[Paper Review] Construction and analysis of sticky reflected diffusions

Martin Grothaus, Robert Voßhall|arXiv (Cornell University)|Dec 12, 2014

Advanced Mathematical Modeling in Engineering3 references3 citations

TL;DR

This paper constructs sticky reflected diffusions on bounded domains in ℝᵈ using Dirichlet form techniques, enabling a diffusion process that spends positive time on the boundary Γ and may evolve along it via a surface SDE. The key contribution is proving the process is a weak solution to a Stratonovich SDE with Wentzell-type boundary conditions, and establishing its ergodicity and 𝒮ᴸᵖ-strong Feller properties under mild regularity assumptions on the boundary and coefficients.

ABSTRACT

We give a Dirichlet form approach for the construction of distorted Brownian motion in a bounded domain $Ω$ of $\mathbb{R}^d$, $d \geq 1$, with boundary $Γ$, where the behavior at the boundary is sticky. The construction covers the case of a static boundary behavior as well as the case of a diffusion on the hypersurface $Γ$ (for $d \geq 2)$. More precisely, we consider the state space $\overlineΩ=Ω\stackrel{.}{\cup} Γ$, the process is a diffusion process inside $Ω$, the occupation time of the process on the boundary $Γ$ is positive and the process may diffuse on $Γ$ as long as it sticks on the boundary. The problem is formulated in an $L^2$-setting and the construction is formulated under weak assumptions on the coefficients and $Γ$. In order to analyze the process we assume a $C^2$-boundary and some weak differentiability conditions. In this case, we deduce that the process is also a solution to a given SDE for quasi every starting point in $\overlineΩ$ with respect to the underyling Dirichlet form. Under the addtional condition that $\{ \varrho =0 \}$ is of capacity zero, we prove ergodicity of the constructed process and consequently, we verify that the boundary behavior is indeed sticky. Moreover, we show ($\mathcal{L}^p$-)strong Feller properties which allow to characterize the constructed process even for every starting point in $\overlineΩ \backslash \{ \varrho=0\}$.

Motivation & Objective

• To construct a diffusion process on the closure of a bounded domain Ω ⊂ ℝᵈ with sticky boundary behavior at ∂Ω = Γ.
• To extend the framework of distorted Brownian motion to include non-trivial boundary dynamics, including diffusion along the boundary when δ=1.
• To rigorously establish the process as a weak solution to a Stratonovich SDE with Wentzell-type boundary conditions.
• To prove ergodicity of the process under the condition that {ϱ=0} has zero capacity, confirming the sticky nature of the boundary behavior.
• To establish 𝒮ᴸᵖ-strong Feller properties, enabling pathwise characterization of the process for all starting points in Ω̅∖{ϱ=0}.

Proposed method

• Uses a Dirichlet form approach on L²(Ω̅; μ), where μ = ϱ(λ + σ) is a reference measure assigning mass to the boundary Γ.
• Constructs the Dirichlet form (ℰ, D(ℰ)) on L²(Ω̅; μ) under weak assumptions on the coefficients α, β and the boundary Γ.
• Imposes a C² boundary condition and weak differentiability assumptions to derive regularity and solve the associated martingale problem.
• Derives the SDE representation via the identification of the generator with a Stratonovich SDE involving interior drift, boundary drift, and a normal reflection term.
• Applies results from [BGS13] to establish the 𝒮ᴸᵖ-strong Feller property by verifying local continuity and point-separating sequences in the domain.
• Uses random time changes to relate the sticky process to reflected diffusions, supporting the non-existence of strong solutions.

Experimental results

Research questions

• RQ1 Can sticky reflected diffusions with boundary diffusion be rigorously constructed via Dirichlet forms under weak regularity assumptions on the coefficients and boundary?
• RQ2 Is the constructed process a weak solution to a Stratonovich SDE with Wentzell-type boundary conditions for quasi-every starting point in Ω̅?
• RQ3 Does the process exhibit true sticky behavior, i.e., positive long-run occupation time on the boundary, under the condition that {ϱ=0} has zero capacity?
• RQ4 What regularity properties, such as strong Feller or 𝒮ᴸᵖ-strong Feller, does the transition semigroup of the process possess?
• RQ5 Can the process be characterized pathwise for every starting point in Ω̅∖{ϱ=0}, even when the boundary is not smooth everywhere?

Key findings

• The constructed process is a weak solution to the Stratonovich SDE (1.1) for quasi-every starting point in Ω̅ with respect to the underlying Dirichlet form.
• The process is ergodic under the assumption that the set {ϱ=0} has zero capacity, confirming its sticky boundary behavior through positive long-run occupation time on Γ.
• The transition semigroup (pₜ)ₜ>0 is 𝒮ᴸᵖ-strong Feller, meaning pₜ maps Lᵖ(Ω̅; μ) into C(Ω̅), and in particular maps bounded measurable functions into continuous functions.
• The process is strong Feller on Ω̅₁ = Ω̅ ∖ {ϱ=0}, allowing pathwise characterization for all starting points in this set.
• The Dirichlet form (ℰ, D(ℰ)) on L²(Ω̅; μ) is associated with the process and satisfies the absolute continuity condition from [FOT11], ensuring the ergodicity result extends to the L²(Ω̅; μ) setting.
• The construction is optimal in the sense that strong solutions do not exist, consistent with known results on sticky Brownian motion as limits of time-scaled reflected diffusions.