Does there exist a finite measure on (0,\infty) du(x) such that

e^{-t}=\int_0^\infty e^{-x^2t^2-xt}du(x) for all t>0?

I tried hard but can not find the answer. This relates to the representation
of a subordinated poisson semigroup (P_s)_s in terms of average of heat
semigroup (T_s)_s.

The known facts are (by the residue theorem)
e^{-t}=\int_0^\infty e^{-xt^2} e^{-1/4x}x^{-3/2}dx for all t>0
and (as a consequence)
e^{-t}=e^{1/2}\int_0^\infty e^{-xt^2-xt} e^{-1/4x}x^{-3/2}e^{-x/4}dx for all t>0

thanks
Tao