# Infinitesimal generator (stochastic processes)
In the theory of stochastic processes, the **infinitesimal generator** of a Markov process is the linear operator that describes how expected values of test functions change over an infinitesimally small time interval. For a Markov process X with transition semigroup (P_t), a function f belongs to the domain D(A) of the generator if the limit
\[ Af = \lim_{t \downarrow 0} \frac{P_t f - f}{t} \]
exists, and the operator A : D(A) → C₀ is then called the infinitesimal generator of the process[1](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf). The semigroup acts on functions by \( P_t f(x) = \mathbb{E}_x[f(X_t)] \), so the generator captures, in one operator, the local movement of the process that the full transition function describes only globally[1](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf) • [2](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95).
| Key fact | Detail |
|---|---|
| Definition | \( Af = \lim_{t \downarrow 0} (P_t f - f)/t \) on the domain D(A) of functions for which the limit exists[1](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf) |
| Probabilistic meaning | For f in D(A), \( \mathbb{E}(f(X_{t+h}) - f(X_t) \mid \mathcal{F}_t) = h\,Af(X_t) + o(h) \) as h ↓ 0[1](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf) |
| Brownian motion | One-dimensional Brownian motion has generator \( Lf(x) = \tfrac{1}{2} f''(x) \) on \( C_0(\mathbb{R}) \cap C^2(\mathbb{R}) \) with f′, f″ in C₀[2](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95) |
| Continuous-time Markov chains | The generator is the matrix with entries \( A_{ij} = \lambda(i,j) \) for i ≠ j and \( A_{ii} = -\lambda(i) \), where \( \lambda(i) = \sum_{j \neq i} \lambda(i,j) \)[3](https://u.math.biu.ac.il/~amirgi/SBA.pdf) |
| Kolmogorov equations | For f in D(A), \( \frac{d}{dt} P_t f = A P_t f = P_t A f \), the backward and forward equations[1](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf) |
| Reconstruction | The Hille–Yosida theorem gives a one-to-one correspondence between Markov generators and Markov semigroups, with \( S_t f = \lim_{n \to \infty} (I - \tfrac{t}{n} L)^{-n} f \)[2](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95) |
| Dynkin's formula | \( \mathbb{E}_x f(X_t) - f(x_0) = \mathbb{E}_x \int_0^t Af(X_s)\,ds \) for f in the domain[4](https://mathweb.tifr.res.in/sites/default/files/publications/ln/tifr75.pdf) |
## Definition and domain
A continuous-time Markov process determines a family of operators (P_t) satisfying the semigroup property \( P_t P_s = P_{t+s} \), together with contraction and positivity properties; for a Markov process on \( \mathbb{R}^n \), \( P_t f(x) = \mathbb{E}_x[f(X_t)] \)[5](https://repozitorij.pmf.unizg.hr/dissertations/pmf%3A5809/show-file/0). The generator is the derivative of this semigroup at time zero, the operator L for which the informal relation \( S_t = e^{tL} \) would hold[6](https://repository.tudelft.nl/file/File_daf19d6a-8366-4ab8-abb6-d4cefa116688).
**The domain is part of the object.** The limit defining Lf does not exist for every function: for the deterministic shift \( S_t f(x) = f(x+t) \) on \( \mathbb{R} \), the limit equals f′(x) only where f is differentiable, so the domain must be restricted to functions for which the limit exists[6](https://repository.tudelft.nl/file/File_daf19d6a-8366-4ab8-abb6-d4cefa116688). The domain carries essential information about the process: two processes can share the same formal differential expression but differ in boundary behavior, and the domain is what distinguishes them[2](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95). In practice the full domain is often hard to characterize, and it suffices to work on a core, a subspace D ⊂ D(L) whose closure under the graph norm recovers the generator[6](https://repository.tudelft.nl/file/File_daf19d6a-8366-4ab8-abb6-d4cefa116688).
## The generator as a local description
For f in the domain, the generator gives the expected instantaneous rate of change of f(X_t): conditionally on the history up to time t, \( \mathbb{E}(f(X_{t+h}) - f(X_t) \mid \mathcal{F}_t) = h\,Af(X_t) + o(h) \) as h ↓ 0[1](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf). Differentiating the semigroup in time yields the Kolmogorov backward and forward equations, \( \frac{d}{dt} P_t f = A P_t f = P_t A f \)[1](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf). Integrating the backward equation gives Dynkin's formula,
\[ \mathbb{E}_x f(X_t) - f(x_0) = \mathbb{E}_x \int_0^t Af(X_s)\,ds, \]
which connects the generator to martingale characterizations of Markov processes[4](https://mathweb.tifr.res.in/sites/default/files/publications/ln/tifr75.pdf).
## Examples
**Brownian motion.** One-dimensional Brownian motion has generator \( Lf(x) = \tfrac{1}{2} f''(x) \), with domain the twice continuously differentiable functions in C₀ whose first and second derivatives also vanish at infinity[2](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95). Changing the boundary behavior changes the domain while keeping the differential expression: [Brownian motion](https://www.edgepedia.org/brownian-motion) on \( [0,\infty) \) absorbed at zero requires f(0) = 0 and \( f''(0^+) = 0 \), while reflected Brownian motion requires \( f'(0^+) = 0 \), and a sticky Brownian motion introduces a stickiness parameter γ in the boundary condition \( \gamma f''(0^+) = f'(0^+) - f'(0^-) \)[2](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95).
**Continuous-time Markov chains.** For a chain with transition intensities λ(i, j) and total rate \( \lambda(i) = \sum_{j \neq i} \lambda(i,j) \), the generator is the matrix with \( A_{ij} = \lambda(i,j) \) off the diagonal and \( A_{ii} = -\lambda(i) \)[3](https://u.math.biu.ac.il/~amirgi/SBA.pdf). Equivalently, for a chain with rates c(x, y) on a state space Ω, the generator acts as \( Lf(x) = \sum_{y \in \Omega} c(x,y)\,(f(y) - f(x)) \)[6](https://repository.tudelft.nl/file/File_daf19d6a-8366-4ab8-abb6-d4cefa116688). On a finite state space the semigroup is the matrix exponential \( S_t = e^{tL} \)[2](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95), and the forward equations take the matrix form \( P'(t) = P(t)A \)[3](https://u.math.biu.ac.il/~amirgi/SBA.pdf). This generator matrix underlies the chemical master equation, the system of differential equations describing stochastic chemical reaction networks[7](https://pmc.ncbi.nlm.nih.gov/articles/PMC2867774/).
**Jump processes.** For a process that waits an exponential time with rate c_x and then jumps to y with probability c(x, y)/c_x, the probability of more than one jump in [0, t] is bounded by \( C t^2 \), which is why the generator involves only single jumps[6](https://repository.tudelft.nl/file/File_daf19d6a-8366-4ab8-abb6-d4cefa116688).
## Reconstructing the process: Hille–Yosida
The generator determines the semigroup, and under suitable regularity conditions the semigroup determines the process: a Markov semigroup satisfying such conditions gives rise to a right-continuous Markov process with \( S_t f(x) = \mathbb{E}_x[f(X_t)] \), for example, a Feller transition semigroup admits a Markov-process realization with càdlàg paths[6](https://repository.tudelft.nl/file/File_daf19d6a-8366-4ab8-abb6-d4cefa116688) • [16](https://d-nb.info/112150812X/34). The [Hille–Yosida theorem](https://www.edgepedia.org/hille-yosida-theorem) makes this correspondence precise in both directions. For a Feller semigroup on C₀(K), the domain of the generator is dense in C₀(K), the resolvent equation \( (\alpha I - A)u = f \) has a unique solution for each α > 0, and the [resolvent operators](https://www.edgepedia.org/resolvent-operator) \( (\alpha I - A)^{-1} \) are non-negative with norm at most 1/α; conversely, an operator satisfying these conditions is the generator of some Feller semigroup[8](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MC/MCde/Taira%20K.%20Boundary%20value%20problems%20and%20Markov%20processes%20%282ed.%2C%20LNM1499%2C%20Springer%2C%202009%29%28ISBN%203642016766%29%28T%29%28O%29%28205s%29_MCde_.pdf). The semigroup is recovered from the generator by the exponential formula \( S_t f = \lim_{n \to \infty} (I - \tfrac{t}{n} L)^{-n} f \), the rigorous meaning of \( e^{tL} \) when the [Taylor series](https://www.edgepedia.org/taylor-series) is unavailable[2](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95).
For Feller semigroups, a characterization via the positive maximum principle applies: an operator A whose domain is dense, satisfying \( Af(z) \leq 0 \) whenever f attains its maximum at z, and whose range condition holds, generates a Feller semigroup; this is the content of the Lumer–Phillips theorem[9](https://people.math.ethz.ch/~jteichma/lecture_7.html). This route is used to construct interacting particle systems, Feller processes arising in statistical physics, biology, economics, traffic flow, and [Monte Carlo](https://www.edgepedia.org/monte-carlo) studies, from their generators[10](https://wt.iam.uni-bonn.de/ferrari/teaching/lectures-homepages/markovprocessesws13).
## Historical development
The analytic theory begins with [Andrey Kolmogorov](https://www.edgepedia.org/andrey-kolmogorov)'s 1931 paper "Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung" (Mathematische Annalen, Vol. 104, pp. 415–458)[11](https://bishtref.com/articles/10.1007/bf01457949). Kolmogorov showed that a wide class of one-dimensional Markov processes is described by a second-order differential equation \( \frac{\partial u}{\partial t} + a\,\frac{\partial^2 u}{\partial x^2} + b\,\frac{\partial u}{\partial x} \), and the question whether higher-order operators could arise remained open until [William Feller](https://www.edgepedia.org/william-feller) resolved it in 1954–1955, showing that every one-dimensional Markov process with continuous paths is governed by an equation involving the generalized second derivative[12](https://epubs.siam.org/doi/abs/10.1137/1101004). Feller's 1940 paper had already investigated solutions of Kolmogorov's equations for jump Markov processes[13](https://arxiv.org/abs/1603.02367).
E. B. Dynkin, whose 1956 paper "Infinitesimal Operators of Markov Processes" appeared in Theory of Probability and Its Applications, developed a method based on path properties that extends to any number of dimensions, proving that every continuous Markov process in arbitrary dimension is governed by a generalized elliptic differential operator of second order[12](https://epubs.siam.org/doi/abs/10.1137/1101004). Dynkin's paper also fixed terminology: the class of processes whose semigroups take continuous functions to continuous functions, which Feller had assumed, are called Feller processes, and every [Feller process](https://www.edgepedia.org/feller-process) is a strong Markov process, though not conversely[12](https://epubs.siam.org/doi/abs/10.1137/1101004).
## Estimation from data
The generator is typically an unbounded operator, which complicates both analysis and statistics. Recent work addresses data-driven learning of the generator of stochastic diffusion processes, introducing frameworks based on energy functionals because conventional techniques for Hilbert–Schmidt operators do not apply to unbounded objects[14](https://arxiv.org/abs/2405.12940). A related practical problem arises for Markov jump processes observed only at discrete times: the generator must be estimated from a finite sample of states, and not every [stochastic matrix](https://www.edgepedia.org/stochastic-matrix) is embeddable as the time-t transition matrix of some generator, the classical embedding problem[15](https://www.sciencedirect.com/science/article/pii/S0021999107003361).
## References
1. [Markov processes, Section 3.5 Generators (lecture notes)](https://public.econ.duke.edu/~get/browse/courses/883/Spr16/COURSE-MATERIALS/Z_Papers/sp_new.pdf)
2. [Essentials of Markov Processes, Appendix A, Delft University of Technology](https://repository.tudelft.nl/file/File_735f08d5-eefd-46ea-991a-2f4ccc234a95)
3. [Continuous Time Markov Chains, Chapter 4 (Bar-Ilan University lecture notes)](https://u.math.biu.ac.il/~amirgi/SBA.pdf)
4. [Lectures on Markov processes, Tata Institute of Fundamental Research](https://mathweb.tifr.res.in/sites/default/files/publications/ln/tifr75.pdf)
5. [Markovian semigroups and generators, University of Zagreb dissertation](https://repozitorij.pmf.unizg.hr/dissertations/pmf%3A5809/show-file/0)
6. [Interacting Particle System Theory, Chapter 2, Delft University of Technology](https://repository.tudelft.nl/file/File_daf19d6a-8366-4ab8-abb6-d4cefa116688)
7. [Solving the chemical master equation using sliding windows, BMC Systems Biology (2010)](https://pmc.ncbi.nlm.nih.gov/articles/PMC2867774/)
8. [Kazuaki Taira, Boundary Value Problems and Markov Processes, Lecture Notes in Mathematics 1499, Springer (2009)](https://www.nzdr.ru/data/media/biblio/kolxoz/M/MC/MCde/Taira%20K.%20Boundary%20value%20problems%20and%20Markov%20processes%20%282ed.%2C%20LNM1499%2C%20Springer%2C%202009%29%28ISBN%203642016766%29%28T%29%28O%29%28205s%29_MCde_.pdf)
9. [Lecture 7: Feller semigroups and the Lumer–Phillips theorem, ETH Zürich](https://people.math.ethz.ch/~jteichma/lecture_7.html)
10. [Markov Processes, WS 2013/14, University of Bonn (Patrik Ferrari)](https://wt.iam.uni-bonn.de/ferrari/teaching/lectures-homepages/markovprocessesws13)
11. [A. Kolmogoroff (1931). Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen 104, 415–458](https://bishtref.com/articles/10.1007/bf01457949)
12. [E. B. Dynkin. Infinitesimal Operators of Markov Processes, Theory of Probability and Its Applications](https://epubs.siam.org/doi/abs/10.1137/1101004)
13. [Feinberg, Mandava, Shiryaev. Kolmogorov's Equations for Jump Markov Processes with Unbounded Jump Rates, arXiv:1603.02367](https://arxiv.org/abs/1603.02367)
14. [Learning the Infinitesimal Generator of Stochastic Diffusion Processes, arXiv:2405.12940](https://arxiv.org/abs/2405.12940)
15. [Generator estimation of Markov jump processes, Journal of Computational Physics](https://www.sciencedirect.com/science/article/pii/S0021999107003361)
16. [d-nb.info](https://d-nb.info/112150812X/34)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Statistics and probability › Stochastic processes › Markov chains and processes › Continuous-time Markov processes › Infinitesimal generators and transition semigroups*
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