Mixing
Pólya's Theorem implies Transience and Recurrence of Brownian motion?
Clash Royale CLAN TAG#URR8PPP
up vote
1
down vote
favorite
For a talk I am going to prove Pólya's theorem, which states that a simple random walk on $mathbbZ^d$ is recurrent for $d=1,2$ and transient for $dge 3$.
I want to show how this relates to Brownian motion, since a similar result holds. Is there an easy way that Pólya's theorem implies transience and recurrence in different dimensions? Brownian motion is often described as the limit of random walks but I am having trouble finding a reference that spells out the correspondence. For example, I looked at the Donsker’s invariance principle but this seems to only give you a Brownian motion defined for finite time.
probability-theory brownian-motion
asked Jul 26 at 19:24
mysatellite
2,08821027
add a comment |Â
up vote
1
down vote
favorite
For a talk I am going to prove Pólya's theorem, which states that a simple random walk on $mathbbZ^d$ is recurrent for $d=1,2$ and transient for $dge 3$.
I want to show how this relates to Brownian motion, since a similar result holds. Is there an easy way that Pólya's theorem implies transience and recurrence in different dimensions? Brownian motion is often described as the limit of random walks but I am having trouble finding a reference that spells out the correspondence. For example, I looked at the Donsker’s invariance principle but this seems to only give you a Brownian motion defined for finite time.
probability-theory brownian-motion
asked Jul 26 at 19:24
mysatellite
2,08821027
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
For a talk I am going to prove Pólya's theorem, which states that a simple random walk on $mathbbZ^d$ is recurrent for $d=1,2$ and transient for $dge 3$.
I want to show how this relates to Brownian motion, since a similar result holds. Is there an easy way that Pólya's theorem implies transience and recurrence in different dimensions? Brownian motion is often described as the limit of random walks but I am having trouble finding a reference that spells out the correspondence. For example, I looked at the Donsker’s invariance principle but this seems to only give you a Brownian motion defined for finite time.
probability-theory brownian-motion
asked Jul 26 at 19:24
mysatellite
2,08821027
For a talk I am going to prove Pólya's theorem, which states that a simple random walk on $mathbbZ^d$ is recurrent for $d=1,2$ and transient for $dge 3$.
I want to show how this relates to Brownian motion, since a similar result holds. Is there an easy way that Pólya's theorem implies transience and recurrence in different dimensions? Brownian motion is often described as the limit of random walks but I am having trouble finding a reference that spells out the correspondence. For example, I looked at the Donsker’s invariance principle but this seems to only give you a Brownian motion defined for finite time.
probability-theory brownian-motion
asked Jul 26 at 19:24
mysatellite
2,08821027
edited Jul 26 at 19:55
asked Jul 26 at 19:24
mysatellite
2,08821027
asked Jul 26 at 19:24
mysatellite
2,08821027
asked Jul 26 at 19:24
mysatellite
2,08821027
2,08821027
add a comment |Â
add a comment |Â
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2863727%2fp%25c3%25b3lyas-theorem-implies-transience-and-recurrence-of-brownian-motion%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password