EIGENVALUES, INEQUALITIES, AND ERGODIC THEORY
CONTENTS CHAPTER AN OVERVIEW OF THE BOOK INTRODUCTION NEW VARIATIONAL FORMULA FOR THE ?RST EIGENVALUE BASIC INEQUALITIES AND NEW FORMS OF CHEEGER’S CONSTANTS A NEW PICTURE OF ERGODIC THEORY AND EXPLICIT CRITERIA CHAPTER OPTIMAL MARKOVIAN COUPLINGS COUPLINGS AND MARKOVIAN COUPLINGS OPTIMALITY WITH RESPECT TO DISTANCES OPTIMALITY WITH RESPECT TO CLOSED FUNCTIONS APPLICATIONS OF COUPLING METHODS NEW VARIATIONAL FORMULAS FOR THE FIRST EIGENVALUE BACKGROUND PARTIAL PROOF IN THE DISCRETE CASE THE THREE STEPS OF THE PROOF IN THE GEOMETRIC CASE TWO DI?CULTIES THE ?NAL STEP OF THE PROOF OF THE FORMULA COMMENTS ON DI?ERENT METHODS PROOF IN THE DISCRETE CASE (CONTINUED) THE ?RST DIRICHLET EIGENVALUE GENERALIZED CHEEGER’S METHOD CHEEGER’S METHOD A GENERALIZATION NEW RESULTS SPLITTING TECHNIQUE AND EXISTENCE CRITERION PROOF OF THEOREM 4 4 LOGARITHMIC SOBOLEV INEQUALITY UPPER BOUNDS NASH INEQUALITY BIRTH–DEATH PROCESSES TEN EXPLICIT CRITERIA IN DIMENSION ONE THREE TRADITIONAL TYPES OF ERGODICITY THE ?RST (NONTRIVIAL) EIGENVALUE (SPECTRAL GAP) THE ?RST EIGENVALUES AND EXPONENTIALLY ERGODIC RATE EXPLICIT CRITERIA EXPONENTIAL ERGODICITY FOR SINGLE BIRTH PROCESSES STRONG ERGODICITY POINCAR´E-TYPE INEQUALITIES IN DIMENSION ONE ORDINARY POINCAR´E INEQUALITIES EXTENSION: NORMED LINEAR SPACES NEUMANN CASE: ORLICZ SPACES NASH INEQUALITY AND SOBOLEV-TYPE INEQUALITY PARTIAL PROOFS OF THEOREM 6 1 FUNCTIONAL INEQUALITIES STATEMENT OF RESULTS SKETCH OF THE PROOFS COMPARISON WITH CHEEGER’S METHOD GENERAL CONVERGENCE SPEED TWO FUNCTIONAL INEQUALITIES ALGEBRAIC CONVERGENCE GENERAL (IRREVERSIBLE) CASE A DIAGRAM OF NINE TYPES OF ERGODICITY STATEMENTS OF RESULTS APPLICATIONS AND COMMENTS PROOF OF THEOREM 1 9 REACTION–DI?USION PROCESSES THE MODELS FINITE-DIMENSIONAL CASE CONSTRUCTION OF THE PROCESSES ERGODICITY AND PHASE TRANSITIONS HYDRODYNAMIC LIMITS STOCHASTIC MODELS OF ECONOMIC OPTIMIZATION 10 1 INPUT–OUTPUT METHOD 10 2 L K HUA’S FUNDAMENTAL THEOREM 10 3 STOCHASTIC MODEL WITHOUT CONSUMPTION 10 4 STOCHASTIC MODEL WITH CONSUMPTION 10 5 PROOF OF THEOREM 10 4 REFERENCES