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The (fractional) Brownian sheet is a simplest example of a Gaussian randomfield X whose covariance is the tensor product of a finite number (d) ofnonnegative correlation functions of self-similar Gaussian processes. Let Y bethe homogeneous Gaussian field obtained by applying to X the Lampertitransform, which involves the exponential change of time and the amplitudenormalization to have unit variance. Under some assumptions, we prove theexistence of the persistence exponents for both fields, X and Y, and find therelation 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 inTG}/T^d:=Q, T>>1, where G is a suitable d-simplex and T is a similaritycoefficient; G can be selected in form [0,c]^d if d=2 . The exponent Q existsfor any continuous Gaussian homogeneous field Y with non-negative covariancewhen G is bounded region with a regular boundary. The problem of the existenceand connection of exponents for X and Y was raised by Li and Shao [Ann. Probab.32:1, 2004] and originally concerned the Brownian sheet.

The persistence exponents of Gaussian random fields connected by the Lamperti transform