TY - JOUR

T1 - Sharp regularity near an absorbing boundary for solutions to second order SPDEs in a half-line with constant coefficients

AU - Ledger, Sean

PY - 2014/3/1

Y1 - 2014/3/1

N2 - 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.

AB - 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.

U2 - 10.1007/s40072-013-0023-1

DO - 10.1007/s40072-013-0023-1

M3 - Article (Academic Journal)

VL - 2

SP - 1

JO - Stochastic Partial Differential Equations: Analysis and Computations

JF - Stochastic Partial Differential Equations: Analysis and Computations

SN - 2194-0401

IS - 1

ER -