## Selected Papers of E.B. Dynkin with CommentaryEugene Dynkin is a rare example of a contemporary mathematician who has achieved outstanding results in two quite different areas of research: algebra and probability. In both areas, his ideas constitute an essential part of modern mathematical knowledge and form a basis for further development. Although his last work in algebra was published in 1955, his contributions continue to influence current research in algebra and in the physics of elementary particles. His work in probability is part of both the historical and the modern development of the topic. This volume presents Dynkin's scientific contributions in both areas. Included are Commentary by recognized experts in the corresponding fields who describe the time, place, role, and impact of Dynkin's research and achievements. Biographical notes and the recollections of his students are also featured. |

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### Contents

Dynkin and Lie theory G M Seitz | 3 |

Classification of simple Lie groups | 23 |

Calculation of the coeffecients in the CampbellHausdorff formula | 31 |

Introduction 246 | 43 |

Chapter 3 Some problems in the theory of linear representations of semi | 58 |

Maximal subgroups of the classical groups 37 | 65 |

Irreducible subgroups of the unimodular group SL N with | 84 |

Irreducible subgroups of the symplectic group SpN with | 96 |

Markov processes and semigroups of operators | 437 |

Strong Markov Processes with A A Yushkevich | 449 |

Infinitesimal operators of Markov processes | 455 |

The natural topology and excessive functions connected with a Markov | 477 |

The optimum choice of the instant for stopping a Markov process | 485 |

Diffusion of tensors | 515 |

Determining functions of Markov processes and corresponding dual regular | 525 |

Comments by I V Evstigneev | 547 |

Supplement Survey of the basic concepts and facts in the theory of lin | 111 |

Comments by A L Onishchik | 171 |

Comments by A L Onishchik | 309 |

Topological characteristics of homomorphisms of compact Lie groups | 317 |

358 | |

Comments by A L Onishchik | 359 |

Tables | 364 |

Comments on the impact of Dynkins work on current research | 365 |

Lie groups in physics K Gottfried | 371 |

Introduction | 374 |

Dynkin and probability theory A A Yushkevich | 385 |

Necessary and sufficient statistics for a family of probability distributions | 393 |

Some limit theorems for sums of independent random variables with infinite | 417 |

Greens and Dirichlet spaces associated with fine Markov processes | 577 |

Markov processes as a tool in field theory | 615 |

Symmetric statistics Poisson point processes and multiple Wiener integrals | 637 |

Gaussian and nonGaussian random fields associated with Markov processes | 645 |

An application of flows to time shift and time reversal in stochastic | 675 |

Authors comments | 683 |

A probabilistic approach to one class of nonlinear differential equations | 711 |

Superdiffusions and parabolic nonlinear differential equations 739 | 761 |

To the history of strong Markov property A A Yushkevich | 773 |

On compactifications of symmetric spaces M A Olshanetsky 111 | 781 |

Dynkins work in mathematical economics I V Evstigneev | 793 |

### Common terms and phrases

adjoint adjoint representation algebra G Appendix arbitrary bilinear called Cartan subalgebra classification component conjugate consider contains Corollary corresponding decomposition defined denote diagram dimension distribution E. B. Dynkin elements equation equivalent exists extreme vector Feller process finite follows formula Gaussian given greatest vector greatest weight Hence homomorphism idempotent inner automorphism integral invariant irreducible representation irreducible with respect isomorphic Lemma Let f Let G Lie groups linear representation linear transformations mapping Markov processes Markov property Math matrices maximal subgroups measure necessary and sufficient obtained operator orthogonal pair positive problem Proof of Theorem proved random variable reducible regular subalgebra relation representation f representation of G roots of G scheme semisimple algebra semisimple Lie algebra semisimple subalgebras simple algebra simple Lie algebras simple roots SL(N solution Sp(N space stochastic subspace sufficient statistic symmetric symplectic system of simple Table Theorem theory three-dimensional subalgebras tion