Hölder versions of Banach space valued random fields (Q2761599)

Let \(B\) be a Banach space and \(T=[0,1]^d\). Any element \(x(t)\) in \(C(T;B)\) admits the expansion \(x(t)=\sum_{j=0}^{\infty} \sum_{v\in V_j}\lambda_{j,v}(x)\Lambda_{j,v}(t),\) using a Rademacher type function system \(\Lambda_{j,v}.\) Consider a B-valued random field \(\xi(t),\;t\in T\), which is continuous in probability. Assume there exist functions \(\sigma\) and \(\Psi\) such that \(P(||\Delta^2_h \xi(t)||_B > r\sigma(|h|)) \leq \Psi(r).\) And consider a modulus of continuity \(\rho\). Put \(R(u)=R(\Psi,\sigma,\rho,u):= \sum_{j=0}^{\infty}2^{jd}\Psi(u{{\rho}\over{\sigma}}(2^{-j}))\). The results are: 1) if \(R(u_0)<\infty\) for some \(u_0\), then \(\xi\) has a version in \(H_\rho (B)\), and 2) if \(R(u)<\infty\) for any \(u\), then \(\xi\) has a version in the space \(H^o_\rho\). There \(H_\rho\) is the space of functions which satify \(||x(t+h)-x(t)||_B = O(\rho(|h|))\), and \(H^o_\rho\) is the space employed \(o\) instead of \(O\).