This was originally posted on Math StackExchange a long time ago, but got no answer (even after a bounty). See https://math.stackexchange.com/questions/1274775/generator-of-wiener-process-and-its-running-maximum

If we let $ W$ be a standard linear Wiener process issued from zero and $ M$ its running maximum $ $ M_t := \sup \{ W_u: u \leq t \}, $ $ then we could show that $ (X,Y):=(M,M-W)$ is a Markov process on $ \mathbb{R}_+ \times \mathbb{R}_+$ with transition kernel $ $ P_t(a,b; dx,dy) = \mathbb{I}_{ \{x \geq a\} } \left[ \delta_a(dx) dy k_t(b,y) + dx dy 2 h_t(x-a+b+y) \right], $ $ where $ k$ is the transition density of a Wiener killed upon reaching the origin, i.e. $ $ k_t(x,y) = \frac{1}{\sqrt{2 \pi t}} \exp \left( -\frac{(x-y)^2}{2t} \right) – \frac{1}{\sqrt{2 \pi t}} \exp \left( -\frac{(x+y)^2}{2t} \right) $ $ and $ t \mapsto h_t(x)$ is the density of the first hitting time of $ x$ by $ W$ , i.e. $ $ h_t(x) = \frac{x}{\sqrt{2 \pi t^3}} \exp \left( -\frac{x^2}{2t} \right). $ $

Assuming $ (X,Y)$ has the Feller property (which I think is true), my question is: what is the infinitesimal generator, say $ G$ , of $ (X,Y)$ ? I am pretty sure that, since $ (X,Y)$ moves as a Wiener only along the $ y$ -direction when it is away from the $ x$ -axis, $ G$ is a degenerate differential operator like $ $ G = \frac12 \partial_{yy}, $ $ so my question is really about the boundary conditions, in particular I think the real issue is on the $ x$ -axis, since in any case the process never moves to the left. Since when $ (X,Y)$ touches the $ x$ -axis (coming from above) it then goes either to the right or it’s pushed back up (and also by thinking of an approximating random walk), I am led to conjecture that the correct boundary condition might be $ $ \partial_{u} v(x,0) := \lim_{h \to 0} \frac{v(x+h,h)}{h} = 0, \quad \forall x, $ $ but I am far from being sure. I tried to derive it more formally with no success, so a real proof and/or a reference (I am sure this can be found somewhere but I didn’t manage) would be really appreciated.

Many thanks in advance.