(Completed) Submartingale Problem for RBM with Drift in a Wedge

We study reflecting Brownian motion with drift constrained to a wedge in the plane. Our first set of results provide necessary and sufficient conditions for existence and uniqueness of a solution to the corresponding submartingale problem with drift, and show that its solution possesses the Markov and Feller properties. Next, we study a version of the problem with absorption at the vertex of the wedge. In this case, we provide a condition for existence and uniqueness of a solution to the problem and some results on the probability of the vertex being reached.

1. Problem Formulation: The Submartingale Approach

Unlike the standard SDE approach, we define Reflected Brownian Motion (RBM) in a wedge domain $\mathcal{S}$ through the Submartingale Problem. Given a drift vector $\mu$ and reflection directions $v_1, v_2$ on the boundaries, we seek a family of probability measures ${\mathbb{P}z}{z \in \mathcal{S}}$ such that for any test function $f \in C_b^2(\mathcal{S})$ satisfying the oblique derivative boundary conditions:

\[\langle \nabla f, v_i \rangle \geq 0 \quad \text{on } \partial \mathcal{S}_i, \quad i=1,2\]

the following process $M_f$ is a $\mathbb{P}_z$-submartingale:

\[M_f(t) = f(Z_t) - \int_0^t \mathcal{L}f(Z_s) ds, \quad t \ge 0\]

where $\mathcal{L} = \frac{1}{2}\Delta + \mu \cdot \nabla$ is the infinitesimal generator of the diffusion.

2. Methodology: Geometric Conformal Invariants

The core of our methodology relies on analyzing the geometric skew-symmetry of the domain.

  • Conformal Transformation: We map the wedge $\mathcal{S}$ with opening angle $\xi$ to a half-plane, allowing us to simplify the boundary operator into a single spectral parameter $\alpha$.
  • $\alpha$-Parameterization: The parameter $\alpha = (\theta_1 + \theta_2)/\xi$ (where $\theta_i$ are reflection angles from inward normals) serves as the fundamental descriptor of the process’s behavior at the vertex.
  • Construction of Lyapunov Functions: We construct specific power-type functions to test the boundaries of the submartingale property and establish hitting time estimates.

3. Research Objectives

  1. Existence-Uniqueness Phase Diagram: To rigorously determine the thresholds for $\alpha$ under which the submartingale problem is well-posed.
  2. Pathological Boundary Analysis: To classify the regularity of the sample paths as they interact with the non-smooth vertex.
  3. Vertex Attainability: To prove whether the process can reach the vertex in finite time and whether it is absorbed or reflected.

4. Key Results & Theorems

Our work establishes the complete sharp criteria for RBM in wedge domains:

  • Existence and Uniqueness: We prove that a unique solution to the submartingale problem exists if and only if $\alpha < 2$.
  • The Semimartingale Threshold: We establish that for $1 \le \alpha < 2$, the process $Z$ is not a semimartingale. In this regime, the reflection “force” is so singular that the local time processes $L^i$ exert infinite variation near the vertex.
  • Vertex as a Trap: We provide necessary and sufficient conditions for the vertex to be a trap (absorbing point) based on the alignment of the drift $\mu$ and the reflection vectors $v_i$.

Selected References

  • Lakner, P., Liu, Z., & Reed, J. (2023). Reflected Brownian motion with drift in a wedge. Queueing Systems.
  • Varadhan, S. R. S., & Williams, R. J. (1985). Diffusions in regions with boundaries.
  • Williams, R. J. (1987). Reflected Brownian motion with skew reflection.