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

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)
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