBanach space of continuous functions on${\displaystyle E}$ vanishing at infinity, equipped with the supremum norm, and${\displaystyle T_{t}f(x)={\mathbb{E}}^{x}f(X_t)=\mathbb{E}(f(X_t)|X_0=x)}$. 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${\displaystyle X}$ is${\displaystyle {\mathbb{R}}^d}$-valued and${\displaystyle D(A)}$ contains thetest functions (compactly supported smooth functions) then^[1]$${\displaystyle Af(x)=-c(x)f(x)+l(x)\cdot \mathrm{\nabla }f(x)+\frac{1}{2}\text{div}Q(x)\mathrm{\nabla }f(x)+{\int }_{{\mathbb{R}}^d\setminus \{0\}}\left(f(x+y)-f(x)-\mathrm{\nabla }f(x)\cdot y\chi (|y|)\right)N(x,dy),}$$where${\displaystyle c(x)\ge 0}$, and${\displaystyle (l(x),Q(x),N(x,\cdot ))}$ is aLévy triplet for fixed${\displaystyle x\in {\mathbb{R}}^d}$.

The generator ofLévy semigroup is of the form$${\displaystyle Af(x)=l\cdot \mathrm{\nabla }f(x)+\frac{1}{2}\text{div}Q\mathrm{\nabla }f(x)+{\int }_{{\mathbb{R}}^d\setminus \{0\}}\left(f(x+y)-f(x)-\mathrm{\nabla }f(x)\cdot y\chi (|y|)\right)\nu (dy)}$$where${\displaystyle l\in {\mathbb{R}}^d,Q\in {\mathbb{R}}^{d\times d}}$ is positive semidefinite and${\displaystyle \nu }$ is aLévy measure satisfying and${\displaystyle 0\le 1-\chi (s)\le \kappa min(s,1)}$for some${\displaystyle \kappa >0}$ with${\displaystyle s\chi (s)}$ is bounded. If we define$${\displaystyle \psi (\xi )=\psi (0)-il\cdot \xi +\frac{1}{2}\xi \cdot Q\xi +{\int }_{{\mathbb{R}}^d\setminus \{0\}}(1-e^{iy\cdot \xi }+i\xi \cdot y\chi (|y|))\nu (dy)}$$for${\displaystyle \psi (0)\ge 0}$ then the generator can be written as$${\displaystyle Af(x)=-\int e^{ix\cdot \xi }\psi (\xi )\hat{f}(\xi )d\xi }$$where${\displaystyle \hat{f}}$ denotes the Fourier transform. So the generator of a Lévy process (or semigroup) is a Fourier multiplier operator with symbol${\displaystyle -\psi }$.

Let$L$ be a Lévy process with symbol${\displaystyle \psi }$ (see above). Let${\displaystyle \mathrm{\Phi }}$ be locally Lipschitz and bounded. The solution of the SDE${\displaystyle d{X}_t=\mathrm{\Phi }(X_{t-})d{L}_t}$ exists for each deterministic initial condition${\displaystyle x\in {\mathbb{R}}^d}$ and yields a Feller process with symbol${\displaystyle q(x,\xi )=\psi ({\mathrm{\Phi }}^{\mathrm{\top }}(x)\xi ).}$

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$d{X}_t=l(X_t)dt+\sigma (X_t)d{W}_t$ with a Brownian motion driving noise. If we assume${\displaystyle l,\sigma }$ are Lipschitz and of linear growth, then for each deterministic initial condition there exists a unique solution, which is Feller with symbol$${\displaystyle q(x,\xi )=-il(x)\cdot \xi +\frac{1}{2}\xi Q(x)\xi .}$$

The meanfirst passage time${\displaystyle T_1}$ satisfies${\displaystyle \mathcal{A}{T}_1=-1}$. 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 theArrhenius equation.^[2]

For finite-state continuous time Markov chains the generator may be expressed as atransition rate matrix.

The general n-dimensional diffusion process${\displaystyle d{X}_t=\mu (X_t,t)dt+\sigma (X_t,t)d{W}_t}$ has generator$${\displaystyle \mathcal{A}f=(\mathrm{\nabla }f{)}^T\mu +tr(({\mathrm{\nabla }}^{2}f)D)}$$where${\displaystyle D=\frac{1}{2}\sigma {\sigma }^T}$ is thediffusion matrix,${\displaystyle {\mathrm{\nabla }}^{2}f}$ is theHessian of the function${\displaystyle f}$, and${\displaystyle tr}$ is thematrix trace. Itsadjoint operator is^[2]$${\displaystyle {\mathcal{A}}^{\ast }f=-\sum _i{\mathrm{\partial }}_i(f{\mu }_i)+\sum _{ij}{\mathrm{\partial }}_{ij}(f{D}_{ij})}$$The following are commonly used special cases for the general n-dimensional diffusion process.

• Standard Brownian motion on${\displaystyle {\mathbb{R}}^n}$, which satisfies the stochastic differential equation${\displaystyle d{X}_t=d{B}_t}$, has generator$\frac{1}{2}\mathrm{\Delta }$, where${\displaystyle \mathrm{\Delta }}$ denotes theLaplace operator.
• The two-dimensional process${\displaystyle Y}$ satisfying:$${\displaystyle \mathrm{d}{Y}_t=(\genfrac{}{}{0ex}{}{\mathrm{d}t}{\mathrm{d}{B}_t})}$$ where${\displaystyle B}$ is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator:$${\displaystyle \mathcal{A}f(t,x)=\frac{\mathrm{\partial }f}{\mathrm{\partial }t}(t,x)+\frac{1}{2}\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2}(t,x)}$$
• TheOrnstein–Uhlenbeck process on${\displaystyle \mathbb{R}}$, which satisfies the stochastic differential equation$d{X}_t=\theta (\mu -X_t)dt+\sigma d{B}_t$, has generator:$${\displaystyle \mathcal{A}f(x)=\theta (\mu -x)f^{\prime }(x)+\frac{{\sigma }^2}{2}{f}^{″}(x)}$$
• Similarly, the graph of the Ornstein–Uhlenbeck process has generator:$${\displaystyle \mathcal{A}f(t,x)=\frac{\mathrm{\partial }f}{\mathrm{\partial }t}(t,x)+\theta (\mu -x)\frac{\mathrm{\partial }f}{\mathrm{\partial }x}(t,x)+\frac{{\sigma }^2}{2}\frac{{\mathrm{\partial }}^{2}f}{\mathrm{\partial }{x}^2}(t,x)}$$
• Ageometric Brownian motion on${\displaystyle \mathbb{R}}$, which satisfies the stochastic differential equation$d{X}_t=r{X}_{t}dt+\alpha X_{t}d{B}_t$, has generator:$${\displaystyle \mathcal{A}f(x)=rx{f}^{\prime }(x)+\frac{1}{2}{\alpha }^2{x}^2{f}^{″}(x)}$$

• Dynkin's formula

• 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. Universitext (Sixth ed.). Berlin: Springer.doi:10.1007/978-3-642-14394-6.ISBN 3-540-04758-1. (See Section 7.3)

• Stochastic differential equations

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