  The (fractional) Brownian sheet is a simplest example of a Gaussian random
field X whose covariance is the tensor product of a finite number (d) of
nonnegative correlation functions of self-similar Gaussian processes. Let Y be
the homogeneous Gaussian field obtained by applying to X the Lamperti
transform, which involves the exponential change of time and the amplitude
normalization to have unit variance. Under some assumptions, we prove the
existence of the persistence exponents for both fields, X and Y, and find the
relation between them. The exponent for X is the limit of ln1/P{X(t)<1, t in
[0,T]^d}/(lnT)^d, T>>1. In terms of Y it has the form lim ln1/P{Y(t)<0,t in
TG}/T^d:=Q, T>>1, where G is a suitable d-simplex and T is a similarity
coefficient; G can be selected in form [0,c]^d if d=2 . The exponent Q exists
for any continuous Gaussian homogeneous field Y with non-negative covariance
when G is bounded region with a regular boundary. The problem of the existence
and connection of exponents for X and Y was raised by Li and Shao [Ann. Probab.
32:1, 2004] and originally concerned the Brownian sheet.
