We have proved that [mathbb{P}left(W_{t} in d x, M_{t} in d y ight)=mathbb{1}_{{y geq 0}} mathbb{1}_{{x leq

We have proved that

\[\mathbb{P}\left(W_{t} \in d x, M_{t} \in d y\right)=\mathbb{1}_{\{y \geq 0\}} \mathbb{1}_{\{x \leq y\}} \frac{1}{\sqrt{t}} g\left(\frac{x}{\sqrt{t}}, \frac{y}{\sqrt{t}}\right) d x d y\]

where

\[g(x, y)=\frac{2(2 y-x)}{\sqrt{2 \pi}} \exp \left(-\frac{(2 y-x)^{2}}{2}\right) .\]

Prove that \(\left(M_{t}, W_{t}, t \geq 0\right)\) is a Markov process and give its semi-group in terms of \(g\).