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Lie groups and Lie algebras: Classification of Dynkin diagrams
The problem in Good Will Hunting - Numberphile
L02 6 Dynkin's identification theorem
Compact Lie groups and Lie algebras - Adjoint representation - Dynkin diagram
Transcription
Statement of the theorem
Let be a Feller process with infinitesimal generator.
For a point in the state-space of , let denote the law of given initial datum , and let denote expectation with respect to .
Then for any function in the domain of , and any stopping time with , Dynkin's formula holds:[1]
The infinitesimal generator of is defined by its action on compactly-supported (twice differentiable with continuous second derivative) functions as[2]
Since this is a Feller process, Dynkin's formula holds.[4]
In fact, if is the first exit time of a bounded set with , then Dynkin's formula holds for all functions , without the assumption of compact support.[4]
Application: Brownian motion exiting the ball
Dynkin's formula can be used to find the expected first exit time of a Brownian motion from the closed ball
which, when starts at a point in the interior of , is given by
This is shown as follows.[5] Fix an integerj. The strategy is to apply Dynkin's formula with , , and a compactly-supported with on . The generator of Brownian motion is , where denotes the Laplacian operator. Therefore, by Dynkin's formula,
Hence, for any ,
Now let to conclude that almost surely, and so
as claimed.
Dynkin, Eugene B.; trans. J. Fabius; V. Greenberg; A. Maitra; G. Majone (1965). Markov processes. Vols. I, II. Die Grundlehren der Mathematischen Wissenschaften, Bände 121. New York: Academic Press Inc. (See Vol. I, p. 133)
Øksendal, Bernt K. (2003). Stochastic Differential Equations: An Introduction with Applications (Sixth ed.). Berlin: Springer. ISBN3-540-04758-1. (See Section 7.4)
This page was last edited on 30 May 2024, at 19:09