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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 equation|stochastic differential equation]]s.<ref>{{cite book|author=Karatzas, Ioannis|author2=Shreve, Steven|title=Brownian motion and stochastic calculus|publisher=Springer|year=2012|pages=215|url=https://books.google.com/books?id=ATNy_Zg3PSsC&pg=PA215}}</ref> (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<ref>{{MathGenealogy|id=29366|title=Hans-Jürgen Engelbert}}</ref> and the probabilist Wolfgang Schmidt<ref>{{MathGenealogy|id=90300|title=Wolfgang Schmidt}}</ref> (not to be confused with the number theorist [[Biography:Wolfgang M. Schmidt|Wolfgang M. Schmidt]]).
==Engelbert–Schmidt 0–1 law==
Let <math>\mathcal{F}</math> be a [[Σ-algebra|σ-algebra]] and let <math>F = (\mathcal{F}_t)_{t \ge 0}</math> be an increasing family of sub-''σ''-algebras of <math>\mathcal{F}</math>. Let <math>(W, F)</math> be a [[Wiener process]] on the [[Probability space|probability space]] <math>(\Omega, \mathcal{F}, P)</math>.
Suppose that <math>f</math> is a Borel measurable function of the real line into [0,∞].
Then the following three assertions are equivalent:

(i) <math> P \Big( \int_0^t f (W_s)\,\mathrm ds < \infty \text{ for all } t \ge 0 \Big) > 0 </math>.

(ii) <math> P \Big( \int_0^t f (W_s)\,\mathrm ds < \infty \text{ for all } t \ge 0 \Big) = 1 </math>.

(iii) <math> \int_K f (y)\,\mathrm dy < \infty \, </math> for all compact subsets <math>K</math> of the real line.<ref>{{cite book|author=Engelbert, H. J.|author2=Schmidt, W.|chapter=On the behavior of certain functionals of the Wiener process and applications to stochastic differential equations|editor=Arató, M.|editor2=Vermes, D.|editor3=Balakrishnan, A. V.|title=Stochastic Differential Systems|series=Lectures Notes in Control and Information Sciences, vol. 36|year=1981|publisher=Springer|location=Berlin; Heidelberg|pages=47–55|doi=10.1007/BFb0006406}}</ref>

==Extension to stable processes==
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|stable process]] of index <math>\alpha = 2</math>.

Let <math>X</math> be a <math>\mathbb R</math>-valued [[Stable process|stable process]] of index <math>\alpha\in(1,2]</math> on the filtered [[Probability space|probability space]] <math>(\Omega, \mathcal{F}, (\mathcal{F}_t), P)</math>.
Suppose that <math>f:\mathbb R \to [0,\infty]</math> is a Borel measurable function.
Then the following three assertions are equivalent:

(i) <math> P \Big( \int_0^t f (X_s)\,\mathrm ds < \infty \text{ for all } t \ge 0 \Big) > 0 </math>.

(ii) <math> P \Big( \int_0^t f (X_s)\,\mathrm ds < \infty \text{ for all } t \ge 0 \Big) = 1 </math>.

(iii) <math> \int_K f (y)\,\mathrm dy < \infty \, </math> for all compact subsets <math>K</math> of the real line.<ref>{{cite journal|author=Zanzotto, P. A.|title=On solutions of one-dimensional stochastic differential equations driven by stable Lévy motion|year=1997|journal=Stochastic Processes and their Applications|volume=68|pages=209–228|doi=10.1214/aop/1023481008|url=https://projecteuclid.org/journals/annals-of-probability/volume-30/issue-2/On-stochastic-differential-equations-driven-by-a-Cauchy-process-and/10.1214/aop/1023481008.pdf}}</ref>

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 (mathematics)|local time]] process associated with stable processes of index <math>\alpha\in(1,2]</math>, which is known to be jointly continuous.<ref>{{cite book|author=Bertoin, J.|title=Lévy Processes, Theorems V.1, V.15|year=1996|publisher=Cambridge University Press}}</ref>

==See also==
*zero-one law

==References==
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[[Category:Probability theorems]]

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