TheEngelbert–Schmidt zero–one law is a theorem that gives a mathematical criterion for an event associated with a continuous, non-decreasing additive functional of Brownian motion to have probability either 0 or 1, without the possibility of an intermediate value. This zero-one law is used in the study of questions of finiteness and asymptotic behavior forstochastic differential equations.^[1] (AWiener process is a mathematical formalization of Brownian motion used in the statement of the theorem.) This 0-1 law, published in 1981, is named after Hans-Jürgen Engelbert^[2] and the probabilist Wolfgang Schmidt^[3] (not to be confused with the number theoristWolfgang M. Schmidt).

Let${\displaystyle \mathcal{F}}$ be aσ-algebra and let${\displaystyle F=({\mathcal{F}}_t{)}_{t\ge 0}}$ be an increasing family of sub-σ-algebras of${\displaystyle \mathcal{F}}$. Let${\displaystyle (W,F)}$ be aWiener process on theprobability space${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$.Suppose that${\displaystyle f}$ is aBorel measurable function of the real line into [0,∞].Then the following three assertions are equivalent:

(i).

(ii).

(iii) for all compact subsets${\displaystyle K}$ of the real line.^[4]

In 1997 Pio Andrea Zanzotto proved the following extension of the Engelbert–Schmidt zero-one law. It contains Engelbert and Schmidt's result as a special case, since the Wiener process is a real-valuedstable process of index${\displaystyle \alpha =2}$.

Let${\displaystyle X}$ be a${\displaystyle \mathbb{R}}$-valuedstable process of index${\displaystyle \alpha \in (1,2]}$ on the filteredprobability space${\displaystyle (\mathrm{\Omega },\mathcal{F},({\mathcal{F}}_t),P)}$.Suppose that${\displaystyle f:\mathbb{R}\to [0,\mathrm{\infty }]}$ is aBorel measurable function.Then the following three assertions are equivalent:

(i).

(ii).

(iii) for all compact subsets${\displaystyle K}$ of the real line.^[5]

The proof of Zanzotto's result is almost identical to that of the Engelbert–Schmidt zero-one law. The key object in the proof is thelocal time process associated with stable processes of index${\displaystyle \alpha \in (1,2]}$, which is known to be jointly continuous.^[6]

• zero-one law

• Theorems in probability theory