×

Ultrametric and tree potential. (English) Zbl 1176.60069

An ultrametric matrix \(U = (U_ {i j} : i, j \in I )\) is a symmetric nonnegative matrix verifying the ultrametric inequality \(U_ {i j} \geq \min \{ U_ {i k}, U_ {k j}\}\) for all \(i, j, k \in I\). Tree matrices are associated with a (locally finite) rooted tree \((I, {\mathcal T})\), with root \(r\) and a strictly increasing function \(w:\{ |k| : k \in I \} \rightarrow \mathbb {R}_ {+}\) by \(U_ {i j} = w_ {|i\wedge j|}\) with \(i\wedge j\) being the farthest vertex from the root that is common to the geodesic from \(i\) and \(j\) to \(r\). Tree matrices are special case of ultrametric matrices. Conversely, each ultrametric matrix posseses a minimal tree matrix extension.
One of the purposes of this paper is to extend to countably infinite ultrametric and tree matrices the results obtained by C. Dellacherie, S. Martinez and J. San Martin [Adv. Appl. Math. 17, No. 2, 169–183 (1996; Zbl 0861.60077)] for the finite case.
The existence of an associated symmetric random walk is proved for each tree matrix. Its Green potential is studied and a representation formula for (increasing) harmonic function is given. For ultrametric matrices, probabilistic conditions are supplied in order to study its potential properties when immersed in its minimal tree matrix extension.

MSC:

60J45 Probabilistic potential theory
60J25 Continuous-time Markov processes on general state spaces

Citations:

Zbl 0861.60077
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Albeverio, S., Karwowski, W.: A random walk on p-adics–the generator and its spectrum. Stoch. Process. Their Appl. 53, 1–22 (1994) · Zbl 0810.60065 · doi:10.1016/0304-4149(94)90054-X
[2] Albeverio, S., Zhao, X.: Measure-valued branching processes associated with random walks on p-adics. Ann. Probab. 28, 1680–1710 (2000) · Zbl 1044.60036 · doi:10.1214/aop/1019160503
[3] Albeverio, S., Zhao, X.: On the relation between different constructions of random walks on p-adics. Markov Processes Relat. Fields 6, 239–255 (2000) · Zbl 0974.60028
[4] Anderson, W.J.: Continuous-Time Markov Chains: An Application-Oriented Approach. Springer Series in Statistics. Springer, Berlin (1991) · Zbl 0731.60067
[5] Benzécri, J.P.: L’Analyse des Données. Dunod, Paris (1973) · Zbl 0297.62039
[6] Berge, C., Ghouila-Houri, A.: Programmes, Jeux et Réseaux de Transport. Dunod, Paris (1962) · Zbl 0111.17302
[7] Bouleau, N.: Autour de la variance comme forme de Dirichlet. In: Séminaire de Théorie du Potentiel 8. Lecture Notes in Mathematics, vol. 1235, pp. 39–53. Springer, Berlin (1989)
[8] Cartier, P.: Fonctions harmoniques sur un arbre. Symposia Mathematica IX. Academic Press, San Diego (1972), pp. 203–270 · Zbl 0283.31005
[9] Cartwright, D., Sawyer, S.: The Martin boundary for general isotropic random walks in a tree. J. Theor. Probab. 4, 111–136 (1991) · Zbl 0728.60013 · doi:10.1007/BF01046997
[10] Chung, K.L., Zhao, Z.: From Brownian Motion to Schrödinger’s Equation. Springer, Berlin (1995) · Zbl 0819.60068
[11] Dartnell, P., Martínez, S., San Martín, J.: Opérateurs filtrés et chaînes de tribus invariantes sur un espace probabilisé dénombrable. In: Séminaire de Probabilités XXII. Lecture Notes in Mathematics, vol. 1321. Springer, Berlin (1988)
[12] Dellacherie, C., Martínez, S., San Martín, J.: Ultrametric matrices and induced Markov chains. Adv. Appl. Math. 17, 169–183 (1996) · Zbl 0861.60077 · doi:10.1006/aama.1996.0009
[13] Dellacherie, C., Martínez, S., San Martín, J., Taïbi, D.: Noyaux potentiels associés à une filtration. Ann. Inst. Henri Poincaré Probab. Stat. 34, 707–725 (1998) · Zbl 0914.60046 · doi:10.1016/S0246-0203(99)80001-6
[14] Dellacherie, C., Stricker, C.: Changement de temps et intégrales stochastiques. In: Séminaire de Probabilités XI. Lectures Notes in Mathematics, vol. 581. Springer, Berlin (1977) · Zbl 0369.60067
[15] Hu, Y.: Potentiel kernels associated with a filtration and forward-backward SDE’s. Potential Anal. 10, 103–118 (1998) · Zbl 0929.60053 · doi:10.1023/A:1008678927573
[16] Hughes, B.: Trees and ultrametric spaces: a categorical equivalence. Adv. Math. 189, 148–191 (2004) · Zbl 1061.57021 · doi:10.1016/j.aim.2003.11.008
[17] Kemeny, J., Snell, J., Knapp, A.: Denumerable Markov Chains, 2nd edn. GTM, vol. 40. Springer, Berlin (1976) · Zbl 0348.60090
[18] Kochubei, A.: Stochastic integrals and stochastic differential equations over the field of p-adic numbers. Potential Anal. 6, 105–125 (1997) · Zbl 0874.60047 · doi:10.1023/A:1017913800810
[19] Lyons, R.: Random walks and percolation on trees. Ann. Probab. 18, 931–958 (1990) · Zbl 0714.60089 · doi:10.1214/aop/1176990730
[20] Lyons, R.: Random walks, capacity and percolation on trees. Ann. Probab. 20, 2043–2088 (1992) · Zbl 0766.60091 · doi:10.1214/aop/1176989540
[21] Lyons, R., Peres, Y.: Probability on Trees and Networks. http://mypage.iu.edu/rdlyons/prbtree/prbtree.html (2005) · Zbl 1376.05002
[22] Lyons, T.: A simple criterion for transience of a reversible Markov chain. Ann. Probab. 11, 393–402 (1983) · Zbl 0509.60067 · doi:10.1214/aop/1176993604
[23] Martínez, S., Michon, G., San Martín, J.: Inverses of ultrametric matrices are of Stieltjes types. SIAM J. Matrix Anal. Appl. 15, 98–106 (1994) · Zbl 0798.15030 · doi:10.1137/S0895479891217011
[24] Martínez, S., Remenik, D., San Martín, J.: Level-wise approximation of a Markov process associated to the boundary of an infinite tree. J. Theor. Probab. 20(3), 561–579 (2007) · Zbl 1132.60055 · doi:10.1007/s10959-007-0073-2
[25] McDonald, J.J., Neumann, M., Schneider, H., Tsatsomeros, M.J.: Inverse M-matrix inequalities and generalized ultrametric matrices. Linear Algebra Appl. 220, 321–341 (1995) · Zbl 0824.15020 · doi:10.1016/0024-3795(94)00077-Q
[26] Nabben, R., Varga, R.S.: A linear algebra proof that the inverse of a strictly ultrametric matrix is a strictly diagonally dominant Stieltjes matrix. SIAM J. Matrix Anal. Appl. 15, 107–113 (1994) · Zbl 0803.15020 · doi:10.1137/S0895479892228237
[27] Nabben, R., Varga, R.S.: Generalized ultrametric matrices–a class of inverse M-matrices. Linear Algebra Appl. 220, 365–390 (1995) · Zbl 0828.15020 · doi:10.1016/0024-3795(94)00086-S
[28] Picardello, M., Woess, W.: Martin boundaries of random walks: ends of trees and groups. Trans. AMS 302, 185–205 (1987) · Zbl 0615.60068 · doi:10.1090/S0002-9947-1987-0887505-2
[29] Rammal, R., Toulouse, G., Virasoro, M.A.: Ultrametricity for physicists. Rev. Mod. Phys. 58, 765–788 (1986) · doi:10.1103/RevModPhys.58.765
[30] Sawyer, S.: Martin boundaries and random walks. Contemp. Math. 206, 17–44 (1997) · Zbl 0891.60073
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.