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.
[1] (A Wiener 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 theorist Wolfgang M. Schmidt).
be a σ-algebra and let
be an increasing family of sub-σ-algebras of
be a Wiener process on the probability space
Suppose that
is a Borel measurable function of the real line into [0,∞].
Then the following three assertions are equivalent: (i)
(ii)
(iii)
for all compact subsets
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-valued stable process of index
α = 2
-valued stable process of index
on the filtered probability space
Suppose that
is a Borel measurable function.
Then the following three assertions are equivalent: (i)
(ii)
(iii)
for all compact subsets
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 the local time process associated with stable processes of index
, which is known to be jointly continuous.