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

(i) $P(\underset{0}{\overset{t}{\int }}f(W_s)\mathrm{d}s<\mathrm{\infty }\text{ for all }t\ge 0)>0$.

(ii) $P(\underset{0}{\overset{t}{\int }}f(W_s)\mathrm{d}s<\mathrm{\infty }\text{ for all }t\ge 0)=1$.

(iii) $\underset{K}{\int }f(y)\mathrm{d}y<\mathrm{\infty }$ 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 $ℝ$-valued stable process of index $\alpha \in (1,2]$ on the filtered probability space $(\mathrm{\Omega },ℱ,({ℱ}_t),P)$. Suppose that $f:ℝ\to [0,\mathrm{\infty }]$ is a Borel measurable function. Then the following three assertions are equivalent:

(i) $P(\underset{0}{\overset{t}{\int }}f(X_s)\mathrm{d}s<\mathrm{\infty }\text{ for all }t\ge 0)>0$.

(ii) $P(\underset{0}{\overset{t}{\int }}f(X_s)\mathrm{d}s<\mathrm{\infty }\text{ for all }t\ge 0)=1$.

(iii) $\underset{K}{\int }f(y)\mathrm{d}y<\mathrm{\infty }$ 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