QUANTUM LÉVY PROCESSES, CLASSICAL PROBABILITY AND PHYSICS

By WILLIAM LEWIS

CONTENTS L´EVY PROCESSES IN EUCLIDEAN SPACES AND GROUPS INTRODUCTION LECTURE 1: IN?NITE DIVISIBILITY AND L´EVY PROCESSES IN EUCLIDEAN SPACE L´EVY PROCESSES LECTURE 2: SEMIGROUPS INDUCED BY L´EVY PROCESSES ANALYTIC DIVERSIONS GENERATORS OF L´EVY PROCESSES LP-MARKOV SEMIGROUPS AND L´EVY PROCESSES LECTURE 3: ANALYSIS OF JUMPS LECTURE 4: STOCHASTIC INTEGRATION LECTURE 5: L´EVY PROCESSES IN GROUPS LECTURE 6: TWO L´EVY PATHS TO QUANTUM STOCHASTICS REFERENCES LOCALLY COMPACT QUANTUM GROUPS ELEMENTARY C*-ALGEBRA THEORY LOCALLY COMPACT QUANTUM GROUPS IN THE C*-ALGEBRA SETTING COMPACT QUANTUM GROUPS WEIGHT THEORY ON VON NEUMANN ALGEBRAS THE DE?NITION OF A LOCALLY COMPACT QUANTUM GROUP EXAMPLES OF LOCALLY COMPACT QUANTUM GROUPS APPENDIX : SEVERAL CONCEPTS QUANTUM STOCHASTIC ANALYSIS – AN INTRODUCTION SPACES AND OPERATORS QS PROCESSES QS INTEGRALS QS DI?ERENTIAL EQUATIONS QS COCYCLES QS DILATION DILATIONS, COCYCLES AND PRODUCT SYSTEMS DILATION THEORY BASICS E0-SEMIGROUPS AND PRODUCT SYSTEMS DOMINATION AND MINIMALITY PRODUCT SYSTEMS: RECENT DEVELOPMENTS L´EVY PROCESSES IN EUCLIDEAN SPACES AND GROUPS SOME BASIC IDEAS OF PROBABILITY IN?NITE DIVISIBILITY EXAMPLES OF L´EVY PROCESSES SUBORDINATORS CONDITIONAL EXPECTATION, FILTRATIONS MARKOV AND FELLER PROCESSES SEMIGROUPS AND GENERATORS THE FOURIER TRANSFORM AND PSEUDO-DI?ERENTIAL OPERATORS SUBORDINATION OF SEMIGROUPS SELF-ADJOINT SEMIGROUPS MARTINGALES C`ADL`AG PATHS STOPPING TIMES THE JUMPS OF A L´EVY PROCESS - POISSON RANDOM MEASURES POISSON INTEGRATION PROCESSES OF FINITE VARIATION THE L´EVY-ITˆO DECOMPOSITION THE INTERLACING CONSTRUCTION ITˆO’S FORMULA QUADRATIC VARIATION AND ITˆO’S PRODUCT FORMULA STOCHASTIC DI?ERENTIAL EQUATIONS 10 1 L´EVY PROCESSES IN LOCALLY COMPACT GROUPS - THE BASICS 10 2 L´EVY PROCESSES IN LCA GROUPS 10 3 L´EVY PROCESSES IN LIE GROUPS 10 4 L´EVY PROCESSES ON LOCALLY COMPACT GROUPS - REPRISE 11 1 PATH 1: - UNITARY REPRESENTATIONS OF L´EVY PROCESSES IN LIE GROUPS 11 2 L´EVY PROCESSES IN FOCK SPACE DISTRIBUTION, THE LOG-NORMAL DISTRIBUTION, THE STABLE LAWS, THE NORMAL INVERSE GAUSSIAN AND THE HYPERBOLIC DISTRIBUTIONS THE BASIC IDEAS OF IN?NITE DIVISI- BILITY CHRYSTALLISED DURING THE HEROIC AGE OF CLASSICAL PROBABILITY IN THE 1920S AND 1930S - THE KEY RESULT IS THE BEAUTIFUL L´EVY-KHINTCHINE FORMULA WHICH GIVES THE GENERAL FORM OF THE CHARACTERISTIC FUNCTION FOR AN IN?NITELY DIVISIBLE PROBABILITY DISTRIBUTION ANOTHER IMPORTANT DISCOVERY FROM THIS ERA IS THAT