ℹ️ Skipped - page is already crawled
| Filter | Status | Condition | Details |
|---|---|---|---|
| HTTP status | PASS | download_http_code = 200 | HTTP 200 |
| Age cutoff | FAIL | download_stamp > now() - 6 MONTH | 7.9 months ago |
| History drop | PASS | isNull(history_drop_reason) | No drop reason |
| Spam/ban | PASS | fh_dont_index != 1 AND ml_spam_score = 0 | ml_spam_score=0 |
| Canonical | PASS | meta_canonical IS NULL OR = '' OR = src_unparsed | Not set |
| Property | Value |
|---|---|
| URL | https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion |
| Last Crawled | 2025-08-20 22:31:15 (7 months ago) |
| First Indexed | not set |
| HTTP Status Code | 200 |
| Meta Title | probability - A counter example of Brownian Motion - Mathematics Stack Exchange |
| Meta Description | null |
| Meta Canonical | null |
| Boilerpipe Text | $U$ is a random variable, but its value is determined before the Brownian motion begins to evolve, we can tell if t equals $U$ or not. Look at the finite dimensional distribution $(\tilde{B}_{t_1}, \cdots, \tilde{B}_{t_n})$ of $\tilde{B}_t$, we have $$(\tilde{B}_{t_1}, \cdots, \tilde{B}_{t_n}) = ({B}_{t_1}, \cdots, {B}_{t_n})$$
on the event $I = \{U\neq t_i,\cdots, U\neq t_n\}$. But $P(I) = 1$, which means $(\tilde{B}_{t_1}, \cdots, \tilde{B}_{t_n}) = ({B}_{t_1}, \cdots, {B}_{t_n})$ almost surely. So we have $E[f(\tilde{B}_{t_1}, \cdots, \tilde{B}_{t_n})] = E[f({B}_{t_1}, \cdots, {B}_{t_n})]$ for any measurable function $f$ |
| Markdown | [Skip to main content](https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion#content)
#### Stack Exchange Network
Stack Exchange network consists of 183 Q\&A communities including [Stack Overflow](https://stackoverflow.com/), the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.
[Visit Stack Exchange](https://stackexchange.com/)
1. - [Tour Start here for a quick overview of the site](https://math.stackexchange.com/tour)
- [Help Center Detailed answers to any questions you might have](https://math.stackexchange.com/help)
- [Meta Discuss the workings and policies of this site](https://math.meta.stackexchange.com/)
- [About Us Learn more about Stack Overflow the company, and our products](https://stackoverflow.co/)
2. ### [current community](https://math.stackexchange.com/)
- [Mathematics](https://math.stackexchange.com/)
[help](https://math.stackexchange.com/help) [chat](https://chat.stackexchange.com/?tab=site&host=math.stackexchange.com)
- [Mathematics Meta](https://math.meta.stackexchange.com/)
### your communities
[Sign up](https://math.stackexchange.com/users/signup?ssrc=site_switcher&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F903892%2Fa-counter-example-of-brownian-motion) or [log in](https://math.stackexchange.com/users/login?ssrc=site_switcher&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F903892%2Fa-counter-example-of-brownian-motion) to customize your list.
### [more stack exchange communities](https://stackexchange.com/sites)
[company blog](https://stackoverflow.blog/)
3. [Log in](https://math.stackexchange.com/users/login?ssrc=head&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F903892%2Fa-counter-example-of-brownian-motion)
4. [Sign up](https://math.stackexchange.com/users/signup?ssrc=head&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F903892%2Fa-counter-example-of-brownian-motion)
[](https://math.stackexchange.com/)
1. 1. [Home](https://math.stackexchange.com/)
2. [Questions](https://math.stackexchange.com/questions)
3. [Unanswered](https://math.stackexchange.com/unanswered)
4. [AI Assist Labs](https://stackoverflow.ai/)
5. [Tags](https://math.stackexchange.com/tags)
6. [Chat](https://chat.stackexchange.com/rooms/158962/stack-exchange-lobby)
7. [Users](https://math.stackexchange.com/users)
2. Teams
Ask questions, find answers and collaborate at work with Stack Overflow for Teams.
[Try Teams for free](https://stackoverflowteams.com/teams/create/free/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams) [Explore Teams](https://stackoverflow.co/teams/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams)
3. [Teams]()
4. Ask questions, find answers and collaborate at work with Stack Overflow for Teams. [Explore Teams](https://stackoverflow.co/teams/?utm_medium=referral&utm_source=math-community&utm_campaign=side-bar&utm_content=explore-teams-compact)
**Teams**
Q\&A for work
Connect and share knowledge within a single location that is structured and easy to search.
[Learn more about Teams](https://stackoverflow.co/teams/)
# [A counter example of Brownian Motion](https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion)
[Ask Question](https://math.stackexchange.com/questions/ask)
Asked
11 years ago
Modified [10 years, 10 months ago](https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion?lastactivity "2014-10-04 10:47:16Z")
Viewed 744 times
4
\$\\begingroup\$
Here is an example in my textbook to illustrate why we need the continuous sample path in the definition of Brownian motion. Let \$(B\_t)\$ be a Brownian motion and \$U\$ be a uniform random variable on \$\[0,1\]\$. Define \$(\\tilde B\_t)\$ by
\$\$ \\tilde B\_t = \\begin{cases} B\_t & \\mbox{if } t\\neq U, \\\\ 0 & \\mbox{if } t = U. \\end{cases} \$\$
Then it is claimed that \$\\tilde B\_t\$ has the same finite-dimensional distribution as \$B\_t\$ but with discontinuous sample path if \$B(U)\\neq 0\$. I do not understand this example. To be more specific,
1. What does it mean by \$t \\neq U\$ or \$t=U\$, please? Here, \$t\$ is a deterministic time and \$U\$ is a random variable. How can we talk about whether the two are equal or not?
2. Since I do not get the definition of \$\\tilde B\_t\$, I cannot see why it has the same finite dimensional distribution and discontinuous sample path. Could anyone explain this with more detail, please? Thank you\!
- [probability](https://math.stackexchange.com/questions/tagged/probability "show questions tagged 'probability'")
- [measure-theory](https://math.stackexchange.com/questions/tagged/measure-theory "show questions tagged 'measure-theory'")
- [probability-theory](https://math.stackexchange.com/questions/tagged/probability-theory "show questions tagged 'probability-theory'")
- [self-learning](https://math.stackexchange.com/questions/tagged/self-learning "show questions tagged 'self-learning'")
- [brownian-motion](https://math.stackexchange.com/questions/tagged/brownian-motion "show questions tagged 'brownian-motion'")
[Share](https://math.stackexchange.com/q/903892 "Short permalink to this question")
Cite
Follow
asked Aug 20, 2014 at 10:39
[](https://math.stackexchange.com/users/103063/latexfan)
[LaTeXFan](https://math.stackexchange.com/users/103063/latexfan)LaTeXFan
3,66811 gold badge2929 silver badges7070 bronze badges
\$\\endgroup\$
1
- 1
\$\\begingroup\$ Could you please tell in which textbook you found this example? \$\\endgroup\$
– [Cm7F7Bb](https://math.stackexchange.com/users/23249/cm7f7bb "17,889 reputation")
Commented Aug 20, 2014 at 10:49
[Add a comment](https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion "Use comments to ask for more information or suggest improvements. Avoid answering questions in comments.") \|
## 2 Answers 2
Sorted by:
[Reset to default](https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion?answertab=scoredesc#tab-top)
4
\$\\begingroup\$
Re 1., the expanded definition (maybe clearer) of the random process \$\\tilde B\$ is that, for every \$\\omega\$ in \$\\Omega\$, \$\\tilde B\_{U(\\omega)}(\\omega)=0\$ and \$\\tilde B\_t(\\omega)=B\_t(\\omega)\$ if \$t\\ne U(\\omega)\$. In short, \$\$ \\tilde B\_t=B\_t\\,\\mathbf 1\_{U\\ne t}. \$\$ Re 2., the almost sure path discontinuity of \$\\tilde B\$ should be obvious. To wit, the event that \$\\tilde B\$ is continuous is \$\[B\_U=0\]\$, the notation being again a shortcut, this time to the event \$\$ \[\\omega\\in\\Omega\\mid B\_{U(\\omega)}(\\omega)=0\]. \$\$ Since \$P(B\_t=0)=0\$ for every \$t\$ and \$U\$ is independent of \$B\$, one gets indeed that \$P(B\_U=0)=0\$.
The equality of the finite marginal distributions of \$B\$ and \$\\tilde B\$ follows from the remark that, for every finite subset \$T\$ of the time interval, the event \$\$ A=\[\\omega\\in\\Omega\\mid\\forall t\\in T, B\_t(\\omega)=\\tilde B\_t(\\omega)\], \$\$ has full probability, since \$P(U=t)=0\$ for every fixed \$t\$ and \$\$ \\Omega\\setminus A\\subseteq\[U\\in T\]=\\bigcup\_{t\\in T}\[U=t\]. \$\$
[Share](https://math.stackexchange.com/a/903903 "Short permalink to this answer")
Cite
Follow
[edited Oct 4, 2014 at 10:47](https://math.stackexchange.com/posts/903903/revisions "show all edits to this post")
answered Aug 20, 2014 at 10:56
[](https://math.stackexchange.com/users/6179/did)
[Did](https://math.stackexchange.com/users/6179/did)Did
284k2727 gold badges332332 silver badges613613 bronze badges
\$\\endgroup\$
[Add a comment](https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \|
1
\$\\begingroup\$
\$U\$ is a random variable, but its value is determined before the Brownian motion begins to evolve, we can tell if t equals \$U\$ or not.
Look at the finite dimensional distribution \$(\\tilde{B}\_{t\_1}, \\cdots, \\tilde{B}\_{t\_n})\$ of \$\\tilde{B}\_t\$, we have \$\$(\\tilde{B}\_{t\_1}, \\cdots, \\tilde{B}\_{t\_n}) = ({B}\_{t\_1}, \\cdots, {B}\_{t\_n})\$\$ on the event \$I = \\{U\\neq t\_i,\\cdots, U\\neq t\_n\\}\$.
But \$P(I) = 1\$, which means \$(\\tilde{B}\_{t\_1}, \\cdots, \\tilde{B}\_{t\_n}) = ({B}\_{t\_1}, \\cdots, {B}\_{t\_n})\$ almost surely. So we have \$E\[f(\\tilde{B}\_{t\_1}, \\cdots, \\tilde{B}\_{t\_n})\] = E\[f({B}\_{t\_1}, \\cdots, {B}\_{t\_n})\]\$ for any measurable function \$f\$
[Share](https://math.stackexchange.com/a/903907 "Short permalink to this answer")
Cite
Follow
answered Aug 20, 2014 at 10:57
[](https://math.stackexchange.com/users/100564/petite-etincelle)
[Petite Etincelle](https://math.stackexchange.com/users/100564/petite-etincelle)Petite Etincelle
15k22 gold badges3333 silver badges6161 bronze badges
\$\\endgroup\$
0
[Add a comment](https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion "Use comments to ask for more information or suggest improvements. Avoid comments like “+1” or “thanks”.") \|
## You must [log in](https://math.stackexchange.com/users/login?ssrc=question_page&returnurl=https%3A%2F%2Fmath.stackexchange.com%2Fquestions%2F903892) to answer this question.
Start asking to get answers
Find the answer to your question by asking.
[Ask question](https://math.stackexchange.com/questions/ask)
Explore related questions
- [probability](https://math.stackexchange.com/questions/tagged/probability "show questions tagged 'probability'")
- [measure-theory](https://math.stackexchange.com/questions/tagged/measure-theory "show questions tagged 'measure-theory'")
- [probability-theory](https://math.stackexchange.com/questions/tagged/probability-theory "show questions tagged 'probability-theory'")
- [self-learning](https://math.stackexchange.com/questions/tagged/self-learning "show questions tagged 'self-learning'")
- [brownian-motion](https://math.stackexchange.com/questions/tagged/brownian-motion "show questions tagged 'brownian-motion'")
See similar questions with these tags.
- Featured on Meta
- [Upcoming initiatives on Stack Overflow and across the Stack Exchange network...](https://meta.stackexchange.com/questions/411578/upcoming-initiatives-on-stack-overflow-and-across-the-stack-exchange-network-j "Upcoming initiatives on Stack Overflow and across the Stack Exchange network - July 2025")
- [Community help needed to clean up goo.gl links (by August 25)](https://math.meta.stackexchange.com/questions/39468/community-help-needed-to-clean-up-goo-gl-links-by-august-25)
#### Related
[7](https://math.stackexchange.com/questions/231914/beginners-question-about-brownian-motion "Question score (upvotes - downvotes)")
[beginner's question about Brownian motion](https://math.stackexchange.com/questions/231914/beginners-question-about-brownian-motion)
[8](https://math.stackexchange.com/questions/683396/a-brownian-motion-b-that-is-discontinuous-at-an-independent-uniformly-distrib "Question score (upvotes - downvotes)")
[A Brownian motion \$B\$ that is discontinuous at an independent, uniformly distributed random variable \$U(0,1)\$](https://math.stackexchange.com/questions/683396/a-brownian-motion-b-that-is-discontinuous-at-an-independent-uniformly-distrib)
[2](https://math.stackexchange.com/questions/904754/invariance-properties-of-brownian-motion "Question score (upvotes - downvotes)")
[Invariance Properties of Brownian Motion](https://math.stackexchange.com/questions/904754/invariance-properties-of-brownian-motion)
[1](https://math.stackexchange.com/questions/1124672/why-is-brownian-motion-b-t-distributed-as-n0-t "Question score (upvotes - downvotes)")
[Why is Brownian Motion B\_t distributed as N(0,t)?](https://math.stackexchange.com/questions/1124672/why-is-brownian-motion-b-t-distributed-as-n0-t)
[5](https://math.stackexchange.com/questions/2635449/finite-dimensional-distributions-of-brownian-motion-changed-at-a-random-point "Question score (upvotes - downvotes)")
[Finite dimensional distributions of brownian motion changed at a random point](https://math.stackexchange.com/questions/2635449/finite-dimensional-distributions-of-brownian-motion-changed-at-a-random-point)
[0](https://math.stackexchange.com/questions/4778087/is-sample-path-of-brownian-motion-deterministic "Question score (upvotes - downvotes)")
[Is sample path of Brownian Motion deterministic?](https://math.stackexchange.com/questions/4778087/is-sample-path-of-brownian-motion-deterministic)
#### [Hot Network Questions](https://stackexchange.com/questions?tab=hot)
- [What the editor meant by 'quick opinion' in this email regarding review report of an article?](https://academia.stackexchange.com/questions/220928/what-the-editor-meant-by-quick-opinion-in-this-email-regarding-review-report-o)
- [Can an AI model license determine which governing law and jurisdiction apply to the users of this model?](https://law.stackexchange.com/questions/110658/can-an-ai-model-license-determine-which-governing-law-and-jurisdiction-apply-to)
- [Let R be a positive integer less than 2017. Exactly R vertices of a regular 2017-gon are red, hence find the number of certain triangles.](https://math.stackexchange.com/questions/5091202/let-r-be-a-positive-integer-less-than-2017-exactly-r-vertices-of-a-regular-2017)
- [Story: humans trying to escape economic slavery by finding aliens' nuclear conversion method](https://scifi.stackexchange.com/questions/298779/story-humans-trying-to-escape-economic-slavery-by-finding-aliens-nuclear-conve)
- [Restaurants with a view of the Alps in Munich](https://travel.stackexchange.com/questions/199993/restaurants-with-a-view-of-the-alps-in-munich)
- [What is causing the LED to turn on instantly as opposed to after a delay in this circuit?](https://electronics.stackexchange.com/questions/754042/what-is-causing-the-led-to-turn-on-instantly-as-opposed-to-after-a-delay-in-this)
- [Do planar transformers need insulation between core and PCB?](https://electronics.stackexchange.com/questions/754081/do-planar-transformers-need-insulation-between-core-and-pcb)
- [Is there a way to find someone to be a co-signer if you have no other option?](https://money.stackexchange.com/questions/166537/is-there-a-way-to-find-someone-to-be-a-co-signer-if-you-have-no-other-option)
- [Where to start with this one area of siding/window?](https://diy.stackexchange.com/questions/322561/where-to-start-with-this-one-area-of-siding-window)
- [What kind of instrument is my door?](https://music.stackexchange.com/questions/141244/what-kind-of-instrument-is-my-door)
- [What should the marked content of multiple nodes be for tagged tikzpictures?](https://tex.stackexchange.com/questions/749945/what-should-the-marked-content-of-multiple-nodes-be-for-tagged-tikzpictures)
- [For us to see an exoplanet transit, what is the maximum angle between that planet's orbital plane and our line of sight?](https://astronomy.stackexchange.com/questions/61577/for-us-to-see-an-exoplanet-transit-what-is-the-maximum-angle-between-that-plane)
- [Pennies to Dollars](https://codegolf.stackexchange.com/questions/283193/pennies-to-dollars)
- [What is the exact wording of anti-Hamas and "pro-Israel" declarations by Saudi Arabia, Egypt, Jordan and Qatar, mentioned by Frédéric Journès?](https://politics.stackexchange.com/questions/93366/what-is-the-exact-wording-of-anti-hamas-and-pro-israel-declarations-by-saudi-a)
- [Is there a tool that can generate possible but non-existent words for a chosen language?](https://linguistics.stackexchange.com/questions/51257/is-there-a-tool-that-can-generate-possible-but-non-existent-words-for-a-chosen-l)
- [Will an aggressive-play kitten grow out of this?](https://pets.stackexchange.com/questions/39495/will-an-aggressive-play-kitten-grow-out-of-this)
- [Daniel 7:3&17 Out of the sea or out of the earth; which is it?](https://hermeneutics.stackexchange.com/questions/107700/daniel-7317-out-of-the-sea-or-out-of-the-earth-which-is-it)
- [What is Trump's position with respect to the famine in Gaza?](https://politics.stackexchange.com/questions/93352/what-is-trumps-position-with-respect-to-the-famine-in-gaza)
- [what is the meaning of "bei noch nicht der Hälfte" and how does it acquire its meaning grammatically and semantically?](https://german.stackexchange.com/questions/80589/what-is-the-meaning-of-bei-noch-nicht-der-h%C3%A4lfte-and-how-does-it-acquire-its-m)
- [Image of affine variety cannot be complete](https://math.stackexchange.com/questions/5091039/image-of-affine-variety-cannot-be-complete)
- [Should theological questions be considered philosophical questions and dealt with by philosophers?](https://philosophy.stackexchange.com/questions/129725/should-theological-questions-be-considered-philosophical-questions-and-dealt-wit)
- [Can I use a PCIe 4.0 M.2 NVMe SSD on a PCIe 3.0 M.2 NVMe slot?](https://superuser.com/questions/1919290/can-i-use-a-pcie-4-0-m-2-nvme-ssd-on-a-pcie-3-0-m-2-nvme-slot)
- [How do I pin a folder to quick access if I don’t have access to the parent folder?](https://superuser.com/questions/1919161/how-do-i-pin-a-folder-to-quick-access-if-i-don-t-have-access-to-the-parent-folde)
- [Why is there no expansion of a local let despite using xdef here?](https://tex.stackexchange.com/questions/749917/why-is-there-no-expansion-of-a-local-let-despite-using-xdef-here)
[more hot questions](https://math.stackexchange.com/questions/903892/a-counter-example-of-brownian-motion)
[Question feed](https://math.stackexchange.com/feeds/question/903892 "Feed of this question and its answers")
# Subscribe to RSS
Question feed
To subscribe to this RSS feed, copy and paste this URL into your RSS reader.
##### [Mathematics](https://math.stackexchange.com/)
- [Tour](https://math.stackexchange.com/tour)
- [Help](https://math.stackexchange.com/help)
- [Chat](https://chat.stackexchange.com/?tab=site&host=math.stackexchange.com)
- [Contact](https://math.stackexchange.com/contact)
- [Feedback](https://math.meta.stackexchange.com/)
##### [Company](https://stackoverflow.co/)
- [Stack Overflow](https://stackoverflow.com/)
- [Teams](https://stackoverflow.co/teams/)
- [Advertising](https://stackoverflow.co/advertising/)
- [Talent](https://stackoverflow.co/advertising/employer-branding/)
- [About](https://stackoverflow.co/)
- [Press](https://stackoverflow.co/company/press/)
- [Legal](https://stackoverflow.com/legal)
- [Privacy Policy](https://stackoverflow.com/legal/privacy-policy)
- [Terms of Service](https://stackoverflow.com/legal/terms-of-service/public)
- Cookie Settings
- [Cookie Policy](https://policies.stackoverflow.co/stack-overflow/cookie-policy)
##### [Stack Exchange Network](https://stackexchange.com/)
- [Technology](https://stackexchange.com/sites#technology)
- [Culture & recreation](https://stackexchange.com/sites#culturerecreation)
- [Life & arts](https://stackexchange.com/sites#lifearts)
- [Science](https://stackexchange.com/sites#science)
- [Professional](https://stackexchange.com/sites#professional)
- [Business](https://stackexchange.com/sites#business)
- [API](https://api.stackexchange.com/)
- [Data](https://data.stackexchange.com/)
- [Blog](https://stackoverflow.blog/?blb=1)
- [Facebook](https://www.facebook.com/officialstackoverflow/)
- [Twitter](https://twitter.com/stackoverflow)
- [LinkedIn](https://linkedin.com/company/stack-overflow)
- [Instagram](https://www.instagram.com/thestackoverflow)
Site design / logo © 2025 Stack Exchange Inc; user contributions licensed under [CC BY-SA](https://stackoverflow.com/help/licensing) . rev 2025.8.20.33080 |
| Readable Markdown | null |
| Shard | 18 (laksa) |
| Root Hash | 8045678284012640218 |
| Unparsed URL | com,stackexchange!math,/questions/903892/a-counter-example-of-brownian-motion s443 |