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.^[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).

Engelbert–Schmidt 0–1 law

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

(i) ${\displaystyle P{\Big (}\int _{0}^{t}f(W_{s})\,\mathrm {d} s<\infty {\text{ for all }}t\geq 0{\Big )}>0}$.

(ii) ${\displaystyle P{\Big (}\int _{0}^{t}f(W_{s})\,\mathrm {d} s<\infty {\text{ for all }}t\geq 0{\Big )}=1}$.

(iii) ${\displaystyle \int _{K}f(y)\,\mathrm {d} y<\infty \,}$ for all compact subsets ${\displaystyle K}$ of the real line.^[4]

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 ${\displaystyle \alpha =2}$.

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

(i) ${\displaystyle P{\Big (}\int _{0}^{t}f(X_{s})\,\mathrm {d} s<\infty {\text{ for all }}t\geq 0{\Big )}>0}$.

(ii) ${\displaystyle P{\Big (}\int _{0}^{t}f(X_{s})\,\mathrm {d} s<\infty {\text{ for all }}t\geq 0{\Big )}=1}$.

(iii) ${\displaystyle \int _{K}f(y)\,\mathrm {d} y<\infty \,}$ for all compact subsets ${\displaystyle K}$ 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 ${\displaystyle \alpha \in (1,2]}$, which is known to be jointly continuous.^[6]

See also

• zero-one law

References