Why is the gaussian free field a distribution but Brownian motion is a function?
up vote
2
down vote
favorite
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?
probability-theory stochastic-processes stochastic-calculus stochastic-pde
asked Nov 18 at 23:12
prdnr
645
add a comment |
up vote
2
down vote
favorite
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?
probability-theory stochastic-processes stochastic-calculus stochastic-pde
asked Nov 18 at 23:12
prdnr
645
To complicate things further, there's also Brownian sheet.
– Zachary Selk
Nov 19 at 6:22
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?
probability-theory stochastic-processes stochastic-calculus stochastic-pde
asked Nov 18 at 23:12
prdnr
645
As I understand it, a GFF is a generalisation of Brownian motion to dimensions greater than one. However, they seem like very different objects. Brownian motion is just a continuous function (even though it is nowhere differentiable). By contrast, the Gaussian free field is not a function but only a distribution (it does not have well defined values at points, but only under integrals).
What makes these two objects 'the same'? and why does moving from dimension one to two lead to such different behaviour?
probability-theory stochastic-processes stochastic-calculus stochastic-pde
probability-theory stochastic-processes stochastic-calculus stochastic-pde
asked Nov 18 at 23:12
prdnr
645
asked Nov 18 at 23:12
prdnr
645
asked Nov 18 at 23:12
prdnr
645
asked Nov 18 at 23:12
prdnr
645
asked Nov 18 at 23:12
prdnr
645
645
To complicate things further, there's also Brownian sheet.
– Zachary Selk
Nov 19 at 6:22
add a comment |
To complicate things further, there's also Brownian sheet.
– Zachary Selk
Nov 19 at 6:22
To complicate things further, there's also Brownian sheet.
– Zachary Selk
Nov 19 at 6:22
To complicate things further, there's also Brownian sheet.
– Zachary Selk
Nov 19 at 6:22
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
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
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004263%2fwhy-is-the-gaussian-free-field-a-distribution-but-brownian-motion-is-a-function%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
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
Required, but never shown
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
Required, but never shown
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
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
To complicate things further, there's also Brownian sheet.
– Zachary Selk
Nov 19 at 6:22