# Gradient systems associated with probability distributions

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## Abstract

Gradient systems on manifolds of various probability distributions are presented. It is shown that the gradient systems can be linearized by the Legendre transformation. It follows that the corresponding flows on the manifolds converge to equilibrium points of potential functions exponentially. It is proved that gradient systems on manifolds of even dimensions are completely integrable Hamiltonian systems. Especially, the gradient system for the Gaussian distribution admits a Lax pair representation.

## Key words

manifolds of probability distributions gradient systems Hamiltonian systems Lax pair representation linear and nonlinear programming problems## Preview

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© JJIAM Publishing Committee 1994