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. (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 and the probabilist Wolfgang Schmidt (not to be confused with the number theorist Wolfgang M. Schmidt).
Engelbert–Schmidt 0–1 law Let F {\displaystyle {\mathcal {F}}} be a σ-algebra and let F = ( F t ) t ≥ 0 {\displaystyle F=({\mathcal {F}}_{t})_{t\geq 0}} be an increasing family of sub-σ-algebras of F {\displaystyle {\mathcal {F}}} . Let ( W , F ) {\displaystyle (W,F)} be a Wiener process on the probability space ( Ω , F , P ) {\displaystyle (\Omega ,{\mathcal {F}},P)} . Suppose that f {\displaystyle f} is a Borel measurable function of the real line into [0,∞]. Then the following three assertions are equivalent: (i) P ( ∫ 0 t f ( W s ) d s < ∞ for all t ≥ 0 ) > 0 {\displaystyle P{\Big (}\int _{0}^{t}f(W_{s})\,\mathrm {d} s<\infty {\text{ for all }}t\geq 0{\Big )}>0} . (ii) P ( ∫ 0 t f ( W s ) d s < ∞ for all t ≥ 0 ) = 1 {\displaystyle P{\Big (}\int _{0}^{t}f(W_{s})\,\mathrm {d} s<\infty {\text{ for all }}t\geq 0{\Big )}=1} . (iii) ∫ K f ( y ) d y < ∞ {\displaystyle \int _{K}f(y)\,\mathrm {d} y<\infty \,} for all compact subsets K {\displaystyle K} of the real line.
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 of index α = 2 {\displaystyle \alpha =2} . Let X {\displaystyle X} be a R {\displaystyle \mathbb {R} } -valued stable process of index α ∈ ( 1 , 2 ] {\displaystyle \alpha \in (1,2]} on the filtered probability space ( Ω , F , ( F t ) , P ) {\displaystyle (\Omega ,{\mathcal {F}},({\mathcal {F}}_{t}),P)} . Suppose that f : R → [ 0 , ∞ ] {\displaystyle f:\mathbb {R} \to [0,\infty ]} is a Borel measurable function. Then the following three assertions are equivalent: (i) P ( ∫ 0 t f ( X s ) d s < ∞ for all t ≥ 0 ) > 0 {\displaystyle P{\Big (}\int _{0}^{t}f(X_{s})\,\mathrm {d} s<\infty {\text{ for all }}t\geq 0{\Big )}>0} . (ii) P ( ∫ 0 t f ( X s ) d s < ∞ for all t ≥ 0 ) = 1 {\displaystyle P{\Big (}\int _{0}^{t}f(X_{s})\,\mathrm {d} s<\infty {\text{ for all }}t\geq 0{\Big )}=1} . (iii) ∫ K f ( y ) d y < ∞ {\displaystyle \int _{K}f(y)\,\mathrm {d} y<\infty \,} for all compact subsets K {\displaystyle K} of the real line. 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 α ∈ ( 1 , 2 ] {\displaystyle \alpha \in (1,2]} , which is known to be jointly continuous.
See also zero-one law
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