DIFFUSIONS AND ELLIPTIC OPERATORS
CONTENTS I STOCHASTIC DIFFERENTIAL EQUATIONS 1 PRELIMINARIES 2 PATHWISE SOLUTIONS 3 LIPSCHITZ COE?CIENTS 4 TYPES OF UNIQUENESS 5 MARKOV PROPERTIES 6 ONE-DIMENSIONAL CASE 7 EXAMPLES 8 SOME ESTIMATES 9 STRATONOVICH INTEGRALS 10 FLOWS 11 SDES WITH RE?ECTION 12 SDES WITH RE?ECTION: PATHWISE RESULTS 13 NOTES II REPRESENTATIONS OF SOLUTIONS 1 POISSON’S EQUATION 2 DIRICHLET PROBLEM 3 CAUCHY PROBLEM 4 SCHR¨ODINGER OPERATORS 5 GIRSANOV TRANSFORMATION 6 THE NEUMANN AND OBLIQUE DERIVATIVE PROBLEMS 7 FUNDAMENTAL SOLUTIONS AND GREEN FUNCTIONS 8 ADJOINTS 9 NOTES III REGULARITY OF SOLUTIONS 1 VARIATION OF PARAMETERS 2 WEIGHTED H¨OLDER NORMS 3 REGULARITY OF HITTING DISTRIBUTIONS 4 SCHAUDER ESTIMATES 5 DIRICHLET PROBLEM 6 EXTENSIONS 7 NEUMANN AND OBLIQUE DERIVATIVE PROBLEM 8 CALDER´ON-ZYGMUND ESTIMATES 9 FLOWS 10 NOTES IV ONE-DIMENSIONAL DIFFUSIONS 1 NATURAL SCALE 2 SPEED MEASURES 3 DI?USIONS AS SOLUTIONS OF SDES 4 BOUNDARIES 5 EIGENVALUE EXPANSIONS 6 NOTES V NONDIVERGENCE FORM OPERATORS 1 DE?NITIONS 2 SOME ESTIMATES 3 EXAMPLES 4 CONVEXITY 5 GREEN FUNCTIONS 6 RESOLVENTS 7 HARNACK INEQUALITY 8 EQUICONTINUITY AND APPROXIMATION VI MARTINGALE PROBLEMS 1 EXISTENCE 2 THE STRONG MARKOV PROPERTY 3 SOME USEFUL TECHNIQUES 4 SOME UNIQUENESS RESULTS 5 CONSEQUENCES OF UNIQUENESS 6 SUBMARTINGALE PROBLEMS 7 NOTES VII DIVERGENCE FORM OPERATORS 2 INEQUALITIES 3 MOSER’S HARNACK INEQUALITY 4 UPPER BOUNDS ON HEAT KERNELS 5 O?-DIAGONAL UPPER BOUNDS 6 LOWER BOUNDS 7 EXTENSIONS 8 PATH PROPERTIES VIII THE MALLIAVIN CALCULUS 1 INTEGRATION BY PARTS FORMULA 2 SMOOTH FUNCTIONALS 3 A CRITERION FOR SMOOTH DENSITIES 4 VECTOR ?ELDS 5 H¨ORMANDER’S THEOREM 6 AN ALTERNATIVE APPROACH