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) ${\displaystyle P\left(\int _{0}^{t}f(W_{s})\,ds<\infty ,{\text{ for all }}t\geq 0\right)>0}$

(ii) ${\displaystyle P\left(\int _{0}^{t}f(W_{s})\,ds<\infty {\text{ for all }}t\geq 0\right)=1}$

(iii) ${\displaystyle \int _{K}f(y)\,dy<\infty \,}$

for all compact subsets ${\displaystyle K}$ of the real line.[4]

See also

• zero-one law

References

1. Karatzas, Ioannis; Shreve, Steven (2012). Brownian motion and stochastic calculus. Springer. p. 215.
2. Hans-Jürgen Engelbert at the Mathematics Genealogy Project
3. Wolfgang Schmidt at the Mathematics Genealogy Project
4. Engelbert, H. J.; Schmidt, W. (1981). "On the behavior of certain functionals of the Wiener process and applications to stochastic differential equations". In Arató, M.; Vermes, D.; Balakrishnan, A. V. (eds.). Stochastic Differential Systems. Lectures Notes in Control and Information Sciences, vol. 36. Berlin; Heidelberg: Springer. pp. 47–55. doi:10.1007/BFb0006406.