Preface -- I Feynman-Kac-type theorems and Gibbs measures -- 1 Heuristics and history -- 1.1 Feynman path integrals and Feynman-Kac formulae -- 1.2 Plan and scope -- 2 Probabilistic preliminaries -- 2.1 An invitation to Brownian motion -- 2.2 Martingale and Markov properties -- 2.2.1 Martingale property -- 2.2.2 Markov property -- 2.2.3 Feller transition kernels and generators -- 2.2.4 Conditional Wiener measure -- 2.3 Basics of stochastic calculus -- 2.3.1 The classical integral and its extensions -- 2.3.2 Stochastic integrals -- 2.3.3 Itô formula -- 2.3.4 Stochastic differential equations and diffusions -- 2.3.5 Girsanov theorem and Cameron-Martin formula -- 2.4 Lévy processes -- 2.4.1 Lévy process and Lévy-Khintchine formula -- 2.4.2 Markov property of Lévy processes -- 2.4.3 Random measures and Lévy-Itô decomposition -- 2.4.4 Itô formula for semimartingales -- 2.4.5 Subordinators -- 2.4.6 Bernstein functions -- 3 Feynman-Kac formulae -- 3.1 Schrödinger semigroups -- 3.1.1 Schrödinger equation and path integral solutions -- 3.1.2 Linear operators and their spectra -- 3.1.3 Spectral resolution -- 3.1.4 Compact operators -- 3.1.5 Schrödinger operators -- 3.1.6 Schrödinger operators by quadratic forms -- 3.1.7 Confining potential and decaying potential -- 3.1.8 Strongly continuous operator semigroups -- 3.2 Feynman-Kac formula for external potentials -- 3.2.1 Bounded smooth external potentials -- 3.2.2 Derivation through the Trotter product formula -- 3.3 Feynman-Kac formula for Kato-class potentials -- 3.3.1 Kato-class potentials -- 3.3.2 Feynman-Kac formula for Kato-decomposable potentials -- 3.4 Properties of Schrödinger operators and semigroups -- 3.4.1 Kernel of the Schrödinger semigroup -- 3.4.2 Number of eigenfunctions with negative eigenvalues.

3.4.3 Positivity improving and uniqueness of ground state -- 3.4.4 Degenerate ground state and Klauder phenomenon -- 3.4.5 Exponential decay of the eigenfunctions -- 3.5 Feynman-Kac-Itô formula for magnetic field -- 3.5.1 Feynman-Kac-Itô formula -- 3.5.2 Alternate proof of the Feynman-Kac-Itô formula -- 3.5.3 Extension to singular external potentials and vector potentials -- 3.5.4 Kato-class potentials and Lp-Lq boundedness -- 3.6 Feynman-Kac formula for relativistic Schrödinger operators -- 3.6.1 Relativistic Schrödinger operator -- 3.6.2 Relativistic Kato-class potentials and Lp-Lq boundedness -- 3.7 Feynman-Kac formula for Schrödinger operator with spin -- 3.7.1 Schrödinger operator with spin -- 3.7.2 A jump process -- 3.7.3 Feynman-Kac formula for the jump process -- 3.7.4 Extension to singular potentials and vector potentials -- 3.8 Feynman-Kac formula for relativistic Schrödinger operator with spin -- 3.9 Feynman-Kac formula for unbounded semigroups and Stark effect -- 3.10 Ground state transform and related diffusions -- 3.10.1 Ground state transform and the intrinsic semigroup -- 3.10.2 Feynman-Kac formula for P(f)1-processes -- 3.10.3 Dirichlet principle -- 3.10.4 Mehler's formula -- 4 Gibbs measures associated with Feynman-Kac semigroups -- 4.1 Gibbs measures on path space -- 4.1.1 From Feynman-Kac formulae to Gibbs measures -- 4.1.2 Definitions and basic facts -- 4.2 Existence and uniqueness by direct methods -- 4.2.1 External potentials: existence -- 4.2.2 Uniqueness -- 4.2.3 Gibbs measure for pair interaction potentials -- 4.3 Existence and properties by cluster expansion -- 4.3.1 Cluster representation -- 4.3.2 Basic estimates and convergence of cluster expansion -- 4.3.3 Further properties of the Gibbs measure -- 4.4 Gibbs measures with no external potential -- 4.4.1 Gibbs measure.

4.4.2 Diffusive behaviour -- II Rigorous quantumfield theory -- 5 Free Euclidean quantum field and Ornstein-Uhlenbeck processes -- 5.1 Background -- 5.2 Boson Fock space -- 5.2.1 Second quantization -- 5.2.2 Segal fields -- 5.2.3 Wick product -- 5.3 ℒ -spaces -- 5.3.1 Gaussian random processes -- 5.3.2 Wiener-Itô-Segal isomorphism -- 5.3.3 Lorentz covariant quantum fields -- 5.4 Existence of ℒ -spaces -- 5.4.1 Countable product spaces -- 5.4.2 Bochner theorem and Minlos theorem -- 5.5 Functional integration representation of Euclidean quantum fields -- 5.5.1 Basic results in Euclidean quantum field theory -- 5.5.2 Markov property of projections -- 5.5.3 Feynman-Kac-Nelson formula -- 5.6 Infinite dimensional Ornstein-Uhlenbeck process -- 5.6.1 Abstract theory of measures on Hilbert spaces -- 5.6.2 Fock space as a function space -- 5.6.3 Infinite dimensional Ornstein-Uhlenbeck-process -- 5.6.4 Markov property -- 5.6.5 Regular conditional Gaussian probability measures -- 5.6.6 Feynman-Kac-Nelson formula by path measures -- 6 The Nelson model by path measures 293 6.1 Preliminaries -- 6.2 The Nelson model in Fock space -- 6.2.1 Definition -- 6.2.2 Infrared and ultraviolet divergences -- 6.2.3 Embedded eigenvalues -- 6.3 The Nelson model in function space -- 6.4 Existence and uniqueness of the ground state -- 6.5 Ground state expectations -- 6.5.1 General theorems -- 6.5.2 Spatial decay of the ground state -- 6.5.3 Ground state expectation for second quantized operators -- 6.5.4 Ground state expectation for field operators -- 6.6 The translation invariant Nelson model -- 6.7 Infrared divergence -- 6.8 Ultraviolet divergence -- 6.8.1 Energy renormalization -- 6.8.2 Regularized interaction -- 6.8.3 Removal of the ultraviolet cutoff -- 6.8.4 Weak coupling limit and removal of ultraviolet cutoff.

7 The Pauli-Fierz model by path measures -- 7.1 Preliminaries -- 7.1.1 Introduction -- 7.1.2 Lagrangian QED -- 7.1.3 Classical variant of non-relativistic QED -- 7.2 The Pauli-Fierz model in non-relativistic QED -- 7.2.1 The Pauli-Fierz model in Fock space -- 7.2.2 The Pauli-Fierz model in function space -- 7.2.3 Markov property -- 7.3 Functional integral representation for the Pauli-Fierz Hamiltonian -- 7.3.1 Hilbert space-valued stochastic integrals -- 7.3.2 Functional integral representation -- 7.3.3 Extension to general external potential -- 7.4 Applications of functional integral representations -- 7.4.1 Self-adjointness of the Pauli-Fierz Hamiltonian -- 7.4.2 Positivity improving and uniqueness of the ground state -- 7.4.3 Spatial decay of the ground state -- 7.5 The Pauli-Fierz model with Kato class potential -- 7.6 Translation invariant Pauli-Fierz model -- 7.7 Path measure associated with the ground state -- 7.7.1 Path measures with double stochastic integrals -- 7.7.2 Expression in terms of iterated stochastic integrals -- 7.7.3 Weak convergence of path measures -- 7.8 Relativistic Pauli-Fierz model -- 7.8.1 Definition -- 7.8.2 Functional integral representation -- 7.8.3 Translation invariant case -- 7.9 The Pauli-Fierz model with spin -- 7.9.1 Definition -- 7.9.2 Symmetry and polarization -- 7.9.3 Functional integral representation -- 7.9.4 Spin-boson model -- 7.9.5 Translation invariant case -- 8 Notes and References -- Bibliography -- Index.