Theory of Dirichlet forms and applications.

*(English)*Zbl 1043.60063
Bernard, Pierre (ed.), Lectures on probability theory and statistics. Ecole d’eté de probabilités de Saint-Flour XXX – 2000. Berlin: Springer (ISBN 3-540-40335-3/pbk). Lect. Notes Math. 1816, 1-106 (2003).

In his lecture notes S. Albeverio reviews the theory of Dirichlet forms, Markov semigroups and associated processes on finite- and infinite-dimensional spaces. The applications in the notes include stochastic differential equations, stochastic dynamics of lattice or continuous classical and quantum systems, quantum fields, and the geometry of loop spaces.

The Introduction in Chapter 0 reviews the history of Dirichlet forms. Chapter 1 reviews the functional analytic background, i.e. semigroups, generators, and resolvents. Chapter 2 investigates the relation between closed symmetric coercive forms and \(C_0\)-contraction semigroups. Chapter 3 presents contraction properties of forms and their relation to positivity preserving and sub-Markovian semigroups. Dirichlet forms are defined and examples are given. Finally, the Beurling-Deny structure theorem for Dirichlet forms is stated. Chapter 4 investigates the relation of Markov processes with Dirichlet forms. Chapter 5 analyzes diffusions and stochastic differential equations associated with classical Dirichlet forms. Chapter 6 presents applications: the stochastic quantization equation and quantum fields, and diffusions on configuration spaces and classical statistical mechanics. For the following further applications, problems and topics connected with Dirichlet forms several references are given: classical spin systems, natural measures and diffusion processes associated with individual and lattice loop spaces, polymers, non-symmetric Dirichlet forms and generalized Dirichlet forms, complex-valued Dirichlet forms, invariant measures for singular processes, subordination of diffusions given by Dirichlet forms, time dependent Dirichlet forms, differential operators and processes with boundary conditions, convergence of Dirichlet forms, Dirichlet forms and geometry, Dirichlet forms and processes on fractals, discrete structures and metric measure spaces, harmonic mappings and non-linear Dirichlet forms, and non commutative and supersymmetric Dirichlet forms and processes. In his lecture notes the author provides more than \(500\) references for further study.

For the entire collection see [Zbl 1015.00013].

The Introduction in Chapter 0 reviews the history of Dirichlet forms. Chapter 1 reviews the functional analytic background, i.e. semigroups, generators, and resolvents. Chapter 2 investigates the relation between closed symmetric coercive forms and \(C_0\)-contraction semigroups. Chapter 3 presents contraction properties of forms and their relation to positivity preserving and sub-Markovian semigroups. Dirichlet forms are defined and examples are given. Finally, the Beurling-Deny structure theorem for Dirichlet forms is stated. Chapter 4 investigates the relation of Markov processes with Dirichlet forms. Chapter 5 analyzes diffusions and stochastic differential equations associated with classical Dirichlet forms. Chapter 6 presents applications: the stochastic quantization equation and quantum fields, and diffusions on configuration spaces and classical statistical mechanics. For the following further applications, problems and topics connected with Dirichlet forms several references are given: classical spin systems, natural measures and diffusion processes associated with individual and lattice loop spaces, polymers, non-symmetric Dirichlet forms and generalized Dirichlet forms, complex-valued Dirichlet forms, invariant measures for singular processes, subordination of diffusions given by Dirichlet forms, time dependent Dirichlet forms, differential operators and processes with boundary conditions, convergence of Dirichlet forms, Dirichlet forms and geometry, Dirichlet forms and processes on fractals, discrete structures and metric measure spaces, harmonic mappings and non-linear Dirichlet forms, and non commutative and supersymmetric Dirichlet forms and processes. In his lecture notes the author provides more than \(500\) references for further study.

For the entire collection see [Zbl 1015.00013].

Reviewer: Stefan Weber (Berlin)