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. (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 [math]\displaystyle{ \mathcal{F} }[/math] be a σ-algebra and let [math]\displaystyle{ F = (\mathcal{F}_t)_{t \ge 0} }[/math] be an increasing family of sub-σ-algebras of [math]\displaystyle{ \mathcal{F} }[/math]. Let [math]\displaystyle{ (W, F) }[/math] be a Wiener process on the probability space [math]\displaystyle{ (\Omega, \mathcal{F}, P) }[/math]. Suppose that [math]\displaystyle{ f }[/math] is a Borel measurable function of the real line into [0,∞]. Then the following three assertions are equivalent:

(i) [math]\displaystyle{ P \Big( \int_0^t f (W_s)\,\mathrm ds \lt \infty \text{ for all } t \ge 0 \Big) \gt 0 }[/math].

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

(iii) [math]\displaystyle{ \int_K f (y)\,\mathrm dy \lt \infty \, }[/math] for all compact subsets [math]\displaystyle{ K }[/math] 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 [math]\displaystyle{ \alpha = 2 }[/math].

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

(i) [math]\displaystyle{ P \Big( \int_0^t f (X_s)\,\mathrm ds \lt \infty \text{ for all } t \ge 0 \Big) \gt 0 }[/math].

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

(iii) [math]\displaystyle{ \int_K f (y)\,\mathrm dy \lt \infty \, }[/math] for all compact subsets [math]\displaystyle{ K }[/math] 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 [math]\displaystyle{ \alpha\in(1,2] }[/math], which is known to be jointly continuous.

See also

• zero-one law