Engelbert–Schmidt zero–one law


The Engelbert–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 for stochastic differential equations. This 0-1 law, published in 1981, is named after Hans-Jürgen Engelbert and the probabilist Wolfgang Schmidt.

Engelbert–Schmidt 0–1 law

Let be a σ-algebra and let be an increasing family of sub-σ-algebras of. Let be a Wiener process on the probability space.
Suppose that is a Borel measurable function of the real line into .
Then the following three assertions are equivalent:



for all compact subsets of the real line.