Reflected Dirichlet Forms

• Reflected Dirichlet forms are an extension of classical Dirichlet forms that model reflection at boundaries rather than absorption, enabling analysis of complex stochastic processes.
• The construction involves truncating with cutoff functions to create a maximal Silverstein extension, preserving jump and killing measures while extending the domain.
• These forms are critical in understanding reflected diffusions, trace Dirichlet spaces on boundaries, and applications in numerical approximations and framed boundary value problems.

A reflected Dirichlet form is a generalization of the classical Dirichlet form framework that accommodates stochastic processes exhibiting “reflection” at the boundary of a domain, rather than being absorbed or killed upon reaching the boundary. Such forms play a central role in the analytic and probabilistic theory of reflected diffusions, nonlocal boundary conditions, and constrained Markov processes. The theory encompasses a wide range of settings—from strongly local forms on domains with regular or fractal boundaries, to non-local jump processes, to infinite-dimensional examples and Dirichlet extensions. The definition, structural properties, decomposition theory, and applications of reflected Dirichlet forms are foundational to modern stochastic analysis and potential theory.

1. Structural Definition and Algebraic Construction

A Dirichlet form $(\mathcal{E}, \mathcal{F})$ on $L^2(X, m)$ is symmetric, closed, and Markovian (it satisfies the contraction, or Markov property). The reflected Dirichlet form extends a “smaller” form (such as the Dirichlet form with absorption at the boundary) to a larger state space or function class, encoding the reflection at the boundary.

The algebraic construction proceeds by truncating with cutoff functions and taking suprema. For a reference form $\mathcal{E}$ and cutoff $\varphi \in \mathcal{F}$ ($0 \leq \varphi \leq 1$):

• Define the truncated form:

$$E_\varphi(f) = \mathcal{E}(\varphi f) - \mathcal{E}(\varphi f^2, \varphi).$$

• The main part is $E^{(M)}(f) = \sup_{\varphi} E_\varphi(f)$ over all $0 \leq \varphi \leq 1$.
• The killing part is

$$\mathcal{E}^{(k)}(f) = \mathcal{E}(f) - E^{(M)}(f),$$

extended by monotonicity.

• The reflected Dirichlet form is

$E^{(\mathrm{refl})}(f) = E^{(M)}(f) + E^{(k)}(f).$

This construction gives an extension of the original form $\mathcal{E}$; the reflected form is the maximal Silverstein extension when there is no killing part (Schmidt, 2017).

2. Analytic and Probabilistic Properties

Maximal Silverstein Extension

A Silverstein extension is a Dirichlet form extension $(\tilde{\mathcal{E}}, \tilde{\mathcal{F}})$ with the property that $\mathcal{F} \subseteq \tilde{\mathcal{F}}$ and $\mathcal{F}$ is an algebraic ideal in $\tilde{\mathcal{F}} \cap L^\infty(m)$. The reflected form $E^{(\mathrm{refl})}$ is the maximal Silverstein extension if there is no killing measure in the Beurling–Deny decomposition.

Markov Property and Reflection

The Markov (contraction) property is crucial:

$$\mathcal{E}(u \wedge 1, u \wedge 1) \leq \mathcal{E}(u,u)$$

for every $u$ in the form domain $\mathcal{F}$. This ensures that the reflection/truncation of a function to $[0,1]$ does not increase energy, encoding the analytic effect of reflection at the boundary (Schmidt, 2017, Kassmann, 2014).

Locality, Jumping, and Killing

The structure is governed by the Beurling–Deny formula:

$$\mathcal{E}(u,v) = \mathcal{E}^{(c)}(u,v) + \int_{X\times X \setminus \text{diag}} (u(x) - u(y))(v(x) - v(y))\, J(dx,dy) + \int_X u(x)v(x) k(dx).$$

Regular subspaces and reflected forms preserve the jumping and killing measures; these are structural invariants. Any extension to a reflected form leaves these measures unchanged, even though the domain may be enlarged (Li et al., 2015).

3. Boundary Behavior, Trace Theory, and Robin Extensions

A central feature of reflected Dirichlet forms is their interaction with the boundary of the state space:

• The trace Dirichlet form is defined on the boundary $\partial D$ of a domain $D$ as the form of the time-changed process $\check X$ obtained by measuring local time or via a smooth boundary measure $\nu$ with full quasi-support. The trace form can be characterized as a Besov-type Dirichlet space (Cao et al., 24 Oct 2024).
• Boundary representations express every Dirichlet form lying between the Dirichlet (absorbing) and Neumann (fully reflected) forms as a sum of an interior energy and a boundary form, e.g.,

$Q(f) = Q^{(D)}(f_0) + q(\mathrm{Tr}\, f),$

where $f = f_0 + f_h$ and $q$ is a Dirichlet form on the Royden boundary (Keller et al., 2017).

• For local Robin-type extensions, the form is represented as

$$a(u,v) = \frac12 \int_\Omega \nabla u \cdot \nabla v \,dx + \int_{\partial \Omega} u v\,\mu(dx),$$

where $\mu$ is a quasi-admissible boundary measure. All such intermediate operators between the Dirichlet and Neumann Laplacians correspond to reflected forms with boundary perturbation (Li et al., 12 Dec 2024).

4. Process-Level Characterization and Martingale Decompositions

Reflected Dirichlet forms correspond to symmetric Markov processes (reflected diffusions). For a process $Z$ solving a stochastic differential equation with reflection, the pathwise decomposition is

$Z(t) = M(t) + A(t)$

where $M$ is a continuous local martingale and $A$ is an adapted, continuous process of zero $p$-variation ($\sum |A(t_{i}) - A(t_{i-1})|^{p} \to 0$ as mesh $\to 0$ for $p > 1$). For $p = 2$, this implies $A$ has zero quadratic variation—$Z$ is a Dirichlet process in the sense of FöLLMer—so that even without semimartingale structure, a generalized stochastic calculus is available (Kang et al., 2010).

In particular, for certain reflected diffusions in polyhedral or curved domains, the constraining term (the boundary push or local time) has infinite total variation but vanishes in quadratic or $p$-variation, placing the process outside the semimartingale class but inside the Dirichlet process class. This provides a rigorous pathwise foundation for stochastic calculus with degenerate reflection, supporting rough paths or zero energy calculus as needed.

5. Trace Forms, Besov Spaces, and Boundary Processes

On irregular or fractal boundaries, the trace (boundary) Dirichlet form for a reflected diffusion is identified with an appropriate Besov space:

$$\|f \|^2_{\Lambda_{\Psi, \sigma}} \lesssim \bar{\mathcal{E}}(f,f)$$

for $f$ continuous in the reflected Dirichlet space. Under volume doubling, Poincaré, and lower scaling properties, restriction and extension theorems yield an analytic characterization of trace spaces in terms of Besov-type norms (Cao et al., 24 Oct 2024). When the harmonic boundary measure is doubling and the jump intensity satisfies appropriate scaling, the Beurling–Deny decomposition of the trace form yields a “stable-like” jump kernel on the boundary, with spectral and heat kernel estimates determined by the geometry of the boundary.

6. Applications and Connections

Reflected Dirichlet forms provide the analytic structure for:

• Stochastic differential equations with reflection and constrained diffusions (e.g., processor sharing networks, reflected Brownian motions in domains with Lipschitz or curved boundaries, infinite particle systems with reflecting boundaries) (Kang et al., 2010, Kawamoto et al., 2017).
• Numerical approximation of boundary value problems on metric measure spaces via discrete (graph) Dirichlet forms and their $\Gamma$-limits, with boundary value matching via Markov locality (Butaev et al., 2023).
• Boundary extension of Markov processes on discrete and metric measure spaces, including “reflected” random walks and nonlocal boundary operators (Dirichlet-to-Neumann maps, jump processes at infinity) (Keller et al., 2017, Mathieu et al., 2018, Li, 2022).
• Operator-theoretic classification of all forms between the Dirichlet and Neumann Laplacians, as in the complete description of forms between absorption and reflection by adding boundary measure contributions (Robin or nonlocal Robin), resolved via the bivariate Revuz (killing) measure (Li et al., 12 Dec 2024).
• Besov trace identifications, heat kernel estimates, and spectral theory for boundary processes of reflected diffusions on fractals or rough domains (Cao et al., 24 Oct 2024).

7. Summary Table

Aspect	Structural Feature	Example/Reference
Extension Type	Maximal Silverstein extension via algebraic truncation	(Schmidt, 2017)
Boundary Interaction	Addition of boundary measure term in form (Robin/nonlocal Robin)	(Li et al., 12 Dec 2024)
Pathwise Decomposition	Martingale plus zero-energy/zero $p$-variation process (non-semimartingale)	(Kang et al., 2010)
Trace Dirichlet Form	Besov-type norm, identifies regularity class of boundary functions	(Cao et al., 24 Oct 2024)
Preservation of Jumps/Killing	Jump and killing measures invariant under regular subspace/reflected construction	(Li et al., 2015)

• A Dirichlet process characterization of a class of reflected diffusions (Kang et al., 2010)
• Geometry and Analysis of Dirichlet forms (Koskela et al., 2012)
• Regular subspaces of Dirichlet forms (Li et al., 2015)
• A note on reflected Dirichlet forms (Schmidt, 2017)
• Boundary representation of Dirichlet forms on discrete spaces (Keller et al., 2017)
• Boundary trace theorems for symmetric reflected diffusions (Cao et al., 24 Oct 2024)
• On domination for (non-symmetric) Dirichlet forms (Li et al., 12 Dec 2024)
• Construction of a Dirichlet form on metric measure spaces of controlled geometry (Butaev et al., 2023)

The theory of reflected Dirichlet forms thus provides a comprehensive and flexible framework for rigorously integrating reflection, constraint, or singular boundary effects into the analysis of Markov processes, partial differential equations, and variational problems in both classical and non-classical (metric, fractal, infinite-dimensional) contexts. The interplay of form-theoretic, probabilistic, and boundary analytic properties underpins a significant fraction of modern approaches to boundary problems and interface phenomena in stochastic analysis and geometric measure theory.