Second order elliptic integro-differential problems.

*(English)*Zbl 1014.45002
Chapman & Hall/CRC Research Notes in Mathematics. 430. Boca Raton, FL: Chapman & Hall/CRC. xv, 221 p. (2002).

The investigation of stochastic processes with jumps has been accomplished recently through the analytical approach, in which the parabolic Green function and the Green spaces play a crucial role.

The first three chapters of the book under review cover the introductory material. In Chapter 1, necessary results, relative to the existence, uniqueness and regularity of classical, strong and weak solutions of second order elliptic problems, are stated. Also included in this chapter is an introduction to Markov-Feller processes. Chapter 2 focuses on integro-differential operators. In view of the nonlocal character of these operators, more difficulties arise in a bounded domain than in the whole space. Here needed assumptions are introduced on the structure of the jumps and nonconvex domains are also considered. Utilizing the localization assumptions introduced in Chapter 2, Chapter 3 gives a precise formulation of the Dirichlet problem in accordance with the class of functions one wants to include in the domain of the integro-differential operator. In this chapter, important a priori estimates, maximum principles, existence, uniqueness, and regularity results for elliptic integro-differential operators in divergence and nondivergence forms are obtained.

In Chapter 4, extending the work of the authors’ earlier book [Green functions for second order integral-differential problems, Pitman Research Notes in Mathematics, No. 275, Longman, Essex (1992; Zbl 0806.45007)], a summary of properties relative to the Green function for parabolic integro-differential operators of second order is provided.

Making use of the Green function constructed in Chapter 4, a study of invariant measure of the semigroup generated by the integro-differential operator and the boundary operator, is discussed in Chapter 5. This study, together with ergodic property, is utilized to consider the asymptotic behavior of the stationary problems as the limit of the solution of the corresponding parabolic problems. In Chapters 6 and 7, two applications of the preceding estimates are discussed, namely, stopping time problems and ergodic control problems of diffusions with jumps.

More than one hundred references are listed in the book. These research notes are timely and provide the required material, and therefore serves for the experts and nonexperts as well.

The first three chapters of the book under review cover the introductory material. In Chapter 1, necessary results, relative to the existence, uniqueness and regularity of classical, strong and weak solutions of second order elliptic problems, are stated. Also included in this chapter is an introduction to Markov-Feller processes. Chapter 2 focuses on integro-differential operators. In view of the nonlocal character of these operators, more difficulties arise in a bounded domain than in the whole space. Here needed assumptions are introduced on the structure of the jumps and nonconvex domains are also considered. Utilizing the localization assumptions introduced in Chapter 2, Chapter 3 gives a precise formulation of the Dirichlet problem in accordance with the class of functions one wants to include in the domain of the integro-differential operator. In this chapter, important a priori estimates, maximum principles, existence, uniqueness, and regularity results for elliptic integro-differential operators in divergence and nondivergence forms are obtained.

In Chapter 4, extending the work of the authors’ earlier book [Green functions for second order integral-differential problems, Pitman Research Notes in Mathematics, No. 275, Longman, Essex (1992; Zbl 0806.45007)], a summary of properties relative to the Green function for parabolic integro-differential operators of second order is provided.

Making use of the Green function constructed in Chapter 4, a study of invariant measure of the semigroup generated by the integro-differential operator and the boundary operator, is discussed in Chapter 5. This study, together with ergodic property, is utilized to consider the asymptotic behavior of the stationary problems as the limit of the solution of the corresponding parabolic problems. In Chapters 6 and 7, two applications of the preceding estimates are discussed, namely, stopping time problems and ergodic control problems of diffusions with jumps.

More than one hundred references are listed in the book. These research notes are timely and provide the required material, and therefore serves for the experts and nonexperts as well.

Reviewer: V.Lakshmikantham (Melbourne)

##### MSC:

45K05 | Integro-partial differential equations |

60J75 | Jump processes (MSC2010) |

45-02 | Research exposition (monographs, survey articles) pertaining to integral equations |

47G20 | Integro-differential operators |