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gnob

New member
Dec 11, 2012
11
Good day! I came across this symbol $dt \otimes dP$-a.e. in the book of Mandrekar (page 72) Stochastic Differential Equations in Infinite Dimensions: With Applications ... - Leszek Gawarecki, Vidyadhar Mandrekar - Google Books.

What does this symbol mean? I understand that in real analysis, given a measure space $(X,\mathcal{A},\mu)$ we say that a property holds $\mu$-a.e. if there is a set $N$ such that $\mu(N)=0$ and the property holds for all $x\in (X\smallsetminus N).$

I am a newbie with the symbols $dt\otimes dP$ since $dt$ and $dP$ aren't measures?
Also, can you suggest a book with detailed explanation on such notation?

Thanks a lot.
 

PaulRS

Member
Jan 26, 2012
37
I have not been able to see page 72 in google books. I am pretty sure that those symbols stand for the product measure though, and that by $dt$ they mean the Lebesgue measure.

You should be able to find a section on the product measure in most Real Analysis books (almost everywhere :D).
 

gnob

New member
Dec 11, 2012
11
I have not been able to see page 72 in google books. I am pretty sure that those symbols stand for the product measure though, and that by $dt$ they mean the Lebesgue measure.

You should be able to find a section on the product measure in most Real Analysis books (almost everywhere :D).
I see. Below is taken from page 72 of the book. It is part of the definition of a strong solution of the semilinear SDE. This is the setting:

Let $K$ and $H$ be real separable Hilbert spaces, and $W_t$ be a $K$-valued $Q$-Wiener process on a complete filtered probability space $\Big(\Omega,\mathcal{F},\{ \mathcal{F}_t\}_{t\leq T},\mathbb{P}\Big)$ with the filtration $\mathcal{F}_t$ satisfying the usual conditions. We consider the semilinear SDEs on $[0,T]$ in $H$ in the general form
\begin{align*}
dX(t) &= (AX(t) +F(t,X))dt + B(t,X)dW_t\\
X(0) &= \xi_0.
\end{align*}
Here, $A: \mathcal{D}(A) \subset H \to H$ is the generator of a $C_0$-semigroup of operators $\{ S_t, t\geq 0\}$ on $H.$ The coefficients $F$ and $B$ are, in general, nonlinear mappings,
\begin{align*}
F&:\Omega\times [0,T] \times C\big([0,T],H\big) \to H\\
B&:\Omega\times [0,T] \times C\big([0,T],H\big) \to \mathcal{L}_{2}(K_Q,H).
\end{align*}
Finally, the initial condition $\xi_0$ is an $\mathcal{F}_0$-measurable $H$-valued random variable.

In the definition of a strong solution of the above SSDE, one requirement is the ff:

$X(t,\omega)\in\mathcal{D}(A)$ $dt\otimes d\mathbb{P}$-a.e.

Does this mean that $X(t,\omega)$ belongs to $\mathcal{D}(A)$, except for a set of measure zero? Which measure will we use? The product measure $Leb\otimes\mathbb{P}.$


Thanks again for further enlightenment.