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Let

  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $(\mathcal F_t)_{t\ge 0}$ be a filtration of $\mathcal A$
  • $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ be a separable Hilbert space
  • $\left(B^{(n)}\right)_{n\in\mathbb N}$ be independent real-valued Brownian motions with respect to $\mathcal F$ on $(\Omega,\mathcal A,\operatorname P)$
  • $\mathfrak L(X)$ be the set of bounded, linear operators on a normed vector space $X$

I've found two different definitions of a cylindrical Brownian motion and don't understand why they are consistent:

Definition 1: Let

  • $Q\in\mathfrak L(U)$ be nonnegative and symmetric
  • $U_0:=Q^{\frac 12}(U)$, $$\langle u,v\rangle_0:=\langle Q^{-\frac 12}u,Q^{-\frac 12}v\rangle\;\;\;\text{for }u,v\in U_0$$ where $Q^{-\frac 12}$ is the pseudo inverse of $Q^{\frac 12}$ and $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U_0$
  • $(U_1,\langle\;\cdot\;,\;\cdot\;\rangle_1)$ be a separable Hilbert space and $$\iota:(U_0,\langle\;\cdot\;,\;\cdot\;\rangle_0)\to(U_1,\langle\;\cdot\;,\;\cdot\;\rangle_1)$$ be a Hilbert-Schmidt embedding, e.g. $(U_1,\langle\;\cdot\;,\;\cdot\;\rangle_1):=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $$\iota u:=\sum_{n\in\mathbb N}\alpha_n\langle u,e_n\rangle_0e_n\;\;\;\text{for }u\in U_0\tag 1$$ for some $(\alpha_n)_{n\in\mathbb N}\subseteq(0,\infty)$ with $\sum_{n\in\mathbb N}\alpha_n^2<\infty$

Then $$W_t:=\sum_{n\in\mathbb N}B_t^{(n)}\iota e_n\;\;\;\text{for }n\in\mathbb N\text{ and }t\ge 0\tag 2$$ is called a cylindrical $Q$-Brownian motion with respect to $\mathcal F$ on $(\Omega,\mathcal A,\operatorname P)$.

[Note that (2) is convergent in $L^2(\Omega; U_1)$.]

Definition 2: Let

  • $(\tilde e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$

Then $$W_t:=\sum_{n\in\mathbb N}B_t^{(n)}\tilde e_n\;\;\;\text{for }n\in\mathbb N\text{ and }t\ge 0\tag 3$$ is called a cylindrical Brownian motion or space-time white noise with respect to $\mathcal F$ on $(\Omega,\mathcal A,\operatorname P)$.

[Note that (3) is convergent in $L^2(\Omega;\tilde U_1)$ if $U_1\supseteq U$ is a separable Hilbert space and there is a Hilbert-Schmidt inclusion $\tilde\iota:U\to\tilde U_1$.]


Definition 2 seems to be a special case of Definition 1, but how do we obtain it?

If we choose $Q:=\operatorname{id}_U$, then $(U_0,\langle\;\cdot\;,\;\cdot\;\rangle_0)=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$). Now, we could choose $(U_1,\langle\;\cdot\;,\;\cdot\;\rangle_1):=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $\iota$ to be $(1)$ (with $\langle\;\cdot\;,\;\cdot\;\rangle_0$ replaced by $\langle\;\cdot\;,\;\cdot\;\rangle$). However, that doesn't yield that $(2)$ and $(3)$ are equal (does it?).


So, how do we obtain a space-time white noise $(3)$ from $(2)$?

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1 Answer 1

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Let $(\tilde e_n)_{n\in\mathbb N}$ be an orthonormal basis of $U$ with $$Q\tilde e_n=\lambda_n\tilde e_n\;\;\;\text{for all }n\in\mathbb N$$ for some $(\lambda_n)_{n\in\mathbb N}\subseteq[0,\infty)$ $\Rightarrow$ $$e_n:=Q^{\frac 12}\tilde e_n=\lambda_n^{\frac 12}\tilde e_n\;\;\;\text{for }n\in\mathbb N\tag 4$$ is an orthonormal basis of $U_0$. Let $(U_1,\langle\;\cdot\;,\;\cdot\;\rangle_1)$ be a separable Hilbert space with $U_0\subseteq U_1$ such that the inclusion $$\iota:(U_0,\langle\;\cdot\;,\;\cdot\;\rangle_0)\to(U_1,\langle\;\cdot\;,\;\cdot\;\rangle_1)$$ is Hilbert-Schmidt. Then $$W_t:=\sum_{n\in\mathbb N}B_t^{(n)}\iota e_n\stackrel{(4)}=\sum_{n\in\mathbb N}\lambda_n^{\frac 12}B_t^{(n)}\tilde e_n\;\;\;\text{for }t\ge 0\tag 5$$ is a cylindrical $Q$-Brownian motion with respect to $\mathcal F$ on $(\Omega,\mathcal A,\operatorname P)$ according to Definition 1. Choose $Q:=\operatorname{id}_U$ $\Rightarrow$ $(U_0,\langle\;\cdot\;,\;\cdot\;\rangle_0)=(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $(\tilde e_n)_{n\in\mathbb N}$ can be any orthonormal basis of $U$ (since $\lambda_n=1$ for all $n\in\mathbb N$) and we obtain that $$W_t=\sum_{n\in\mathbb N}B_t^{(n)}\tilde e_n\;\;\;\text{for all }t\ge 0\tag 6$$ is a cylindrical Brownian motion or space-time white noise with respect to $\mathcal F$ on $(\Omega,\mathcal A,\operatorname P)$ according to Definition 2.

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