The Lévy Laplacian

The Lévy Laplacian is an infinite-dimensional generalization of the well-known classical Laplacian. The theory has become well developed in recent years and this book was the first systematic treatment of the Lévy–Laplace operator. The book describes the infinite-dimensional analogues of finite-dimensional results, and more especially those features which appear only in the generalized context. It develops a theory of operators generated by the Lévy Laplacian and the symmetrized Lévy Laplacian, as well as a theory of linear and nonlinear equations involving it. There are many problems leading to equations with Lévy Laplacians and to Lévy–Laplace operators, for example superconductivity theory, the theory of control systems, the Gauss random field theory, and the Yang–Mills equation. The book is complemented by an exhaustive bibliography. The result is a work that will be valued by those working in functional analysis, partial differential equations and probability theory.

Introduction; 1. The Lévy Laplacian; 2. Lévy–Laplace operators; 3. Symmetric Lévy–Laplace operators; 4. Harmonic functions of infinitely many variables; 5. Linear elliptic and parabolic equations with Lévy Laplacians; 6. Quasilinear and nonlinear elliptic equation with Lévy Laplacians; 7. Nonlinear parabolic equations with Lévy Laplacians; 8. Appendix. Lévy–Dirichlet forms and associated Markov processes; Bibliography; Index.

The Lévy Laplacian is an infinite-dimensional generalization of the well-known classical Laplacian. The theory has become well developed in recent years and this 2005 book was the first systematic treatment. With an extensive bibliography, the work will be valued by those working in functional analysis, partial differential equations and probability theory.