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Measure on $C([0,1]^n)$
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We can define a measure on $C[0,1]$ by viewing a continuous function as a path of 1-dimensional Brownian motion. This is called classical Wiener measure. I am wondering if there is a generalization to function space of higher dimension? Of course by the addition
$+:C[0,1]timescdotstimes C[0,1]rightarrow C([0,1]^n)$, $(f_{1},...,f_{n})mapsto f_{1}+cdots+f_{n}$
we can obtain a pushforward measure on $C([0,1]^n)$, and it does have properties such the set of as nowhere differentiable functions has measure $1$ (I do not know if Hölder condition also holds), but I am not sure if this is a natural generalization since it does not seem to possess intuitive meaning.
I knew from Wikipedia that there is something called abstract Wiener space, which uses a canonical Gaussian cylinder set measure. However I do not know what is the "quotient inner product" mentioned in the construction. Could anyone explain for me what is a quotient inner product, and how does it coincide with the classical case?
probability functional-analysis measure-theory stochastic-processes
asked 18 hours ago
1830rbc03
39045
add a comment |
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We can define a measure on $C[0,1]$ by viewing a continuous function as a path of 1-dimensional Brownian motion. This is called classical Wiener measure. I am wondering if there is a generalization to function space of higher dimension? Of course by the addition
$+:C[0,1]timescdotstimes C[0,1]rightarrow C([0,1]^n)$, $(f_{1},...,f_{n})mapsto f_{1}+cdots+f_{n}$
we can obtain a pushforward measure on $C([0,1]^n)$, and it does have properties such the set of as nowhere differentiable functions has measure $1$ (I do not know if Hölder condition also holds), but I am not sure if this is a natural generalization since it does not seem to possess intuitive meaning.
I knew from Wikipedia that there is something called abstract Wiener space, which uses a canonical Gaussian cylinder set measure. However I do not know what is the "quotient inner product" mentioned in the construction. Could anyone explain for me what is a quotient inner product, and how does it coincide with the classical case?
probability functional-analysis measure-theory stochastic-processes
asked 18 hours ago
1830rbc03
39045
To 1: Of course, there is a simple generalization to the $d$-dimensional case. Just take the $d$-dimensional browinan motion. The $d$-dimensional BM can be defined by setting $B_t = (B_t^{(1)}, ldots B_t^{(d)})$, where $B_t^{(1)},ldots B_t^{(d)}$ are continuous independent Brownian Motions.
– p4sch
14 hours ago
@p4sch But isn't that a measure on $R$ to $R^{n}$ functions instead of $R^{n}$ to $R$ functions?
– 1830rbc03
11 hours ago
Oh, you are right!
– p4sch
11 hours ago
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
We can define a measure on $C[0,1]$ by viewing a continuous function as a path of 1-dimensional Brownian motion. This is called classical Wiener measure. I am wondering if there is a generalization to function space of higher dimension? Of course by the addition
$+:C[0,1]timescdotstimes C[0,1]rightarrow C([0,1]^n)$, $(f_{1},...,f_{n})mapsto f_{1}+cdots+f_{n}$
we can obtain a pushforward measure on $C([0,1]^n)$, and it does have properties such the set of as nowhere differentiable functions has measure $1$ (I do not know if Hölder condition also holds), but I am not sure if this is a natural generalization since it does not seem to possess intuitive meaning.
I knew from Wikipedia that there is something called abstract Wiener space, which uses a canonical Gaussian cylinder set measure. However I do not know what is the "quotient inner product" mentioned in the construction. Could anyone explain for me what is a quotient inner product, and how does it coincide with the classical case?
probability functional-analysis measure-theory stochastic-processes
asked 18 hours ago
1830rbc03
39045
We can define a measure on $C[0,1]$ by viewing a continuous function as a path of 1-dimensional Brownian motion. This is called classical Wiener measure. I am wondering if there is a generalization to function space of higher dimension? Of course by the addition
$+:C[0,1]timescdotstimes C[0,1]rightarrow C([0,1]^n)$, $(f_{1},...,f_{n})mapsto f_{1}+cdots+f_{n}$
we can obtain a pushforward measure on $C([0,1]^n)$, and it does have properties such the set of as nowhere differentiable functions has measure $1$ (I do not know if Hölder condition also holds), but I am not sure if this is a natural generalization since it does not seem to possess intuitive meaning.
I knew from Wikipedia that there is something called abstract Wiener space, which uses a canonical Gaussian cylinder set measure. However I do not know what is the "quotient inner product" mentioned in the construction. Could anyone explain for me what is a quotient inner product, and how does it coincide with the classical case?
probability functional-analysis measure-theory stochastic-processes
probability functional-analysis measure-theory stochastic-processes
asked 18 hours ago
1830rbc03
39045
asked 18 hours ago
1830rbc03
39045
asked 18 hours ago
1830rbc03
39045
asked 18 hours ago
1830rbc03
39045
asked 18 hours ago
1830rbc03
39045
39045
To 1: Of course, there is a simple generalization to the $d$-dimensional case. Just take the $d$-dimensional browinan motion. The $d$-dimensional BM can be defined by setting $B_t = (B_t^{(1)}, ldots B_t^{(d)})$, where $B_t^{(1)},ldots B_t^{(d)}$ are continuous independent Brownian Motions.
– p4sch
14 hours ago
@p4sch But isn't that a measure on $R$ to $R^{n}$ functions instead of $R^{n}$ to $R$ functions?
– 1830rbc03
11 hours ago
Oh, you are right!
– p4sch
11 hours ago
add a comment |
To 1: Of course, there is a simple generalization to the $d$-dimensional case. Just take the $d$-dimensional browinan motion. The $d$-dimensional BM can be defined by setting $B_t = (B_t^{(1)}, ldots B_t^{(d)})$, where $B_t^{(1)},ldots B_t^{(d)}$ are continuous independent Brownian Motions.
– p4sch
14 hours ago
@p4sch But isn't that a measure on $R$ to $R^{n}$ functions instead of $R^{n}$ to $R$ functions?
– 1830rbc03
11 hours ago
Oh, you are right!
– p4sch
11 hours ago
To 1: Of course, there is a simple generalization to the $d$-dimensional case. Just take the $d$-dimensional browinan motion. The $d$-dimensional BM can be defined by setting $B_t = (B_t^{(1)}, ldots B_t^{(d)})$, where $B_t^{(1)},ldots B_t^{(d)}$ are continuous independent Brownian Motions.
– p4sch
14 hours ago
To 1: Of course, there is a simple generalization to the $d$-dimensional case. Just take the $d$-dimensional browinan motion. The $d$-dimensional BM can be defined by setting $B_t = (B_t^{(1)}, ldots B_t^{(d)})$, where $B_t^{(1)},ldots B_t^{(d)}$ are continuous independent Brownian Motions.
– p4sch
14 hours ago
@p4sch But isn't that a measure on $R$ to $R^{n}$ functions instead of $R^{n}$ to $R$ functions?
– 1830rbc03
11 hours ago
@p4sch But isn't that a measure on $R$ to $R^{n}$ functions instead of $R^{n}$ to $R$ functions?
– 1830rbc03
11 hours ago
Oh, you are right!
– p4sch
11 hours ago
Oh, you are right!
– p4sch
11 hours ago
add a comment |
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To 1: Of course, there is a simple generalization to the $d$-dimensional case. Just take the $d$-dimensional browinan motion. The $d$-dimensional BM can be defined by setting $B_t = (B_t^{(1)}, ldots B_t^{(d)})$, where $B_t^{(1)},ldots B_t^{(d)}$ are continuous independent Brownian Motions.
– p4sch
14 hours ago
@p4sch But isn't that a measure on $R$ to $R^{n}$ functions instead of $R^{n}$ to $R$ functions?
– 1830rbc03
11 hours ago
Oh, you are right!
– p4sch
11 hours ago