Sharp regularity near an absorbing boundary for solutions to second order SPDEs in a half-line with constant coefficients

TL;DR

This paper establishes the existence, uniqueness, and detailed regularity properties of solutions to a class of second-order stochastic partial differential equations in a half-line, revealing degeneracy near the boundary and relating it to a two-dimensional Brownian motion in a wedge.

Contribution

It provides a rigorous analysis of the regularity and degeneracy of solutions to second-order SPDEs with absorbing boundary conditions, including a probabilistic representation and sharp regularity results.

Findings

Solution has a smooth density with degeneracy near boundary

Regularity characterized in weighted Sobolev spaces

Behavior linked to a 2D Brownian motion in a wedge

Abstract

We prove that the weak version of the SPDE problem \begin{align*} dV_{t}(x) & = [-\mu V_{t}'(x) + \frac{1}{2} (\sigma_{M}^{2} + \sigma_{I}^{2})V_{t}"(x)]dt - \sigma_{M} V_{t}'(x)dW^{M}_{t}, \quad x > 0, \\ V_{t}(0) &= 0 \end{align*} with a specified bounded initial density, $V_{0}$, and $W$ a standard Brownian motion, has a unique solution in the class of finite-measure valued processes. The solution has a smooth density process which has a probabilistic representation and shows degeneracy near the absorbing boundary. In the language of weighted Sobolev spaces, we describe the precise order of integrability of the density and its derivatives near the origin, and we relate this behaviour to a two-dimensional Brownian motion in a wedge whose angle is a function of the ratio $\sigma_{M}/\sigma_{I}$. Our results are sharp: we demonstrate that better regularity is unattainable.