# Extending Lévy's characterisation of brownian motion

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## Abstract

Extending Lévy's characterisation of Brownian motion SÉMINAIRE DE PROBABILITÉS (STRASBOURG) P. MC GILL BHASKARAN RAJEEV B. V. RAO Extending Lévy’s characterisation of Brownian motion Séminaire de probabilités (Strasbourg), tome 22 (1988), p. 163-165. <http://www.numdam.org/item?id=SPS_1988__22__163_0> © Springer-Verlag, Berlin Heidelberg New York, 1988, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://www-irma. u-strasbg.fr/irma/semproba/index.shtml), implique l’accord avec les conditions gé- nérales d’utilisation (http://www.numdam.org/legal.php). Toute utilisation commer- ciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Extending Lévy’s characterisation of Brownian motion P. McGill*, B. Rajeev~ and B. V. Rao~ * Deport de Math., Université Louis Pasteur, 67000 Strasbourg, France. tlndian Stat. Inst., 203 Barrackpore Trunk Road, Calcutta 700 035, India. Let f(x, t) be a solution of the heat equation. Then if Bt is a real Brownian motion, a simple application of Ito’s formula shows that f(Bt t) is a local martingale. Paul Levy gave a converse of this using the parabolic function f(x, t) = x2 - t, namely he showed that every continuous local martingale Xt for which is also a local martingale has to be a Brownian motion. It is natural to ask if there exist other parabolic functions with this property. The short answer is yes; it is the rule, not the exception. The purpose of this note is to prove the following extension of Levy’s characteri- sation. In the statement we use the notation m for Lebesgue measure on the line, and if D is measurable we write m JL D to mean that m(D) = 0. Proposition Let Xt be a continuous local martingale which verifies the following condition. (A) There exists a soluti

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