Infinitesimal generator (stochastic processes)

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 ${\displaystyle \partial _{t}\rho ={\mathcal {A}}^{*}\rho }${\displaystyle \partial _{t}\rho ={\mathcal {A}}^{*}\rho }, where ${\displaystyle \rho }${\displaystyle \rho } is the probability density function, and ${\displaystyle {\mathcal {A}}^{*}}${\displaystyle {\mathcal {A}}^{*}} 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 ${\displaystyle (X_{t})_{t\geq 0}}${\displaystyle (X_{t})_{t\geq 0}} with Feller semigroup ${\displaystyle T=(T_{t})_{t\geq 0}}${\displaystyle T=(T_{t})_{t\geq 0}} and state space ${\displaystyle E}${\displaystyle E} the generator ${\displaystyle (A,D(A))}${\displaystyle (A,D(A))} is defined as^[1]

$${\displaystyle {\begin{aligned}D(A)&=\left\{f\in C_{0}(E):\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}}{\text{ exists as uniform limit}}\right\},\\Af&=\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}},~~{\text{ for any }}f\in D(A).\\\end{aligned}}}$${\displaystyle {\begin{aligned}D(A)&=\left\{f\in C_{0}(E):\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}}{\text{ exists as uniform limit}}\right\},\\Af&=\lim _{t\downarrow 0}{\frac {T_{t}f-f}{t}},~~{\text{ for any }}f\in D(A).\\\end{aligned}}}

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

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

Let ${\textstyle L}${\textstyle L} be a Lévy process with symbol ${\displaystyle \psi }${\displaystyle \psi } (see above). Let ${\displaystyle \Phi }${\displaystyle \Phi } be locally Lipschitz and bounded. The solution of the SDE ${\displaystyle dX_{t}=\Phi (X_{t-})dL_{t}}${\displaystyle dX_{t}=\Phi (X_{t-})dL_{t}} exists for each deterministic initial condition ${\displaystyle x\in \mathbb {R} ^{d}}${\displaystyle x\in \mathbb {R} ^{d}} and yields a Feller process with symbol ${\displaystyle q(x,\xi )=\psi (\Phi ^{\top }(x)\xi ).}${\displaystyle q(x,\xi )=\psi (\Phi ^{\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 ${\textstyle dX_{t}=l(X_{t})dt+\sigma (X_{t})dW_{t}}${\textstyle dX_{t}=l(X_{t})dt+\sigma (X_{t})dW_{t}} with a Brownian motion driving noise. If we assume ${\displaystyle l,\sigma }${\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 $${\displaystyle q(x,\xi )=-il(x)\cdot \xi +{\frac {1}{2}}\xi Q(x)\xi .}$${\displaystyle q(x,\xi )=-il(x)\cdot \xi +{\frac {1}{2}}\xi Q(x)\xi .}

The mean first passage time ${\displaystyle T_{1}}${\displaystyle T_{1}} satisfies ${\displaystyle {\mathcal {A}}T_{1}=-1}${\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 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 ${\displaystyle dX_{t}=\mu (X_{t},t),円dt+\sigma (X_{t},t),円dW_{t}}${\displaystyle dX_{t}=\mu (X_{t},t),円dt+\sigma (X_{t},t),円dW_{t}} has generator$${\displaystyle {\mathcal {A}}f=(\nabla f)^{T}\mu +tr((\nabla ^{2}f)D)}$${\displaystyle {\mathcal {A}}f=(\nabla f)^{T}\mu +tr((\nabla ^{2}f)D)}where ${\displaystyle D={\frac {1}{2}}\sigma \sigma ^{T}}${\displaystyle D={\frac {1}{2}}\sigma \sigma ^{T}} is the diffusion matrix, ${\displaystyle \nabla ^{2}f}${\displaystyle \nabla ^{2}f} is the Hessian of the function ${\displaystyle f}${\displaystyle f}, and ${\displaystyle tr}${\displaystyle tr} is the matrix trace. Its adjoint operator is^[2] $${\displaystyle {\mathcal {A}}^{*}f=-\sum _{i}\partial _{i}(f\mu _{i})+\sum _{ij}\partial _{ij}(fD_{ij})}$${\displaystyle {\mathcal {A}}^{*}f=-\sum _{i}\partial _{i}(f\mu _{i})+\sum _{ij}\partial _{ij}(fD_{ij})}The following are commonly used special cases for the general n-dimensional diffusion process.

• Standard Brownian motion on ${\displaystyle \mathbb {R} ^{n}}${\displaystyle \mathbb {R} ^{n}}, which satisfies the stochastic differential equation ${\displaystyle dX_{t}=dB_{t}}${\displaystyle dX_{t}=dB_{t}}, has generator ${\textstyle {1 \over {2}}\Delta }${\textstyle {1 \over {2}}\Delta }, where ${\displaystyle \Delta }${\displaystyle \Delta } denotes the Laplace operator.
• The two-dimensional process ${\displaystyle Y}${\displaystyle Y} satisfying: $${\displaystyle \mathrm {d} Y_{t}={\mathrm {d} t \choose \mathrm {d} B_{t}}}$${\displaystyle \mathrm {d} Y_{t}={\mathrm {d} t \choose \mathrm {d} B_{t}}} where ${\displaystyle B}${\displaystyle B} is a one-dimensional Brownian motion, can be thought of as the graph of that Brownian motion, and has generator: $${\displaystyle {\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)}$${\displaystyle {\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+{\frac {1}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)}
• The Ornstein–Uhlenbeck process on ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} }, which satisfies the stochastic differential equation ${\textstyle dX_{t}=\theta (\mu -X_{t})dt+\sigma dB_{t}}${\textstyle dX_{t}=\theta (\mu -X_{t})dt+\sigma dB_{t}}, has generator: $${\displaystyle {\mathcal {A}}f(x)=\theta (\mu -x)f'(x)+{\frac {\sigma ^{2}}{2}}f''(x)}$${\displaystyle {\mathcal {A}}f(x)=\theta (\mu -x)f'(x)+{\frac {\sigma ^{2}}{2}}f''(x)}
• Similarly, the graph of the Ornstein–Uhlenbeck process has generator: $${\displaystyle {\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+\theta (\mu -x){\frac {\partial f}{\partial x}}(t,x)+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)}$${\displaystyle {\mathcal {A}}f(t,x)={\frac {\partial f}{\partial t}}(t,x)+\theta (\mu -x){\frac {\partial f}{\partial x}}(t,x)+{\frac {\sigma ^{2}}{2}}{\frac {\partial ^{2}f}{\partial x^{2}}}(t,x)}
• A geometric Brownian motion on ${\displaystyle \mathbb {R} }${\displaystyle \mathbb {R} }, which satisfies the stochastic differential equation ${\textstyle dX_{t}=rX_{t}dt+\alpha X_{t}dB_{t}}${\textstyle dX_{t}=rX_{t}dt+\alpha X_{t}dB_{t}}, has generator: $${\displaystyle {\mathcal {A}}f(x)=rxf'(x)+{\frac {1}{2}}\alpha ^{2}x^{2}f''(x)}$${\displaystyle {\mathcal {A}}f(x)=rxf'(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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