Chapter 1. Introduction -- Chapter 2. Elements of potential theory -- 2.1 Notation -- 2.2 Harmonic 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 interpretations -- 2.10 Duality -- Chapter 3. Markov processes and harmonic structures -- 3.1 Markov processes and Brelot spaces -- 3.2 Markov processes and Bauer spaces -- 3.3 Projective sequences of harmonic spaces: examples, definitions, statements of theorems -- 3.4 Projective sequences of harmonic spaces: proofs of theorems -- 3.5 Projective sequences of harmonic spaces: some remarks on harmonic functions on a Wiener space -- Chapter 4. Markov processes and harmonic structures on a group -- 4.1 Harmonic groups -- 4.2 Space-homogeneous processes and harmonic functions -- 4.3 Space homogeneous processes and harmonic functions: quasidiagonal case -- 4.4 Bony's theorem on the group ℝp ×T∞ -- 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 T∞ -- Chapter 6. Special classes of harmonic functions and potentials -- 6.1 Spaces Mp⃗ of martingales with mixed norm -- 6.2 Classes hp⃗ of harmonic functions in the semispace T∞+ -- 6.3 Mp⃗ -estimates of potentials. Sobolev inequality on group T∞ -- Chapter 7. Some thoughts on probability and analysis on locally compact groups -- 7.1 Dichotomy problem -- 7.2 Harmonic functions on a group -- 7.3 The problem of hypoellipticity -- 7.4 "Can one hear the shape of a drum?" -- 7.5 Geometry on a group -- Bibliography -- Index.