I am wondering about how to prove a modification of Kolmogrov's continuity criterion in order to also being able to quantify the growth behaviour of the process. In particular, I am interested in the space $C_{tem} =\lbrace f \colon \mathbb R \to \mathbb R , \text{continuous} \vert \sup_{x\in \mathbb R } e-^{\lambda |x|} |f(x)|<\infty \text{ for all } \lambda>0 \rbrace $. Now, in Shiga's 1988 paper (https://www.cambridge.org/core/journals/canadian-journal-of-mathematics/article/two-contrasting-properties-of-solutions-for-onedimensional-stochastic-partial-differential-equations/4FE47A56FA2457F2EE32AAC74A0EEDCB) he claims the following.

Let $X \colon \Omega \times [0,t] \times \mathbb R \to \mathbb R$ be a stochastic process. If for every $\lambda>0$ there exists $p>0, d>2$ such that $$\mathbb E\left[|X(t,x)-X(s,y)|^{2p} \right] \leq C_\lambda (|t-s|^d+|x-y|^d) e^{\lambda |x|}$$ for all $0\leq s,t\leq 1$ and $x,y$ with $|x-y|\leq 1$ then $X(t,x)$ has a continuous $C_{tem}$-valued version.

Now, the reference he gives is not very insightful. Does anyone have an idea how to modify the proof of Kolmogorov's original theorem to get this result or a more helpful reference?


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