Planar diffusions with a point interaction on a finite time horizon

TL;DR

The article builds a finite-horizon, axiomatic framework for planar diffusions with a point interaction at the origin, modeling the interaction through time-dependent, singular drifts arising from Doob transforms of point-interaction Schrödinger kernels. By introducing admissible driving families $h^{\vartheta}$ and establishing a robust set of regularity hypotheses, it constructs a Markov family, derives an SDE with a singular drift away from the origin, and analyzes the origin-hitting behavior via a submartingale structure. The paper then formalizes several concrete driving families, including ground-state, Lebesgue-, Dirac-, and Gaussian-driven diffusions, and derives their survival probabilities, conditioned laws, and SDE representations, providing a unified view of how delta-like point interactions manifest on a finite horizon. Kolmogorov continuity arguments and Jacobian/geometry analyses of the transforming map ensure pathwise regularity, enabling weak solutions and a detailed hitting-time theory. The results generalize canonical examples and offer a coherent framework for studying zero-range interactions in planar diffusions with finite-time horizons, with potential connections to Brownian-bridge-type conditioning and zero-range quantum Hamiltonians.

Abstract

The skew-product diffusion [Ann. Appl. Probab. 35, 3150--3214 (2025)] and exponentially tilted planar Brownian motion [Electron. J. Probab. 30, 1--97 (2025)] are canonical examples of planar diffusions with a point interaction at the origin in the sense that their drifts are singular only at the origin and allow visits there with positive probability. However, in this article we propose an axiomatic framework for such diffusions on a finite time horizon. We isolate admissibility conditions and additional regularity hypotheses on a general driving family under which the associated diffusion, constructed as a Doob transform of point-interaction Schrödinger semigroup kernels, exhibits the same point interaction structure. In particular, for the ground-state driving family, we obtain a heuristic alternative construction of the skew-product diffusion based on Kolmogorov continuity arguments. We also consider formal applications of this framework to some driving families generated by measures, including the Lebesgue-driven diffusion, which is the exponentially tilted planar Brownian motion.

Figures (3)

• Figure 1: Here $R_t:=|X_t|$ is the radial process and $\tau:=\inf\{t\ge0:X_t=0\}$. The one-dimensional Brownian motion $\bar{W}^{T,\vartheta}$ is defined by radial projection of $W^{T,\vartheta}$ via $d\bar{W}_t^{T,\vartheta}=\frac{X_t}{|X_t|}\cdot dW_t^{T,\vartheta}$ for $t<\tau$, and the radial drift is the corresponding projection $\bar{b}_t^\vartheta(r):=\frac{x}{|x|}\cdot b_t^\vartheta(x)$ for $x\neq0$ with $|x|=r$.
• Figure 2: Here $\bar{b}^{\vartheta,\textup{GSt}}(|x| ):= \frac{x}{|x|}\cdot b^{\vartheta,\textup{GSt}}(x)$ for $x\neq0$ and $\bar{W}_t=\int_0^{t\wedge\tau}\frac{X_s}{|X_s|}\cdot dW_s$ for $t\in[0,T]$.
• Figure 3: The above diagram summarizes the relations among the laws appearing in Section \ref{['ExamplesDiffusion']} and represents a formal application of Theorems \ref{['ThmExistenceRelativeh']} and \ref{['ThmPropertiesRelativeh']}; the notation is gathered in Notation \ref{['LawsNotation']} for convenience.

Theorems & Definitions (27)

• Definition 1.1: Planar diffusion with a point interaction
• Theorem 1.2: Existence of point-interaction planar diffusions
• Theorem 1.3: Structural properties of planar $h^\vartheta$-diffusions
• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• Remark 2.5
• Proposition 2.6
• Proposition 2.7