Law of a step function and its generalization to two dimensions on an appropriate spaces

Question

Let's consider two discontinuous functions defined on $D$ and $D \times [0,T]$, respectively:

• A step function: $u_1(x)=\begin{cases} u_{L}, x<c_1, \\[2ex] u_{R}, x>c_1, \end{cases}$
• A "generalization to two dimensions": $u_2(x,t)=\begin{cases} u_{L}, x<c_2\cdot t, \\[2ex] u_{R}, x>c_2\cdot t. \end{cases}$

Here $x \in D \subseteq \mathbb{R}$, $t \in [0,T]$ and $c_1,c_2,u_{L},u_{R}$ are constants. Additionaly in the book "Stochastic equations in infinite dimensions, Da Prato G., Zabczyk J., 1992", we could find the definition of the law:

If $X$ is a random variable from $(\Omega,\mathcal{F})$ to $(E,\mathcal{S})$ and $P$ a probability measure on $\Omega$, then by $\mathcal{L}(X)$ we will denote the image of $P$ under the mapping $X$: $\mathcal{L}(X)(A)=P\{\omega \in \Omega:x(\omega)\in A\},\forall A\in \mathcal{S}.$ The measure $ \mathcal{L} (X)$ is called the distribution or the law of $X$.

As far as I know the probability law of $u_1$ and $u_2$ is the Dirac mass at $u_1$ and $u_2$, respectively, seen as a measure on an appropriate function space. Although I can't remember where I have read this in the literature. If my recollection is wrong, please correct me.

If we assume that the last paragraph above is correct, my question is: What would be those appropriate spaces in the cases of the functions $u_1$ and $u_2$?

Example I got last week for the $u_1$: the law of the function $u_1$ is the Dirac measure concentrated at $u_1$ on the space of cadlag function from [0,1] to $\mathbb{R}$. But for the problem I have in my mind cadlag functions probably won't work.

For the function $u_2$ I don't have any examples. I think that the space of $BV$ functions from $(D\times [0,T])$ to $\mathbb{R}$ should be one of the appropriate spaces. But my ideal appropriate spaces should look as $C([0,T];\mathcal{M}(D))$ or similar - they should be Banach space-valued.

I work on one problem for a few weeks that concerns these two functions and in order to apply the technique that was recommended me (in order to solve it), I need to consider the laws of this two functions. I need help with this. Any appropriate space you recommend me is welcome. Thanks everyone in advance.

Answer 1

Your functions $u_1$ and $u_2$ are not completely defined. For instance, $u_1(c_1)$ is undefined.

If you do not need to distinguish functions differing only on a set of Lebesgue measure $0$, then you may consider $u_1$ a point in the Banach space $B_1:=L^\infty(D)$, and $u_2$ a point in the Banach space $B_2:=L^\infty(D\times[0,T])$, where $D$ is any nonempty Lebesgue-measurable subset of $\mathbb R$.

For $(\Omega,\mathcal F,P)$, you can take any probability space. For each $i\in\{1,2\}$, you can take any $\sigma$-algebra $\mathcal S_i$ over $B_i$, and then the probability law of the (nonrandom) random element $\Omega\ni\omega\mapsto u_i\in B_i$ will be the Dirac measure on $B_i$ at $u_i$.

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Added in response to a comment by the OP: Of course, you can identify any function $w\colon X\times Y\to Z$ with the function $\tilde w\colon Y\to Z^X$ by the formula $\tilde w(y)(x):=w(x,y)$ for all $(x,y)\in X\times Y$.

In the case of your function $u_2\colon D\times[0,T]\to\mathbb R$, the corresponding function $\tilde u_2$ may be considered as follows: $$[0,T]\ni t\mapsto\tilde u_2\in \tilde B_2:=L^\infty\big([0,T],L^\infty(\mathbb R)\big),$$ where $D\ni x\mapsto\tilde u_2(t):=u_2(x,t)\in\mathbb R$. Then the probability law of the (nonrandom) random element $\Omega\ni\omega\mapsto\tilde u_2\in\tilde B_2$ will be the Dirac measure on the Banach space $\tilde B_2$ at the point $\tilde u_2$.