A growth estimate for continuous random fields. (English) Zbl 0879.60039

Manthey, Ralph; Mittman, Katrin

Math. Bohem. 121, No. 4, 397-413 (1996).

The authors prove that if a random field \((Z(x), x\in \mathbb{R} ^{d})\) satisfies the assumption of the Kolmogorov continuity test on every cube \([-n,n]^{d}\) with a constant growing polynomially in \(n\), then \(Z\) itself growths at most polynomially almost surely. More precisely: let there exist constants \(\nu >1\), \(\kappa \geq 0\), \(\alpha >d\) and \(c>0\) such that \(E|Z(x)-Z(y)|^{\nu}\leq cn^{\kappa}|x-y|^{\alpha}\) holds for any \(x,y\in [-n,n]^{d}\) and \(n\in \mathbb{N}\). Then (the continuous modification of) \(Z\) satisfies for each \(\delta >1\) the estimate \(|Z(x)|\leq \eta _{\delta}(1+|x|^{(\alpha +\delta +\kappa) /\nu})\) for any \(x\in \mathbb{R}^{d}\) with a random variable \(\eta _{\delta}\) almost surely finite. Applications of this estimate are given, in particular, to mild solutions of the Cauchy problem for a linear stochastic reaction-diffusion equation.

Reviewer: J.Seidler (Praha)

MSC:

60G17	Sample path properties
60G15	Gaussian processes
60H15	Stochastic partial differential equations (aspects of stochastic analysis)

Keywords:

asymptotic behaviour of paths; Wiener field; stochastic diffusion equation