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This paper is concerned with
random series of the form ∑
n=0∞Xn(ω)anvn(x,t) where the Xn’s are random
variables, the an’s are real numbers, and the vn’s are heat polynomials as introduced
by P. C. Rosenbloom and D. V. Widder. The sequences {an} are assumed to satisfy
limsupn→∞|an|2∕n(2n∕e) = 1 which implies ∑
n=0∞anvn(x,t) has |t| < 1 as its strip
of convergence, i.e., it converges to a C2-solution of the heat equation in |t| < 1 and
does not converge everywhere in any larger open strip. Associated with each sequence
{an} is its classification number from [0,1] which indicates how rapidly an tends to
zero. Assumptions are placed on the random variables which imply that for
almost every ω the series ∑
n=0∞Xn(ω)anvn(x,t) has |t| < 1 as its strip of
convergence.
The main results of the paper are two theorems. The first states that if {an}
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∞Xn(ω)anvn(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∞Xn(ω)anvn(x,t).
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