Streched exponential in non-linear stochastic filed theories

We consider the time dependent two point function, <ϕ_q (t) ϕ_-q (0)> in non-linear stochastic field theories, for which the KPZ equation serves as a prototype. In particular we consider the small q's and long times such that ω_q t>>1 (ω_q being the corresponding decay rate). We find that, since the generic case has ω_q \propto q^μfor small q where μ>1, the decay of the two point function is given by a streched exponential in ω_qt multiplied by a factor of t, <ϕ_q (t) ϕ_-q (0)> \propto t^{β_d}exp[-γ(ω_qt)^{1/μ}], where β_d=(d-1)/2μ, d is the dimensionality of space and γa dimensionless constant.

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