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

(i) $P \Big( \int_0^t f (W_s)\,\mathrm ds < \infty \text t \ge 0 \Big) > 0$.

(ii) $P \Big( \int_0^t f (W_s)\,\mathrm ds < \infty \text t \ge 0 \Big) = 1$.

(iii) $\int_K f (y)\,\mathrm dy < \infty \,$ 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 $(\Omega, \mathcal, (\mathcal_t), P)$.
Suppose that f:\mathbb R \to ,\infty/math> is a Borel measurable function.
Then the following three assertions are equivalent:

(i) $P \Big( \int_0^t f (X_s)\,\mathrm ds < \infty \text t \ge 0 \Big) > 0$.

(ii) $P \Big( \int_0^t f (X_s)\,\mathrm ds < \infty \text t \ge 0 \Big) = 1$.

(iii) $\int_K f (y)\,\mathrm dy < \infty \,$ 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

References

{{DEFAULTSORT:Engelbert-Schmidt zero-one law
Probability theorems