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I would like to consider a generalization of the classical Wiener process, also known as Brownian Motion.

While the Brownian Motion is a process $W \colon [0,\infty) \times \Omega → \mathbb{R}$, I would like a Wiener process defined on a multi-dimensional domain, that is: $$f \colon \mathbb{R}^n \times \Omega → \mathbb{R} $$

I use the notation $f$, since I see it as a random function, so I would like $f(\cdot, \omega) \in C(\mathbb{R}^n, \mathbb{R})$.

I read here at section 4.2 that it is possible to define a Gaussian measure also on non-separable Banach spaces, but it is always concentrated on a separable subspace.

Now, the space $C(\mathbb{R}^n,\mathbb{R})$ is not even a Banach space, but it is a complete metric space with the topology of uniform convergence in compact sets. With this metric (which is not a norm), it should also be separable.

These are my questions:

  1. Can one construct a Wiener Process on the space $C(\mathbb{R}^n,\mathbb{R})$? Does one use the topology of uniform convergence in compact sets?

I would like that for each $x \in \mathbb{R}^n$, $f(x)$ had a Gaussian distribution with mean 0 and variance $|x|$. The covariance between $f(x)$ and $f(y)$ could be e.g. $\prod_{i=1}^n \min(x_i,y_i)$.

  1. What can we say about the (separable) support of this Wiener process?
  2. What properties can we show about the realizations of this process? For instance, are they continuous a.s.? Are they non-differentiable a.s. like the 1-dim. Brownian Motion?
  3. For the 1-dimensional Brownian Motion, it holds the following formula: $$ \lim_{t → \infty} \frac{W_t}{t} = 0 \qquad \textrm{ a.s. .}$$ Can we say something about the asymptotic behaviour of our abstract Wiener process on $\mathbb{R}^n$, for instance that $$ \lim_{|x| → \infty} \frac{f(x)}{|x|} = 0 \qquad \textrm{ a.s. ?}$$

Thank you for your help!

EDIT: I think that the Brownian Sheets may be the thing I am looking for. Does anyone know a good reference for them?

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  • $\begingroup$ The way the question is written is not helpful to the reader who would like to contribute an answer. 1) There are pure distractions: why take $\mathbb{R}^n$? If you understand $n=1$, just take $n$ independent copies and you have the $\mathbb{R}^n$- valued case. 2) Lots of things are said, but the only thing that matters is missing: what explicit finite-dimensional marginals do you want? There are lots of sensible options here. Finally, did you look up "Brownian sheet"? $\endgroup$ Mar 17, 2021 at 13:10
  • $\begingroup$ You are right, I changed the co-domain and I added a possibility for the finite-dimensional marginals. @AbdelmalekAbdesselam $\endgroup$
    – A. Pesare
    Mar 17, 2021 at 15:21
  • $\begingroup$ I think that Brownian Sheets may be the thing I am looking for. Do you know a good reference for them? @AbdelmalekAbdesselam $\endgroup$
    – A. Pesare
    Mar 17, 2021 at 22:41
  • $\begingroup$ I put a couple in my answer. $\endgroup$ Mar 18, 2021 at 15:09

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With the indicated choice of covariance then it seems this is indeed the Brownian sheet. I think a good introduction to this type of random object is the book "Multiparameter Processes: An Introduction to Random Fields" by Davar Khoshnevisan. For a freely accessible reference, see also his lecture notes http://www.math.utah.edu/~davar/UW/notes.pdf

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