This paper is concerned with
random series of the form ∑
_{n=0}^{∞}X_{n}(ω)a_{n}v_{n}(x,t) where the X_{n}’s are random
variables, the a_{n}’s are real numbers, and the v_{n}’s are heat polynomials as introduced
by P. C. Rosenbloom and D. V. Widder. The sequences {a_{n}} are assumed to satisfy
limsup_{n→∞}a_{n}^{2∕n}(2n∕e) = 1 which implies ∑
_{n=0}^{∞}a_{n}v_{n}(x,t) has t < 1 as its strip
of convergence, i.e., it converges to a C^{2}solution of the heat equation in t < 1 and
does not converge everywhere in any larger open strip. Associated with each sequence
{a_{n}} is its classification number from [0,1] which indicates how rapidly a_{n} tends to
zero. Assumptions are placed on the random variables which imply that for
almost every ω the series ∑
_{n=0}^{∞}X_{n}(ω)a_{n}v_{n}(x,t) has t < 1 as its strip of
convergence.
The main results of the paper are two theorems. The first states that if {a_{n}}
has its classification number in [0,1∕2), then for almost every ω the lines
t = 1 and t = −1 form the natural boundary for ∑
_{n=0}^{∞}X_{n}(ω)a_{n}v_{n}(x,t).
The second is concerned with sequences having their classification numbers
in (1/2.1]. The conclusion implies that for almost every ω no interval of
either of the lines t = 1 or t = −1 is part of the natural boundary for
∑
_{n=0}^{∞}X_{n}(ω)a_{n}v_{n}(x,t).
