##
**Green functions for second order parabolic integro-differential problems.**
*(English)*
Zbl 0806.45007

Pitman Research Notes in Mathematics Series. 275. Harlow: Longman Scientific & Technical. New York: Wiley. xvi, 417 p. (1992).

This book is concerned with the fundamental solution for the parabolic integro-differential equation \({\mathcal A} u = {\mathcal L} u - Iu = 0\) where
\[
{\mathcal A} \equiv \partial_ t - \sum^ d_{i,j=1} a_{ij} \partial_{ij} - \sum^ d_{i=1} a_ i \partial_ i + a_ 0
\]
is a linear parabolic differential operator of second order and \(I\) is the integral operator
\[
I\varphi (x,t) = \int_ F \biggl[ \varphi \bigl( x + j(x,t,\zeta) \bigr) - \varphi (x,t) \biggr] m(x,t, \zeta) \pi (d \zeta)
\]
or
\[
I \varphi (x,t) = \int_ F \biggl[ \varphi \bigl( x + j(x,t, \zeta) \bigr) - \varphi (x) - j(x,t,\zeta) \cdot \nabla \varphi (x) \biggr] m(x,t, \zeta) \pi (d \zeta),
\]
\(j(x,t, \zeta)\), \(m(x,t, \zeta)\) being measurable functions of \((x,t, \zeta) \in \mathbb{R}^ d \times [0,1] \times F\) and \(\pi (\cdot)\) being a \(\sigma\)-finite measure on the measure space \((F,{\mathcal F})\). The authors also construct the Green and Poisson functions for the Dirichlet, Neumann and oblique derivative initial- boundary value problems. These functions play a role of fundamental importance in the study of Markov-Feller processes.

Chapter I is devoted to the fundamental materials of basic function spaces, the a priori estimates and maximum principles for linear parabolic differential operators of second order, Markov-Feller processes, Ito’s formula, etc.

In Chapter II the integral operator \(I\) is investigated. The estimates for \(I\) depending on its order are established in various function spaces. It is shown that the maximum principles hold for the operator \({\mathcal A}\). Next, the authors establish the solvability of the initial value problem for \({\mathcal A}\) in \(\mathbb{R}^ d\) and that of the initial- boundary value problems for \({\mathcal A}\) in a general domain.

In Chapter III as an introductory problem the authors construct the fundamental solution of the initial value problem for the simple operator \(\partial_ t - {1 \over 2} \Delta-I\), and the Green and Poisson functions for the Dirichlet and Neumann problem of the same operator, in case where \(I\) is the jump operator \[ I \varphi (x) = \lambda \bigl[ \varphi (x+j) - \varphi (x) \bigr], \quad \lambda > 0, \quad j \in \mathbb{R}^ d, \] or an operator of the form \[ I \varphi (x) = \int_ F \biggl[ \varphi \bigl( x + j(\zeta) \bigr) - \varphi (x) \biggr] \pi (d \zeta). \] It is explained that the integro-differential term propagates the singularity at the origin unlike the case of differential operators. The authors describe a stochastic representation by showing that the transition probability of a Markov-Feller process is expressed as an integral operator with the fundamental solution, Green function or Poisson function constructed above as an integral kernel.

In Chapter IV the definition of the fundamental solution, Green function and Poisson function for the operator \({\mathcal A}\) is given. Their main properties such as the semigroup property, positivity, etc. are described. It is also shown that these functions are unique and it is explained how they are used to express the solutions of the initial value problem or the initial boundary value problem.

In Chapter V the fundamental solution of the differential operator \({\mathcal L}\) is constructed by Levi’s parametrix method. The solutions of the Cauchy problem are given using the fundamental solution. Next, the solutions of the Neumann problem and an oblique derivative problem are expressed as a single layer potential and the solution of the Dirichlet boundary value problem as a double layer potential.

Chapter VI is devoted to the construction of the Green and Poisson functions for the differential operator \({\mathcal L}\). The authors begin with the case of the heat equation in a half space as a model problem, and successively proceed to the general case. It is shown that the Green and Poisson functions exist uniquely for the Dirichlet problem, the Neumann problem and the oblique derivative problem and satisfy various desired estimates.

In Chapter VII a family of Banach spaces called Green (function) spaces is described. The introduction of such a family is necessitated by the presence of the propagation of the singularity explained in Chapter III. The method of successive approximations to construct the Green function of the integro-differential operator involves the solution of an equation of Volterra type. The solution is given as a series of kernels each of whose terms belongs to a Green space of an appropriate order. Integral transformations are investigated in this family of function spaces. Among other things the very important result of the commutativity of an integral operator and an integro-differential operator is established.

Under the preliminaries in the previous sections the Green function is constructed in Chapter VIII. The Green function has the form \(G = G_ 0 + G_ 0 \bullet Q\), where \(G_ 0\) is the principal part with the highest order singularity and \(G_ 1 = G_ 0 \bullet Q\) is the additional term, \(\bullet\) being a convolution in some sense. It is also shown that there exists a Markov-Feller process with the Green function thus constructed as the density of the transition function.

The aim of Chapter IX is to establish estimates of the derivatives of the Green function constructed in Chapter VIII. For that purpose the authors obtain estimates for \(\partial_ i G_ 1\), \(\partial_{ij} G_ 1\), \(\partial_ t G_ 1\) and the HĂ¶lder estimates for \(\partial_ i G_ 1\) with respect to the variable \(t\).

Chapter I is devoted to the fundamental materials of basic function spaces, the a priori estimates and maximum principles for linear parabolic differential operators of second order, Markov-Feller processes, Ito’s formula, etc.

In Chapter II the integral operator \(I\) is investigated. The estimates for \(I\) depending on its order are established in various function spaces. It is shown that the maximum principles hold for the operator \({\mathcal A}\). Next, the authors establish the solvability of the initial value problem for \({\mathcal A}\) in \(\mathbb{R}^ d\) and that of the initial- boundary value problems for \({\mathcal A}\) in a general domain.

In Chapter III as an introductory problem the authors construct the fundamental solution of the initial value problem for the simple operator \(\partial_ t - {1 \over 2} \Delta-I\), and the Green and Poisson functions for the Dirichlet and Neumann problem of the same operator, in case where \(I\) is the jump operator \[ I \varphi (x) = \lambda \bigl[ \varphi (x+j) - \varphi (x) \bigr], \quad \lambda > 0, \quad j \in \mathbb{R}^ d, \] or an operator of the form \[ I \varphi (x) = \int_ F \biggl[ \varphi \bigl( x + j(\zeta) \bigr) - \varphi (x) \biggr] \pi (d \zeta). \] It is explained that the integro-differential term propagates the singularity at the origin unlike the case of differential operators. The authors describe a stochastic representation by showing that the transition probability of a Markov-Feller process is expressed as an integral operator with the fundamental solution, Green function or Poisson function constructed above as an integral kernel.

In Chapter IV the definition of the fundamental solution, Green function and Poisson function for the operator \({\mathcal A}\) is given. Their main properties such as the semigroup property, positivity, etc. are described. It is also shown that these functions are unique and it is explained how they are used to express the solutions of the initial value problem or the initial boundary value problem.

In Chapter V the fundamental solution of the differential operator \({\mathcal L}\) is constructed by Levi’s parametrix method. The solutions of the Cauchy problem are given using the fundamental solution. Next, the solutions of the Neumann problem and an oblique derivative problem are expressed as a single layer potential and the solution of the Dirichlet boundary value problem as a double layer potential.

Chapter VI is devoted to the construction of the Green and Poisson functions for the differential operator \({\mathcal L}\). The authors begin with the case of the heat equation in a half space as a model problem, and successively proceed to the general case. It is shown that the Green and Poisson functions exist uniquely for the Dirichlet problem, the Neumann problem and the oblique derivative problem and satisfy various desired estimates.

In Chapter VII a family of Banach spaces called Green (function) spaces is described. The introduction of such a family is necessitated by the presence of the propagation of the singularity explained in Chapter III. The method of successive approximations to construct the Green function of the integro-differential operator involves the solution of an equation of Volterra type. The solution is given as a series of kernels each of whose terms belongs to a Green space of an appropriate order. Integral transformations are investigated in this family of function spaces. Among other things the very important result of the commutativity of an integral operator and an integro-differential operator is established.

Under the preliminaries in the previous sections the Green function is constructed in Chapter VIII. The Green function has the form \(G = G_ 0 + G_ 0 \bullet Q\), where \(G_ 0\) is the principal part with the highest order singularity and \(G_ 1 = G_ 0 \bullet Q\) is the additional term, \(\bullet\) being a convolution in some sense. It is also shown that there exists a Markov-Feller process with the Green function thus constructed as the density of the transition function.

The aim of Chapter IX is to establish estimates of the derivatives of the Green function constructed in Chapter VIII. For that purpose the authors obtain estimates for \(\partial_ i G_ 1\), \(\partial_{ij} G_ 1\), \(\partial_ t G_ 1\) and the HĂ¶lder estimates for \(\partial_ i G_ 1\) with respect to the variable \(t\).

Reviewer: H.Tanabe (Toyonaka)

### MSC:

45K05 | Integro-partial differential equations |

47G20 | Integro-differential operators |

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

60J35 | Transition functions, generators and resolvents |

60J25 | Continuous-time Markov processes on general state spaces |