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

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

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

(iii) $\int _{K}f(y)\,\mathrm {d} y<\infty \,$ for all compact subsets $K$ 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 $\alpha =2$.

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

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

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

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

See also

• zero-one law