Uniqueness in the Cauchy Problem for the Hermite Heat Equation

We prove the uniqueness of solutions to the Cauchy problem for the Hermite heat equation as follows: Let U(x, t) be a continuous function on R x [0, T] satisfying the following (i) (partial derivative/partial derivative t - partial derivative(2)/partial derivative x(2) + x(2))U(x, t) = 0 in R x (0, T), (ii) For every epsilon > 0, there exists a constant C := C(epsilon) > 0 and 0 < alpha < 1 such that sup(x) (is an element of R)vertical bar U(x, t)vertical bar <= C exp[(epsilon/t)(alpha)] for 0 < t < T, (iii) U(x, 0) = 0 for x is an element of R. Then U(x, t) = 0 in R x [0, T]. Also the condition 0 < alpha < 1 is optimal.