# Joint distribution for sticky Brownian motion

$$\newcommand{\R}{\mathbb R}$$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.

• Markdown note: as annoying as it is, $\newcommand{\R}{\mathbb R}$ cannot be on its own line, but must be run up against the start of the post, as $\newcommand{\R}{\mathbb R}$The or so. Otherwise, the end of line is interpreted as leading whitespace, which is forced into the rendered post. I have edited accordingly. Mar 23 at 23:01
• than you @LSpice I keep forgetting about this. Old LaTeX habits die hard, I guess Mar 24 at 8:55