Potential Theory on Infinite-Dimensional Abelian Groups

Contents
Chapter 1. Introduction
Chapter 2. Elements of potential theory . .
2.1 Notation . . . . . . . . . . . . . . . .
2.2 Ilarmonic and hyperharmonic sheaves.
2.3 The generalized Dirichlet problem.
2.4 Harmonic spaces . . . .
2.5 Brelot and Bauer spaces
2.6 Smooth Bauer spaces .
2.7 Markov processes . . .
2.8 Markov processes on harmonic spaces
2.9 Probability interprctations
2.10 Duality . . . . . . . . . . . . . . . . .
Chapter 3. Markov processes and harmonic structures 19
3.1 Markov processes and Brelot spaces ....... 19
3.2 Markov processes and Bauer spaces. . . . . . . . 24
3.3 Projective scqucnccs of hannonic spaces: examplcs. definitions, state-
ments of theorems . . . . . . . . . . . . . . . . . . . . . . . . . . .. 33
3.4 Projective sequences of hannonic spaces: proofs of theorems ..... 43
3.5 Projective sequences of hannonic spaces: some remarks on hannonic
functions on a Wiener space. . . . . . . . . . . . . . . . . . . . . .. 67
Chapter 4. Markov processes and harmonic structures on a group 72
4.1 Harmonic groups. . . . . . . . . . . . . . . . . . . . . . . . 73
4.2 Space-homogeneous processes and harmonic functions . . . . 81
4.3 Space homogeneous processes and hannonic functions: quasidiagonal
case .................. 92
4.4 Bony's theorem on the group IRP x Toe:: . . . . . . . . . . . . . . . . . 101
Chapter 5. Elliptic equations on a group
5.1 Admissible distributions and multipliers. . . . .
5.2 Weak solutions of elliptic equations (Lp-theory)
5.3 Weyl's lemma and the hypoelliptic property
5.4 Bessel potentials on group '])'00 .........
Chapter 6. Special classes of harmonic functions and potentials
6.1 Spaces At,:; of martingales with mixed norm ........
6.2 Classes hp of harmonic functions in the semispace . . .
6.3 ..U p -estimates of potentials. Sobolev inequality on group -roo
Chapter 7. Some thouRhts on probability and analysis on locally compact
groups . . . . . . . . . 166
7. ] Dichotomy problem . . . . . . 166
7.2 Hannonic functions on a group 169
7.3 The problem of hypoellipticity 170
7.4 "Can one hear the shape of a drum?" 171
7.5 Geometry on a group ........ ]72
Bibliography. 175
Index. . . . . 183