$\newcommand{\R}{\mathbb R}$The one-dimensional Sticky Brownian Motion (SBM in short) is an $\R$-valued Markov process given by \begin{gather*} dX_t=1_{[X_t\neq 0]}dB_t\\ L_t(X)=\int_0^t 1_{[X_s=0]}ds, \end{gather*} where $B$ is a standard Brownian motion and $L_t=\lim\limits_{\epsilon\to 0}\frac{1}{2\epsilon}\int_0^t1_{|X_s|\leq \epsilon}ds$ denotes the local time at $x=0$. This definition is kind of formal (one should talk about weak solutions, strong solutions apparently cannot exist). Roughly speaking, SBM is just standard Brownian motion outside of $X=0$, and sticks at the origin in a nontrivial way (in particular the set $\{t:\,X_t=0\}$ is almost-surely Cantor-like with positive measure). Another way to put it is that $X_t$ is a time-changed Brownian motion, where the new clock only slows down when the process crosses $X=0$. There is much more than meets the eye here, but for the sake of simplicity I will not elaborate any further here. Let me just mention that the transition kernel $p_t(x;dy)$ has an atom at $y=0$ due to the speed measure being $2dx+\delta_0(dx)$.
In some project of mine I'm interested in particular higher-dimensional variations of the SBM, and for some technical reason (mainly a computation by disintegration) I need the joint distribution of $$ (X_t,L_t) $$ under $\mathbb P_x$ (process started at $x\in \R$). Following the paper [1] by Karatzas and Shreve on trivariate distributions for Brownian motion, the density of my desired bivariate distribution $\mathbb P_x(X_t\in dy,L_t\in dl)$ is computed explicitly in [2] (Theorem 3.3 gives a more general trivariate density and corollary 3.4 gives the bivariate one by taking a 2-marginal). Unfortunately the density is only computed for $y>0$ (or $y<0$), and the necessarily atomic part at $y=0$ is not provided (such an atomic part must exist otherwise the transition kernel $p_t(x;dy)$ would be absolutely continuous). As far as I can tell the method of proof absolutely does not give access to this atomic part, and since I am not really a probabilist I am completely stuck.
Has anyone either a reference (I would be surprised if this had never been looked at) or a clue at least on how to get started?
[1] Karatzas, I., & Shreve, S. E. (1984). Trivariate density of Brownian motion, its local and occupation times, with application to stochastic control. The Annals of Probability, 819-828.
[2] Touhami, W. (2021). On skew sticky Brownian motion. Statistics & Probability Letters, 173, 109086.
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