In mathematics, a Rota–Baxter algebra is an algebra, usually over a field k, together with a particular k-linear map R which satisfies the weight-θ Rota–Baxter identity. It appeared first in the work of the American mathematician Glen E. Baxter in the realm of probability theory. Baxter's work was further explored from different angles by Gian-Carlo Rota, Pierre Cartier, and Frederic V. Atkinson, among others. Baxter’s derivation of this identity that later bore his name emanated from some of the fundamental results of the famous probabilist Frank Spitzer in random walk theory.
Definition and first properties
Let A be a k-algebra with a k-linear map R on A and let θ be a fixed parameter in k. We call A a Rota-Baxter k-algebra and R a Rota-Baxter operator of weight θ if the operator R satisfies the following Rota–Baxter relation of weight θ:
The operator R:= θ id − R also satisfies the Rota–Baxter relation of weight θ.
Integration by Parts
Integration by parts is an example of a Rota–Baxter algebra of weight 0. Let be the algebra of continuous functions from the real line to the real line. Let : be a continuous function. Define integration as the Rota–Baxter operator
Let G(x) = I(g)(x) and F(x) = I(f)(x). Then the formula for integration for parts can be written in terms of these variables as
In other words
which shows that I is a Rota–Baxter algebra of weight 0.
The Spitzer identity appeared is named after the American mathematician Frank Spitzer. It is regarded as a remarkable stepping stone in the theory of sums of independent random variables in fluctuation theory of probability. It can naturally be understood in terms of Rota–Baxter operators.
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- Spitzer, F. (1956). "A combinatorial lemma and its application to probability theory". Trans. Amer. Math. Soc. 82 (2): 323–339. doi:10.1090/S0002-9947-1956-0079851-X.
- Spitzer, F. (1976). "Principles of random walks". Graduate Texts in Mathematics. 34 (Second ed.). New York, Heidelberg: Springer-Verlag.
- Li Guo. WHAT IS...a Rota-Baxter Algebra? Notices of the AMS, December 2009, Volume 56 Issue 11