HERMITE MARTINGALES

P.J. Fitzsimmons

Department of Mathematics University of California, San Diego 9500 Gilman Drive La Jolla, CA 92093-0112 USA

The Hermite polynomials h_{n}, n ∈ N, defined by the Rodrigues formulae

h_{n}(x) := (−1)^{n} exp(x^{2}/2)\frac{d^{n}}{dx^{n}} exp(−x^{2}/2),  x ∈ R,  (1)

play an important role in the theory of Brownian motion; see, for example, [3], [4], [6]. In particular, if (Ω, F, F_{t}, P) is a filtered probability space on which is defined a standard one-dimensional Brownian motion {B_{t}; t ≥ 0} with B_{0} = 0, then \{t^{n/2}h_{n}(B_{t}/\sqrt{t}); t ≥ 0\}, is a martingale for every n ∈ N.

An interesting converse, characterizing the Hermite polynomials, has recently been discovered by A. Plucińska [5]: If n ≥ 0 is an integer, h : R → R is real analytic, and t ↦ t^{n/2}h(B_{t}/\sqrt{t}) is a martingale, then h is proportional to h_{n}. Strictly speaking, this assertion is true only if we alter the initial state of the Brownian motion to ensure that P[B_{0} = 0] < 1. Indeed, for every real p > 0 there is a non-polynomial real analytic h such that \{t^{p/2}h(B_{t}/\sqrt{t}); t ≥ 0\} is a martingale, provided the Brownian motion satisfies P[B_{0} = 0] = 1; see part (b) of Theorem 1 below. Our purpose in this note is to give a new proof of (an extension of) Plucińska’s Theorem.

As preparation we collect some known results concerning the connection between space-time harmonic functions and martingale functions of space-time Brownian motion. Let

p_{t}(x, y) := [2πt]^{−1/2} exp(−(y − x)^{2}/2t)

denote the Brownian transition kernel, and define the corresponding semigroup of transition operators by

\begin{aligned}P_{t}f(x) : & = ∫_{R} p_{t}(x, y)f(y) dy \\ & = P^{x}[f(B_{t})] = P[f(x + B_{t})],  x ∈ R, t ≥ 0. \end{aligned}  (2)

Here P^{x} denotes both the law of Brownian motion started at x and the associated expectation operator.

Lemma 1. If H : R × (0, ∞) → R is Borel measurable, then the following statements are equivalent:

(a) P_{t−s}[H(·, t)](x) = H(x, s) for all x ∈ R and all 0 < s < t;

(b) P_{t−s}[H(·, t)](x) = H(x, s) for Lebesgue a.e. x ∈ R, for all 0 < s < t,

and P^{x}|H(B_{t}, t + r)| < ∞ for all x ∈ R and all r, t > 0;
(c) t ↦ H(B_{t}, t + r) is a P^{x} martingale, for all x ∈ R and all r > 0.

Proof. The implication (a)⇒(b) is trivial, and (b)⇒(c) follows easily because the P^{x}-distribution of B_{s} is absolutely continuous with respect to Lebesgue measure for all x ∈ R and all s > 0:

P^{x}[H(B_{t}, t + r)|F_{s}] = P_{t−s}[H(·, t + r)](B_{s}) = H(B_{s}, s + r), P^{x}-a.s.

Finally, if (c) holds then for x ∈ R and r, t > 0,

H(x, r) = P^{x}[H(B_{0}, 0 + r)] = P^{x}[H(B_{t}, t + r)] = P_{t}[H(·, t + r)](x),

which yields (a) after a change of variables.

Lemma 2. Let H : R × (0, ∞) → R be a function of class C^{2,1}.

(i) The process t ↦ H(B_{t}, t + r) is a P^{x} local martingale for all (x, r) ∈ R × (0, ∞) if and only if ∂H/∂t + \frac{1}{2}∂^{2}H/∂x^{2} ≡ 0.

(ii) Suppose that ∂H/∂t + \frac{1}{2}∂^{2}H/∂x^{2} ≡ 0 and that for each T > 0 there is a constant C_{T} such that |H(x, t)| ≤ C_{T} exp(x^{2}/2t) for all (x, t) ∈ R × (0, T]. Then t ↦ H(B_{t}, t + r) is a P^{x} martingale for all x ∈ R and all r > 0.

Proof. Assertion (i) follows immediately from Itô’s formula. Assertion (ii) is a consequence of classical theorems on the well-posedness of the Cauchy problem. Let us fix T > 0 and r > 0, and define

K(x, t) := P_{T −t}[H(·, T + r)](x), (x, t) ∈ R × [0, T].

Then K is a C^{2,1} solution of ∂H/∂t+ \frac{1}{2}∂^{2}H/∂x^{2} ≡ 0 on R×[0, T) with K(x, T) = H(x, T + r) for all x ∈ R, and

|K(x, t)| ≤ C exp(k · x^{2}), (x, t) ∈ R × [0, T],

for some constant k > 0; see Theorem 12 in Chapter 1 of [2]. By Theorem 16 loc. cit., K(x, t) = H(x, t + r) for all (x, t) ∈ R × [0, T]. That is

P_{T −t}[H(·, T + r)](x) = H(x, t + r)

for all (x, t) ∈ R × [0, T]. Since T > 0 and r > 0 were arbitrary, part (ii) follows from Lemma 1.

Here is the main result of this note. One could relax the conditions imposed on α and h in part (a) (measurability and local boundedness would suffice); we leave this extension to the reader.

Theorem 1. (a) Let h : R → R be of class C^{2}, and let α and β be C^{1} mappings of (0, ∞) into itself such that

α(1) = β(1) = 1  and  β(0+) = 0.  (3)

Define

H(x, t) := α(t) · h(x/β(t)),  t > 0, x ∈ R,  (4)

and suppose that

t ↦ H(B_{t}, t + r) is a P^{x} local martingale, for all x ∈ R and all r > 0. (5)

Then one of the following statements is true:

(i) h is constant and α ≡ 1. (ii) h(x) = Const. · x and α ≡ β. (iii) β(t) = \sqrt{t} for t > 0, there is a real number p such that α(t) = t^{p/2} for

t > 0, and h satisfies the Hermite equation

h′′(x) − x · h′(x) + p · h(x) = 0,  ∀x ∈ R.  (6)

(b) Conversely, if h is a C^{2} function satisfying (6), then t ↦ H(B_{t}, t + r) is a P^{x} martingale for every x ∈ R and every r > 0, where H(x, t) := t^{p/2}h(x/\sqrt{t}). If, in addition, p > 0, then t ↦ H(B_{t}, t) is a P^{0} martingale.

(c) If h is a C^{2} function such that t ↦ t^{p/2}h(B_{t}/\sqrt{t}) is a P^{x} martingale for some x ≠ 0, then p is a non-negative integer and h is proportional to the Hermite polynomial h_{p}.

Proof. (a) By Lemma 2(i), H satisfies the (dual) heat equation ∂H/∂t + \frac{1}{2}∂^{2}H/∂x^{2} ≡ 0; consequently,

\frac{1}{2}h′′(x) − β(t)β′(t)xh′(x) + [β(t)]^{2} \frac{α′(t)}{α(t)}h(x) = 0,  ∀t > 0, x ∈ R. (7)

If ββ′ is non-constant then there are times s, t > 0 such that c := β(t)β′(t) − β(s)β′(s) is non-zero. Fix such times and define b := [β(t)]^{2} \frac{α′(t)}{α(t)} − [β(s)]^{2} \frac{α′(s)}{α(s)}; then (7) implies

c · xh′(x) = b · h(x),  ∀x ∈ R.  (8)

Any solution of (8) must be of the form h(x) = Const. · x^{γ} for x > 0, where γ := b/c. For an h of this form to satisfy (7) (for x > 0) we must have γ = 0 or γ = 1. If γ = 0 then the C^{2} solutions of (8) are constant; this is case (i) of part (a) of Theorem 1. If γ = 1 then h(x) = Const. · x, which is case (ii).

Thus, with the exception of the trivial cases (i) and (ii), β(t)β′(t) is constant, which means that β(t) = \sqrt{t} for t ≥ 0, because of (3). Inserting this expression for β into (7) we arrive at

h′′(x) − xh′(x) + 2t\frac{α′(t)}{α(t)}h(x) = 0.  (9)

Unless h is identically 0 (which case has already been dealt with), (9) implies that t ↦ tα′(t)/α(t) is constant. In this case α(t) = t^{p/2} for some p ∈ R, and (9) simplifies to (6).

(b) Fix p ∈ R, let h solve (6), and define H(x, t) := t^{p/2}h(x/\sqrt{t}). The function h, being a solution of (6), can be expressed as c_{1}Y_{1}(x) + c_{2}Y_{2}(x), where

Y_{1}(x) := M(−\frac{1}{2}p, \frac{1}{2}, \frac{1}{2}x^{2}), Y_{2}(x) := xM(−\frac{1}{2}(p − 1), \frac{3}{2}, \frac{1}{2}x^{2})  (10)

are linearly independent solutions of (6); here z ↦ M(a, b, z) is the solution of Kummer’s equation

zw′′(z) + (b − z)w′(z) − aw(z) = 0

given by

M(a, b, z) = ∑_{n=0}^{∞} \frac{a(a + 1) ⋯ (a + n − 1)}{b(b + 1) ⋯ (b + n − 1)} \frac{z^{n}}{n!} .  (11)

See 13.1.1, 13.1.2, 19.2.1 and 19.2.3 in [1]. For b > 0 as in the present situation, M(a, b, z) is an entire function of z. Moreover, Y_{1} (resp. Y_{2}) is a polynomial if and only if p is an even (resp. odd) non-negative integer. The asymptotic behavior of M is known [1; 13.1.4], and yields the estimate

|h(x)| ≤ Const. · exp(x^{2}/2) · [1 + |x|]^{−p−1}.  (12)

Clearly (12) implies the bound appearing in part (ii) of Lemma 2. Moreover, because h satisfies (6), H satisfies ∂H/∂t + \frac{1}{2}∂^{2}H/∂x^{2} ≡ 0. The first assertion therefore follows from Lemma 2(ii). Turning to the second assertion, if p > 0, then P^{0}|H(B_{t}, t)| < ∞ by (12). The family {H(B_{t}, t); t > 0} of P^{0}-integrable random variables is a martingale because of Lemma 2(ii). By the backward martingale convergence theorem, the limit lim_{t↓0} H(B_{t}, t) exists P^{0}-a.s. and in L^{1}(P^{0}); the P^{0}-a.s. limit is easily seen to be 0, by (12) and the law of the iterated logarithm. Consequently, if H(B_{0}, 0) is understood to be 0, then {H(B_{t}, t); t ≥ 0} is a P^{0} martingale.

(c) Let h be a C^{2} function such that t ↦ t^{p/2}h(B_{t}/\sqrt{t}) is a P^{x} martingale for some x ≠ 0. Then h satisfies (6), and unless h is a polynomial the estimate (12) can be strengthened to an asymptotic equivalence:

|h(x)| ∼ Const. · exp(x^{2}/2) · |x|^{−p−1}, |x| → ∞.

See 13.1.4 in [1]. The P^{x} integrability of h(B_{t}/\sqrt{t}), for t = 1, implies that for N sufficiently large

∞ > ∫_{R} |h(y)| exp(−(y − x)^{2}/2) dy
≥ Const. · exp(−x^{2}/2) ∫_{|y|≥N} exp(xy)|y|^{−p−1} dy,

which is clearly absurd because x ≠ 0. Thus, h must be a polynomial. In view of (10) and (11), the only polynomial solutions of (6) occur when p is a non-negative integer, and any such polynomial solution is proportional to h_{p}.

Remark. Only the local martingale property of t^{p/2}h(B_{t}/\sqrt{t}) and the integrability of h(B_{1}) were used in the proof of (c). An alternative proof, which uses more fully the hypothesis that t^{p/2}h(B_{t}/\sqrt{t}) is a martingale, was suggested by the referee: If t^{p/2}h(B_{t}/\sqrt{t}) is a P^{x} martingale for some x ≠ 0, then lim_{t↓0} t^{p/2}h(B_{t}/\sqrt{t}) exists P^{x} almost surely. This implies the existence of lim_{t↓0} t^{p/2}h(x/\sqrt{t}), which forces the (entire!) function h to have a pole (of order at most p) at infinity. In other words, h must be a polynomial.

Acknowledgment. I thank the referee for helpful comments, especially those concerning the confluent hypergeometric function M.

References

[1] M. Abramowitz and A. Stegun: Handbook of Mathematical Functions (Reprint of the 1972 edition). Dover, New York, 1992.

[2] A. Friedman: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs, N.J., 1964.

[3] S. Janson: Gaussian Hilbert Spaces. (Cambridge University Press, Cambridge, 1997.

[4] D. Nualart: The Malliavin Calculus and Related Topics. Springer-Verlag, New York, 1995.

[5] A. Plucińska: A stochastic characterization of Hermite polynomials, J. Math. Sci. 89 (1998) 1541–1544.

[6] D.W. Stroock: Probability Theory. Cambridge University Press, Cambridge,