  We consider a random variable X satisfying almost-sure conditions involving
G:=<DX,-DL^{-1}X> where DX is X's Malliavin derivative and L^{-1} is the
inverse Ornstein-Uhlenbeck operator. A lower- (resp. upper-) bound condition on
G is proved to imply a Gaussian-type lower (resp. upper) bound on the tail
P[X>z]. Bounds of other natures are also given. A key ingredient is the use of
Stein's lemma, including the explicit form of the solution of Stein's equation
relative to the function 1_{x>z}, and its relation to G. Another set of
comparable results is established, without the use of Stein's lemma, using
instead a formula for the density of a random variable based on G, recently
devised by the author and Ivan Nourdin. As an application, via a Mehler-type
formula for G, we show that the Brownian polymer in a Gaussian environment
which is white-noise in time and positively correlated in space has deviations
of Gaussian type and a fluctuation exponent \chi=1/2. We also show this
exponent remains 1/2 after a non-linear transformation of the polymer's
Hamiltonian.
