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).
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 [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 .
Let be a -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. [6]
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).
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 [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 .
Let be a -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. [6]