(To appear in the Séminaire de Probabilités)
Following up on work of A. Plucinska [A stochastic characterization of Hermite
polynomials, J. Math. Sci.89 (1998) 1541-1544]
we examine the class of Brownian martingales of the form
a(t)h(B(t)/b(t)) where B(.) is a standard one-dimensional Brownian motion.
Apart from trivial cases, if a(t)h(B(t)/b(t)) is a martingale then a(t)=tp/2,
h''-xh'+ph=0 (Hermite equation). Moreover, if the Brownian motion B(.) satisfies
P[B(0) = 0]
<1, then p must be a non-negative integer and h must be proportional to the Hermite polynomial of degree p.
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version of June 27, 2000