Potentials of random walks on trees Dellacherie, Claude; Martinez, Servet; San Martin, Jaime
Linear algebra and its applications,
07/2016, Volume:
501
Journal Article
Peer reviewed
Open access
In this article we characterize inverse M-matrices and potentials whose inverses are supported on trees. In the symmetric case we show they are a Hadamard product of tree ultrametric matrices, ...generalizing a result by Gantmacher and Krein 12 done for inverse tridiagonal matrices. We also provide an algorithm that recognizes when a positive matrix W has an inverse M-matrix supported on a tree. This algorithm has quadratic complexity. We also provide a formula to compute W−1, which can be implemented with a linear complexity. Finally, we also study some stability properties for Hadamard products and powers.
In this article we study stability properties of
g
O
, the standard Green kernel for
O
an open regular set in
ℝ
d
. In
d
≥ 3 we show that
g
O
β
is again a Green kernel of a Markov Feller process, for ...any power
β
∈ 1,
d
/(
d
− 2)). In dimension
d
= 2, we show the same result for
g
O
β
, for any
β
≥ 1 and for kernels
exp
(
α
g
O
)
,
exp
(
α
g
O
)
−
1
, for
α
∈ (0,2
π
), when
O
is an open Greenian regular set whose complement contains a ball.
Inverse M-matrix, a new characterization Dellacherie, Claude; Martínez, Servet; San Martín, Jaime
Linear algebra and its applications,
06/2020, Volume:
595
Journal Article
Peer reviewed
Open access
In this article we present a new characterization of inverse M-matrices, inverse row diagonally dominant M-matrices and inverse row and column diagonally dominant M-matrices, based on the positivity ...of certain inner products.
In this article we characterize the closed cones respectively generated by the symmetric inverse M-matrices and by the inverses of symmetric row diagonally dominant M-matrices. We show the latter has ...a finite number of extremal rays, while the former has infinitely many extremal rays. As a consequence we prove that every potential is the sum of ultrametric matrices.
Hadamard Functions of Inverse M -Matrices Dellacherie, Claude; Martinez, Servet; San Martin, Jaime
SIAM journal on matrix analysis and applications,
01/2009, Volume:
31, Issue:
2
Journal Article
Peer reviewed
The authors prove that the class of generalized ultrametric matrices (GUM) is the largest class of bipotential matrices stable under Hadamard increasing functions. They also show that any power ... , ...in the sense of Hadamard functions, of an inverse M-matrix is also inverse M-matrix. This was conjectured for ... by Neumann in (Linear Algebra Appl., 285 (1998), pp. 277-290), and solved for integer ... by Chen in (Linear Algebra Appl., 381 (2004), pp. 53-60). They study the class of filtered matrices, which include naturally the GUM matrices, and present some sufficient conditions for a filtered matrix to be a bipotential.(ProQuest: ... denotes formulae/symbols omitted.)
Cauchy problem for derivors in finite dimension Jean-Francois Couchouron; Claude Dellacherie; Michel Grandcolas
Electronic journal of differential equations,
05/2001, Volume:
2001, Issue:
32
Journal Article
Peer reviewed
Open access
In this paper we study the uniqueness of solutions to ordinary differential equations which fail to satisfy both accretivity condition and the uniqueness condition of Nagumo, Osgood and Kamke. The ...evolution systems considered here are governed by a continuous operators $A$ defined on $mathbb{R}^N$ such that $A$ is a derivor; i.e., $-A$ is quasi-monotone with respect to $(mathbb{R}^{+})^N$.
Ultrametric and Tree Potential Dellacherie, Claude; Martinez, Servet; San Martin, Jaime
Journal of theoretical probability,
06/2009, Volume:
22, Issue:
2
Journal Article
Peer reviewed
In this article we study which infinite matrices are potential matrices. We tackle this problem in the ultrametric framework by studying infinite tree matrices and ultrametric matrices. For each tree ...matrix, we show the existence of an associated symmetric random walk and study its Green potential. We provide a representation theorem for harmonic functions that includes simple expressions for any increasing harmonic function and the Martin kernel. For ultrametric matrices, we supply probabilistic conditions to study its potential properties when immersed in its minimal tree matrix extension.
In this article we present a new characterization of inverse M-matrices, inverse row diagonally dominant M-matrices and inverse row and column diagonally dominant M-matrices, based on the positivity ...of certain inner products.
Powers of Brownian Green Potentials Dellacherie, Claude; Duarte, Mauricio; Martínez, Servet ...
arXiv (Cornell University),
04/2020
Paper, Journal Article
Open access
In this article we study stability properties of \(g_O\), the standard Green kernel for \(O\) an open regular set in \(R^d\). In \(d\ge 3\) we show that \(g_O^\beta\) is again a Green kernel of a ...Markov Feller process, for any power \(\beta\in 1,d/(d-2))\). In dimension \(d=2\), if \(O\) is an open Greenian regular set, we show the same result for \(g_O^\beta\), for any \(\beta\ge 1\) and for the kernel \(\exp(\alpha g_O)\), when \(\alpha \in (0,2\pi)\).
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