Kolmogorov continuity criteria

Last updated: 2026-04-12

The Kolmogorov continuity criteria is a simple moment condition that ensures the existence of a Hölder-continuous version of a given process in a complete metric space. It has many applications in Probability Theory. For example, one can show the Brownian Motion is almost surely locally Hölder continuous of order pp, for every p∈(0,12)p \in(0, \frac{1}{2})
Theorem (The Kolmogorov continuity criteria)
Let XX be a process on Rd\mathbb{R}^d with values in a complete metric space (S, ρ)(S, \, \rho) such that
E{ρ(Xs,Xt)}a≤c∣s−t∣d+b,s,t∈Rd,\begin{equation}E \left\{ \rho \left(X_s, X_t\right) \right\}^a \leq c|s-t|^{d+b}, \quad s, t \in \mathbb{R}^d,\end{equation}
for some constants a,b>0a, b > 0. Then there exists a modification X~\tilde{X} of XX that is locally Hölder continuous of order pp, for every p∈(0,ba)p \in(0, \frac{b}{a}), i.e. process X~ ⁣:[0,∞) × Ω\tilde{X} \colon [0, \infty)~\times~\Omega such that X~\tilde{X} is locally Hölder continuous and
∀t≥0,P(X~t=Xt)=1\begin{equation*}\forall t \geq 0, \quad \mathbb{P} (\tilde{X}_t = X_t) = 1\end{equation*}

Motivation

The fundamental motivation for Kolmogorov's continuity criterion is to provide a practical way to prove that random processes have continuous sample paths. Here's why this is important:
Problem formulation

Direct Verification Challenge

When working with stochastic processes, directly verifying continuity of sample paths can be extremely difficult because:
  • We need to check continuity at every point
  • Random processes often have complex dependencies
  • Traditional continuity arguments may not work well with randomness

Moment-Based Alternative

Kolmogorov's insight was that instead of checking continuity directly, we could look at the moments of increments:
For a process XtX_t, if we can show that for some p>0p > 0 and α>0\alpha > 0:
E{ρ(Xs,Xt)}a≤c∣s−t∣d+b,s,t∈Rd,\begin{equation*}E \left\{ \rho \left(X_s, X_t\right) \right\}^a \leq c|s-t|^{d+b}, \quad s, t \in \mathbb{R}^d,\end{equation*}
Then under mild additional conditions, the process has a continuous modification.

Practical Applications for Brownian Motion

This criterion is particularly valuable for Brownian Motion B(t)B(t). We have:
E[∣B(t)−B(s)∣4]=3∣t−s∣2\begin{equation*}E[|B(t) - B(s)|^4] = 3|t-s|^2\end{equation*}
Taking p=4p=4 and α=1\alpha=1, the criterion immediately gives continuity.

Proof

We can consider the restriction of XX to [0,1]d[0,1]^d without the loss of generality. We can define the approximation dataset GnG_n of points [0,1]d[0, 1]^d such that the coordinates of 2nx2^n x are positive integers and ∣Gn∣=(2n+1)d| G_n | = (2^n + 1)^d. The for each u∈[0,1]du \in [0, 1]^d we choose πn(u)∈Gn\pi_n (u) \in G_n as close to uu as possible, so that ∣πn(u)−u∣≤2−n| \pi_n (u) - u | \leq 2^{-n} and ∣πn(u)−πn−1(u)∣≤∣πn(u)−u∣+∣u−πn−1(u)∣≤32−n|\pi_n (u) - \pi_{n-1} (u)| \leq | \pi_n (u) - u | + | u - \pi_{n-1} (u) | \leq 3 2^{-n}
Then consider the random variable
Yn=max⁡{ρ(Xs,Xt);∣s−t∣≤32n}\begin{equation*}Y_n = \max \left\{ \rho(X_s, X_t); \quad |s-t| \leq \frac{3}{2^n} \right\}\end{equation*}
and since Gn−1⊂GnG_{n-1} \subset G_n, which means ∀u∈[0,1]d ⁣:πn−1(u),πn(u)∈Gn\forall u \in [0,1]^d \colon \pi_{n-1} (u), \pi_{n} (u) \in G_n. So we get ∀u∈[0,1]d\forall u \in [0,1]^d
ρ(Xπn(u),Xπn−1(u))≤Yn\begin{equation*}\rho(X_{\pi_{n}(u)}, X_{\pi_{n-1}(u)}) \leq Y_n\end{equation*}
For the set G=⋃n≥0GnG = \bigcup_{n\geq0} G_n we claim that
sup⁡s,t∈G;∣s−t∣≤2−kρ(Xs,Xt)≤3∑n≥kYn\begin{equation}\sup_{s, t \in G; \newline |s-t| \leq 2^{-k}} \rho(X_s, X_t) \leq 3 \sum_{n\geq k} Y_n\end{equation}
To prove this, consider n≥kn \geq k such that s,t∈Gns, t \in G_n, so that s=πn(s)s = \pi_n (s) and t=πn(t)t = \pi_n (t). Assuming ∣s−t∣≤2−k| s - t| \leq 2^{-k}, we have
∣πk(s)−πk(t)∣≤∣πk(u)−s∣+∣s−t∣+∣t−πk(t)∣≤32−k\begin{equation*}|\pi_k (s) - \pi_{k} (t)| \leq | \pi_k (u) - s | + |s - t| + | t - \pi_k (t) | \leq 3 2^{-k}\end{equation*}
and thus
ρ(Xπk(s),Xπk(t))≤Yk\begin{equation*}\rho(X_{\pi_{k}(s)}, X_{\pi_k(t)}) \leq Y_k\end{equation*}
Next, for u∈{s,t}u \in \{ s, t \},
ρ(Xu,Xπk(u))=ρ(Xπn(u),Xπk(u))=∑k≤l≤nρ(Xπl+1(u),Xπl(u)),\begin{equation*}\rho(X_u, X_{\pi_k(u)}) = \rho(X_{\pi_{n}(u)}, X_{\pi_k(u)}) = \sum_{k \leq l \leq n} \rho(X_{\pi_{l+1}(u)}, X_{\pi_l(u)}),\end{equation*}
and since ρ(Xπl+1(u),Xπl(u))≤Yl+1\rho(X_{\pi_{l+1}(u)}, X_{\pi_l(u)}) \leq Y_{l+1}, we obtain
ρ(Xu,Xπk(u))≤∑l≥kYl+1\begin{equation*}\rho(X_u, X_{\pi_k(u)}) \leq \sum_{ l \geq k} Y_{l+1}\end{equation*}
We then use the previous inequalities and get
ρ(Xs,Xt)≤ρ(Xs,Xπk(s))+ρ(Xπk(s),Xπk(t))+ρ(Xπk(t),Xt)≤3∑l≥kYl+1\begin{equation*}\rho(X_s, X_t) \leq \rho(X_s, X_{\pi_k(s)}) + \rho(X_{\pi_k(s)}, X_{\pi_k(t)}) + \rho(X_{\pi_k(t)}, X_t) \leq 3 \sum_{ l \geq k} Y_{l+1}\end{equation*}

Construction approximation sequence

We can consider the restriction of XX to [0,1]d[0,1]^d without the loss of generality. We can define the approximation dataset GnG_n of points [0,1]d[0, 1]^d such that the coordinates of 2nx2^n x are positive integers and ∣Gn∣=(2n+1)d| G_n | = (2^n + 1)^d. The for each u∈[0,1]du \in [0, 1]^d we choose πn(u)∈Gn\pi_n (u) \in G_n as close to uu as possible, so that ∣πn(u)−u∣≤2−n| \pi_n (u) - u | \leq 2^{-n} and ∣πn(u)−πn−1(u)∣≤∣πn(u)−u∣+∣u−πn−1(u)∣≤32−n|\pi_n (u) - \pi_{n-1} (u)| \leq | \pi_n (u) - u | + | u - \pi_{n-1} (u) | \leq 3 2^{-n}
Then consider the random variable
Yn=max⁡{ρ(Xs,Xt);∣s−t∣≤32n}\begin{equation*}Y_n = \max \left\{ \rho(X_s, X_t); \quad |s-t| \leq \frac{3}{2^n} \right\}\end{equation*}
and since Gn−1⊂GnG_{n-1} \subset G_n, which means ∀u∈[0,1]d ⁣:πn−1(u),πn(u)∈Gn\forall u \in [0,1]^d \colon \pi_{n-1} (u), \pi_{n} (u) \in G_n. So we get ∀u∈[0,1]d\forall u \in [0,1]^d
ρ(Xπn(u),Xπn−1(u))≤Yn\begin{equation*}\rho(X_{\pi_{n}(u)}, X_{\pi_{n-1}(u)}) \leq Y_n\end{equation*}
For the set G=⋃n≥0GnG = \bigcup_{n\geq0} G_n we claim that
sup⁡s,t∈G;∣s−t∣≤2−kρ(Xs,Xt)≤3∑n≥kYn\begin{equation}\sup_{s, t \in G; \newline |s-t| \leq 2^{-k}} \rho(X_s, X_t) \leq 3 \sum_{n\geq k} Y_n\end{equation}
To prove this, consider n≥kn \geq k such that s,t∈Gns, t \in G_n, so that s=πn(s)s = \pi_n (s) and t=πn(t)t = \pi_n (t). Assuming ∣s−t∣≤2−k| s - t| \leq 2^{-k}, we have
∣πk(s)−πk(t)∣≤∣πk(u)−s∣+∣s−t∣+∣t−πk(t)∣≤32−k\begin{equation*}|\pi_k (s) - \pi_{k} (t)| \leq | \pi_k (u) - s | + |s - t| + | t - \pi_k (t) | \leq 3 2^{-k}\end{equation*}
and thus
ρ(Xπk(s),Xπk(t))≤Yk\begin{equation*}\rho(X_{\pi_{k}(s)}, X_{\pi_k(t)}) \leq Y_k\end{equation*}
Next, for u∈{s,t}u \in \{ s, t \},
ρ(Xu,Xπk(u))=ρ(Xπn(u),Xπk(u))=∑k≤l≤nρ(Xπl+1(u),Xπl(u)),\begin{equation*}\rho(X_u, X_{\pi_k(u)}) = \rho(X_{\pi_{n}(u)}, X_{\pi_k(u)}) = \sum_{k \leq l \leq n} \rho(X_{\pi_{l+1}(u)}, X_{\pi_l(u)}),\end{equation*}
and since ρ(Xπl+1(u),Xπl(u))≤Yl+1\rho(X_{\pi_{l+1}(u)}, X_{\pi_l(u)}) \leq Y_{l+1}, we obtain
ρ(Xu,Xπk(u))≤∑l≥kYl+1\begin{equation*}\rho(X_u, X_{\pi_k(u)}) \leq \sum_{ l \geq k} Y_{l+1}\end{equation*}
We then use the previous inequalities and get
ρ(Xs,Xt)≤ρ(Xs,Xπk(s))+ρ(Xπk(s),Xπk(t))+ρ(Xπk(t),Xt)≤3∑l≥kYl+1\begin{equation*}\rho(X_s, X_t) \leq \rho(X_s, X_{\pi_k(s)}) + \rho(X_{\pi_k(s)}, X_{\pi_k(t)}) + \rho(X_{\pi_k(t)}, X_t) \leq 3 \sum_{ l \geq k} Y_{l+1}\end{equation*}

The inequality for maximum

The number of points on which we consider the maximum process YnY_n can be evaluated as follow
∣{(s,t)∈Gn ⁣:∣s−t∣≤32n}∣≤3d2nd\begin{equation*}\left| \left\{(s,t) \in G_n \colon |s-t| \leq \frac{3}{2^n} \right\} \right| \leq 3^d 2^{nd}\end{equation*}
Then using Chebyshev inequality and condition (1):
P{Yn>12pn}=P{⋃(s,t) ⁣:∣s−t∣≤32−nρ(Xs,Xt)>12pn}≤∑(s,t) ⁣:∣s−t∣≤32−nP{ρ(Xs,Xt)>12pn}≤≤∑(s,t) ⁣:∣s−t∣≤32−nP{ρ(Xs,Xt)a>12pna}≤C(d)2n(pa−b)\begin{align*}P \{ Y_n > \frac{1}{2^{pn}}\} &= P \left\{ \bigcup_{(s,t) \colon |s-t| \leq 3 2^{-n}} \rho(X_s, X_t) > \frac{1}{2^{pn}} \right\} \leq \sum_{(s,t) \colon |s-t| \leq 3 2^{-n} } P \left\{ \rho(X_s, X_t) > \frac{1}{2^{pn}}\right\} \leq \\&\leq \sum_{(s,t) \colon |s-t| \leq 3 2^{-n} } P \left\{ \rho(X_s, X_t)^a > \frac{1}{2^{pna}}\right\} \leq C(d) 2^{n(pa - b)}\end{align*}
where the constant C(d)C(d) depends only on dd.
Now we can verify that XX
If we choose p∈(0,ba)p \in (0, \frac{b}{a})
∑n=1∞P{Yn≥12pn}<∞\begin{equation*}\sum_{n=1}^{\infty} P \left\{ Y_n \geq \frac{1}{2^{pn}} \right\} < \infty\end{equation*}
By Borel-Cantelli Lemma, we get that ∃N0 ∀n≥N0 ⁣:Yn≤2−pn \exists N_0 \, \forall n \geq N_0 \colon Y_n \leq 2^{-p n} a.s. This implies convergence of the series ∑n≥0Yn\sum_{n\geq0} Y_n a.s.

Hölder continuous of XX

Thus
sup⁡{ρ(Xs,Xt) ⁣:s,t∈G,∣s−t∣≤2−m}≤3∑n≥mYn≤3∑n≥m2−pn≤2−pm1−2−p\begin{equation}\sup \left\{ \rho(X_s, X_t) \colon s, t \in G, |s-t| \leq 2^{-m} \right\} \leq 3 \sum_{n\geq m} Y_n \leq 3 \sum_{n\geq m} 2^{-pn} \leq \frac{2^{-pm}}{1 - 2^{-p}}\end{equation}
showing that XX is a.s. Hölder continuous on GG of order pp.
In particular, XX agrees a.s. on ⋃nDn\bigcup_n \mathcal{D}_n with a continuous process X~\tilde{X} on [0,1]d[0,1]^d, and we note that the Hölder continuity of X~\tilde{X} on GG extends with the same order pp to the entire cube [0,1]d[0,1]^d.
To show that X~\tilde{X} is a version of XX, fix any t∈[0,1]dt \in[0,1]^d, and choose t1,t2,…∈Gt_1, t_2, \ldots \in G with tn→tt_n \rightarrow t. Then Xtn=X~tnX_{t_n}=\tilde{X}_{t_n} a.s. for each nn. Since also Xtn→PXtX_{t_n} \stackrel{P}{\rightarrow} X_t by (1) and X~tn→X~t\tilde{X}_{t_n} \rightarrow \tilde{X}_t a.s. by continuity, we get Xt=X~tX_t=\tilde{X}_t a.s.

References