Potential theory on graphs and manifolds.
(Théorie du potentiel sur les graphes et les variétés.)

*(French)*Zbl 0719.60074
Calcul de probabilités, Éc. d’Été XVIII, Saint-Flour/Fr. 1988, Lect. Notes Math. 1427, 5-112 (1990).

This expository paper of the potential theory on the Riemannian manifolds and graphs constitutes from five chapters and the related references.

In Chapter 1, the fundamental concepts of the potential theory such as harmonic functions, potentials, balayage, etc. for an elliptic or parabolic operator \(L\) on a manifold \(M\) are stated.

In Chapter 2, for a Greenian pair \((M,L)\), the concepts of the Martin boundary and its properties including the Fatou-Doob-Naïm limit theorem concerning the existence of fine limit at the minimal boundary points of the function of the form \(u/v\) for \(u\in\mathcal S_+\), \(v\in\mathcal H_+\) are stated. Also, when \(L(1)=0\), its probabilistic interpretation is stated. A countable set \(X\) with symmetric, reflexive relation \(\Gamma\) is called a graph if, for any \(a,b\in X\), there exist \(x_0=a\), \(x_1,\ldots,x_m=b\), such that \(x_j\Gamma x_{j+1}\).

In Chapter 3, the author concerns to the recurrence and transience of the graphs and the manifolds. For a selfadjoint operator \(L\), \(a(\phi,\psi)\) be the Dirichlet form on \(L^2(M: \sigma)\). Then \((M,L)\) is transient if and only if there exists an open set \(V\) and a constant \(C<0\) such that

\[ (\int_{V}\phi \,d\sigma)^2 \leq Ca(\phi,\psi)\quad \forall \phi \in C^2_0(M),\quad \phi >0. \]

This characterization is applied to show the stability of transience under some mappings. It is also used to give a characterization of recurrent groups.

Chapter 4 is devoted to the study of the Liouville property and its applications to the Laplace-Beltrami operator on \(M\). Some sufficient conditions for the property are given.

In Chapter 5, under conditions on \((M,L)\), including the condition that \(L+tI\) is transient for some \(t>0\), the Harnack inequality at infinity (there exists a constant \(c\) such that \(G(x_m,x_1) \leq cG(x_m,x_k)G(x_k,x_1)\) for any “\(\phi\)-chain” \(\{x_k\})\) is proved, where \(G\) is the Green function. In particular if \(M\) is hyperbolic, then for any geodesic segment \(xz\) and any point \(y\) on it such that \(\min[d(x,y),d(y,z)] \geq 1\), it holds that

\[ c^{-1}G(x,y)G(y,z) \leq G(x,z) \leq cG(x,y)G(y,z). \]

As an application of this result, the identification of the Martin compactification and the geometric one is shown. Some other applications are also given.

[For the entire collection see Zbl 0708.00013.]

In Chapter 1, the fundamental concepts of the potential theory such as harmonic functions, potentials, balayage, etc. for an elliptic or parabolic operator \(L\) on a manifold \(M\) are stated.

In Chapter 2, for a Greenian pair \((M,L)\), the concepts of the Martin boundary and its properties including the Fatou-Doob-Naïm limit theorem concerning the existence of fine limit at the minimal boundary points of the function of the form \(u/v\) for \(u\in\mathcal S_+\), \(v\in\mathcal H_+\) are stated. Also, when \(L(1)=0\), its probabilistic interpretation is stated. A countable set \(X\) with symmetric, reflexive relation \(\Gamma\) is called a graph if, for any \(a,b\in X\), there exist \(x_0=a\), \(x_1,\ldots,x_m=b\), such that \(x_j\Gamma x_{j+1}\).

In Chapter 3, the author concerns to the recurrence and transience of the graphs and the manifolds. For a selfadjoint operator \(L\), \(a(\phi,\psi)\) be the Dirichlet form on \(L^2(M: \sigma)\). Then \((M,L)\) is transient if and only if there exists an open set \(V\) and a constant \(C<0\) such that

\[ (\int_{V}\phi \,d\sigma)^2 \leq Ca(\phi,\psi)\quad \forall \phi \in C^2_0(M),\quad \phi >0. \]

This characterization is applied to show the stability of transience under some mappings. It is also used to give a characterization of recurrent groups.

Chapter 4 is devoted to the study of the Liouville property and its applications to the Laplace-Beltrami operator on \(M\). Some sufficient conditions for the property are given.

In Chapter 5, under conditions on \((M,L)\), including the condition that \(L+tI\) is transient for some \(t>0\), the Harnack inequality at infinity (there exists a constant \(c\) such that \(G(x_m,x_1) \leq cG(x_m,x_k)G(x_k,x_1)\) for any “\(\phi\)-chain” \(\{x_k\})\) is proved, where \(G\) is the Green function. In particular if \(M\) is hyperbolic, then for any geodesic segment \(xz\) and any point \(y\) on it such that \(\min[d(x,y),d(y,z)] \geq 1\), it holds that

\[ c^{-1}G(x,y)G(y,z) \leq G(x,z) \leq cG(x,y)G(y,z). \]

As an application of this result, the identification of the Martin compactification and the geometric one is shown. Some other applications are also given.

[For the entire collection see Zbl 0708.00013.]

Reviewer: Yoichi Oshima (Kumamoto)

##### MSC:

60J45 | Probabilistic potential theory |

31C12 | Potential theory on Riemannian manifolds and other spaces |

31C35 | Martin boundary theory |