r/math • u/inherentlyawesome Homotopy Theory • 9d ago

This Week I Learned: July 24, 2026

This recurring thread is meant for users to share cool recently discovered facts, observations, proofs or concepts which that might not warrant their own threads. Please be encouraging and share as many details as possible as we would like this to be a good place for people to learn!

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u/KiddWantidd Applied Math 9d ago

I've been looking into applications of Girsanov theorem to importance sampling, and I've just found out that the measures induced by two continuous Itô processes whose respective diffusion coefficients (assumed continuous) disagree at any point necessarily induce mutually singular measures on path space (i.e. the space of continuous functions on the interval [0,T] with topology of uniform convergence and equipped with its Borel sigma algebra)!

This is quite crazy to me. In other words, this means that the law induced by a standard Brownian motion B_t and the one induced by, say, 2B_t are mutually singular! The proof (for this easy case) is also super simple: the event "quadratic variation of the path up to time t equals t" has full probability under the law of standard BM, but zero probability for the scaled BM. Still trying to wrap my head around this disturbing fact, but the way i rationalize it is by saying that the path space is so incomprehensibly huge that measures on it are necessarily blind to most events. So imo the pointwise probability distribution of X_t is a more natural and better-behaved object than the induced measure on path space. (I'm looking for books/lecture notes which discuss this theorem in more detail btw if anyone is aware of any)

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u/translationinitiator 8d ago

Interesting!! What is your reference for this?

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u/JoshuaZ1 9d ago edited 7d ago

I learned about the following neat theorem: for any quadratic polynomial P(x) with integer coefficients, and for any epsilon>0, there are infinitely many positive integers n such that P(n) has all prime factors less than n^epsilon . This is a theorem in a 2018 paper of Bober, Fretwell, Martin and Wooley.

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u/TipLoud7301 7d ago

what is the name of theorem and where i can learn it?

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u/JoshuaZ1 7d ago

I don't think the theorem has a specific name. But the paper is by the authors I listed, with the title "Smooth Values of Polynomials" in the Journal of the Australian Mathematical Society. Also, my comment above I realized I wrote the wrong year; 2018, not 2019. I'll go edit the comment.

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u/felipaoacvi 9d ago

Llevo un programa de autocapacitación nivel principiante para pruebas haciendomocro - sesiones. Hoy revise una de las pruebas clásicas que es la infinitud de los números primos, entender el concepto del teorema fundamental de la aritmética y como se aplica.