Mixing
Why is the canonical projection Lipschitz?
$begingroup$
Let $A={w:[0,infty)->mathbb{R}^d, wtext{ is continuous and } w(0)=0}$, and let's use this metric of locally uniform convergence:
$rho(w,v)=sum_{ngeq 1} (1wedge sup_{iin [0,n]}|w(i)-v(i)|)2^{-n} $.
Let also $pi_t(w):=w(t)$(canonical projection).
Is the $pi_t$ Lipschitz? Why?
I'm reading, Brownian Motion by Schlling and Partzsch, and they state that it is, without any proof. I've tried, but couldn't, and in fact I'm convinced it's not. Maybe they were thinking of Locally Lipschitz?
real-analysis probability-theory stochastic-processes
asked Feb 1 at 11:51
An old man in the sea.An old man in the sea.
1,65711135
$endgroup$
add a comment |
$begingroup$
Let $A={w:[0,infty)->mathbb{R}^d, wtext{ is continuous and } w(0)=0}$, and let's use this metric of locally uniform convergence:
$rho(w,v)=sum_{ngeq 1} (1wedge sup_{iin [0,n]}|w(i)-v(i)|)2^{-n} $.
Let also $pi_t(w):=w(t)$(canonical projection).
Is the $pi_t$ Lipschitz? Why?
I'm reading, Brownian Motion by Schlling and Partzsch, and they state that it is, without any proof. I've tried, but couldn't, and in fact I'm convinced it's not. Maybe they were thinking of Locally Lipschitz?
real-analysis probability-theory stochastic-processes
asked Feb 1 at 11:51
An old man in the sea.An old man in the sea.
1,65711135
$endgroup$
$begingroup$
What is $t$? An integer? Same question for $i$.
$endgroup$
– mathcounterexamples.net
Feb 1 at 12:18
add a comment |
$begingroup$
Let $A={w:[0,infty)->mathbb{R}^d, wtext{ is continuous and } w(0)=0}$, and let's use this metric of locally uniform convergence:
$rho(w,v)=sum_{ngeq 1} (1wedge sup_{iin [0,n]}|w(i)-v(i)|)2^{-n} $.
Let also $pi_t(w):=w(t)$(canonical projection).
Is the $pi_t$ Lipschitz? Why?
I'm reading, Brownian Motion by Schlling and Partzsch, and they state that it is, without any proof. I've tried, but couldn't, and in fact I'm convinced it's not. Maybe they were thinking of Locally Lipschitz?
real-analysis probability-theory stochastic-processes
asked Feb 1 at 11:51
An old man in the sea.An old man in the sea.
1,65711135
$endgroup$
Let $A={w:[0,infty)->mathbb{R}^d, wtext{ is continuous and } w(0)=0}$, and let's use this metric of locally uniform convergence:
$rho(w,v)=sum_{ngeq 1} (1wedge sup_{iin [0,n]}|w(i)-v(i)|)2^{-n} $.
Let also $pi_t(w):=w(t)$(canonical projection).
Is the $pi_t$ Lipschitz? Why?
I'm reading, Brownian Motion by Schlling and Partzsch, and they state that it is, without any proof. I've tried, but couldn't, and in fact I'm convinced it's not. Maybe they were thinking of Locally Lipschitz?
real-analysis probability-theory stochastic-processes
real-analysis probability-theory stochastic-processes
asked Feb 1 at 11:51
An old man in the sea.An old man in the sea.
1,65711135
asked Feb 1 at 11:51
An old man in the sea.An old man in the sea.
1,65711135
edited Feb 1 at 12:01
An old man in the sea.
asked Feb 1 at 11:51
An old man in the sea.An old man in the sea.
1,65711135
asked Feb 1 at 11:51
An old man in the sea.An old man in the sea.
1,65711135
asked Feb 1 at 11:51
An old man in the sea.An old man in the sea.
1,65711135
1,65711135
$begingroup$
What is $t$? An integer? Same question for $i$.
$endgroup$
– mathcounterexamples.net
Feb 1 at 12:18
add a comment |
$begingroup$
What is $t$? An integer? Same question for $i$.
$endgroup$
– mathcounterexamples.net
Feb 1 at 12:18
$begingroup$
What is $t$? An integer? Same question for $i$.
$endgroup$
– mathcounterexamples.net
Feb 1 at 12:18
$begingroup$
What is $t$? An integer? Same question for $i$.
$endgroup$
– mathcounterexamples.net
Feb 1 at 12:18
add a comment |
1 Answer
1
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It's certainly not globally Lipschitz, since $rho(w,v)le 1$ always, but $pi_t$ is unbounded. Yes, I suspect they mean locally Lipschitz.
answered Feb 1 at 12:55
Robert IsraelRobert Israel
331k23221477
$endgroup$
$begingroup$
math.stackexchange.com/questions/3085502/… Robert, I wrote an answer to a question equal to this. In my answer I try to prove that's it's locally lipschitz. Could you please check if it's ok? I hope I'm not being bothersome. ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
Also, Thanks for your answer. =D ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
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oldest
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$begingroup$
It's certainly not globally Lipschitz, since $rho(w,v)le 1$ always, but $pi_t$ is unbounded. Yes, I suspect they mean locally Lipschitz.
answered Feb 1 at 12:55
Robert IsraelRobert Israel
331k23221477
$endgroup$
$begingroup$
math.stackexchange.com/questions/3085502/… Robert, I wrote an answer to a question equal to this. In my answer I try to prove that's it's locally lipschitz. Could you please check if it's ok? I hope I'm not being bothersome. ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
Also, Thanks for your answer. =D ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
add a comment |
$begingroup$
It's certainly not globally Lipschitz, since $rho(w,v)le 1$ always, but $pi_t$ is unbounded. Yes, I suspect they mean locally Lipschitz.
answered Feb 1 at 12:55
Robert IsraelRobert Israel
331k23221477
$endgroup$
$begingroup$
math.stackexchange.com/questions/3085502/… Robert, I wrote an answer to a question equal to this. In my answer I try to prove that's it's locally lipschitz. Could you please check if it's ok? I hope I'm not being bothersome. ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
Also, Thanks for your answer. =D ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
add a comment |
$begingroup$
It's certainly not globally Lipschitz, since $rho(w,v)le 1$ always, but $pi_t$ is unbounded. Yes, I suspect they mean locally Lipschitz.
answered Feb 1 at 12:55
Robert IsraelRobert Israel
331k23221477
$endgroup$
It's certainly not globally Lipschitz, since $rho(w,v)le 1$ always, but $pi_t$ is unbounded. Yes, I suspect they mean locally Lipschitz.
answered Feb 1 at 12:55
Robert IsraelRobert Israel
331k23221477
answered Feb 1 at 12:55
Robert IsraelRobert Israel
331k23221477
answered Feb 1 at 12:55
Robert IsraelRobert Israel
331k23221477
answered Feb 1 at 12:55
Robert IsraelRobert Israel
331k23221477
331k23221477
$begingroup$
math.stackexchange.com/questions/3085502/… Robert, I wrote an answer to a question equal to this. In my answer I try to prove that's it's locally lipschitz. Could you please check if it's ok? I hope I'm not being bothersome. ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
Also, Thanks for your answer. =D ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
add a comment |
$begingroup$
math.stackexchange.com/questions/3085502/… Robert, I wrote an answer to a question equal to this. In my answer I try to prove that's it's locally lipschitz. Could you please check if it's ok? I hope I'm not being bothersome. ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
Also, Thanks for your answer. =D ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
math.stackexchange.com/questions/3085502/… Robert, I wrote an answer to a question equal to this. In my answer I try to prove that's it's locally lipschitz. Could you please check if it's ok? I hope I'm not being bothersome. ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
math.stackexchange.com/questions/3085502/… Robert, I wrote an answer to a question equal to this. In my answer I try to prove that's it's locally lipschitz. Could you please check if it's ok? I hope I'm not being bothersome. ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
Also, Thanks for your answer. =D ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
$begingroup$
Also, Thanks for your answer. =D ;)
$endgroup$
– An old man in the sea.
Feb 1 at 14:41
add a comment |
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$begingroup$
What is $t$? An integer? Same question for $i$.
$endgroup$
– mathcounterexamples.net
Feb 1 at 12:18