Pointwise explicit estimates for derivatives of solutions to linear parabolic PDEs with Neumann boundary conditions

TL;DR

This paper provides explicit pointwise bounds for the derivatives of solutions to linear parabolic PDEs with Neumann boundary conditions, crucial for optimal control applications and derived using probabilistic methods.

Contribution

It introduces a fully explicit probabilistic approach to estimate derivatives of PDE solutions with Neumann boundary conditions, extending to time-inhomogeneous diffusions and spectral analysis.

Findings

Derived explicit bounds depending only on PDE coefficients and domain

Extended results to time-inhomogeneous reflected diffusions

Provided conditions for boundedness of derivatives as time approaches infinity

Abstract

We derive explicit pointwise bounds for the spatial derivative $\left| \frac{\partial V}{\partial x} \right|$ of solutions to linear parabolic PDEs with Neumann boundary conditions. The bound is fully explicit in the sense that it depends only on the coefficients of the PDE and the domain, including closed-form expression for all constants. The proof is purely probabilistic. We first extend to time inhomogeneous diffusions a result concerning the derivative of the solution of a reflected SDE. Then, we combine it with the spectral expansion of the law of the first hitting time to a boundary for a reflected diffusion. The main motivation comes from optimal control where, in order to apply verification theorems, precise gradient estimates are often required when closed-form solutions of the Hamilton-Jacobi-Bellman equation. This result will be used in a forthcoming work to rigorously prove that the conjectured optimal strategy for the sailboat trajectory optimization problem is indeed optimal far from the buoy. We also state a sufficient condition for $\limsup_{t\rightarrow \infty} \left| \frac{\partial V}{\partial x}(t,x) \right|$ to be bounded, which only involves the coefficients of the problem and the first eigenvalue of the spectral expansion.