Feller–Wentzell Interface Diffusions

• Feller–Wentzell interface diffusions are advanced models that extend traditional diffusion by incorporating nonlocal interface conditions to manage reflection, absorption, and jump phenomena.
• They are constructed using rigorous semigroup techniques and coupled Volterra integral equations to ensure the existence and uniqueness of solutions.
• These models bridge classical and fractional dynamics, offering a unified framework for translating microscopic scattering rules to macroscopic interface laws in heterogeneous media.

Feller–Wentzell interface diffusions generalize classical diffusion processes by incorporating nonlocal interface (or boundary) conditions that couple local diffusion dynamics with probabilistic transitions—such as partial reflection, jump-like exits, absorption, and transmission—at specified interfaces. These processes arise as semigroups associated with generators whose domains are specified not only by local regularity but by Feller–Wentzell-type conditions at moving or fixed interfaces. The theory unifies classical and nonlocal (e.g., fractional) models and offers a rigorous framework for modeling transport and random motion across heterogeneous media, including phenomena observed in interfaces with complex reflection, absorption, or transmission behaviors.

1. Model Setup and Interface Domains

The foundational setting for Feller–Wentzell interface diffusions is a Markov process on a state space $\mathbb{R}$ partitioned at a (potentially moving) interface, or "membrane," $x = h(s)$ for $0 \leq s \leq T$. The space is decomposed as follows, with time-dependent domains:

• $D_1(s) = (-\infty, h(s))$ and $D_2(s) = (h(s), +\infty)$,
• $D(s) = D_1(s) \cup D_2(s)$.

On each half-line $D_i(s)$, the process follows a uniformly parabolic diffusion with time- and space-dependent generator: $$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$ where $b_i(s, x) > 0$, $a_i$, and $x = h(s)$0 are bounded and Hölder continuous of order $x = h(s)$1 in the domain. The interface curve $x = h(s)$2 is continuous, Hölder of order $x = h(s)$3, ensuring the nondegeneracy of subdomains in the space–time plane (Kopytko et al., 2019).

In superdiffusive kinetic scaling, the bulk equation is a fractional heat equation $x = h(s)$4 with $x = h(s)$5 (for scattering rates $x = h(s)$6) on each side of a static interface, e.g., $x = h(s)$7 (Komorowski et al., 2019).

2. Infinitesimal Generators, Domain, and Interface (Feller–Wentzell) Conditions

The infinitesimal generator $x = h(s)$8 of these interface diffusions acts as $x = h(s)$9 or $0 \leq s \leq T$0 away from the interface. The critical point is the domain $0 \leq s \leq T$1: functions $0 \leq s \leq T$2 in $0 \leq s \leq T$3 satisfying matched value and a generalized Feller–Wentzell conjugation condition at $0 \leq s \leq T$4:

$0 \leq s \leq T$5

Here, $0 \leq s \leq T$6, $0 \leq s \leq T$7, and $0 \leq s \leq T$8 is a Borel measure satisfying integrability and regularity constraints. The nonlocal term models jump-like exits from the interface into the domain with a distribution $0 \leq s \leq T$9. The alias notation,

$D_1(s) = (-\infty, h(s))$0

absorbs the $D_1(s) = (-\infty, h(s))$1-term into the measure $D_1(s) = (-\infty, h(s))$2, where $D_1(s) = (-\infty, h(s))$3, $D_1(s) = (-\infty, h(s))$4 (Kopytko et al., 2019).

For nonlocal (e.g., fractional) equations, the generator at $D_1(s) = (-\infty, h(s))$5 is

$D_1(s) = (-\infty, h(s))$6

with domain determined by Feller–Wentzell interface conditions enforcing “jump-flux” balance at $D_1(s) = (-\infty, h(s))$7 (Komorowski et al., 2019).

3. Construction of the Semigroup and Transition Kernel

Given smooth initial data, the nonlocal parabolic problem

$D_1(s) = (-\infty, h(s))$8

coupled with Feller–Wentzell interface conditions at $D_1(s) = (-\infty, h(s))$9, admits a unique classical solution $D_2(s) = (h(s), +\infty)$0 for $D_2(s) = (h(s), +\infty)$1. The associated two-parameter semigroup $D_2(s) = (h(s), +\infty)$2 is defined by $D_2(s) = (h(s), +\infty)$3 and forms a strongly continuous, positive contraction family (Kopytko et al., 2019).

The transition probability kernel $D_2(s) = (h(s), +\infty)$4 admits representation via Poisson (bulk) potentials and parabolic single-layer potentials centered at the moving membrane. The latter are determined by a $D_2(s) = (h(s), +\infty)$5 Volterra integral equation system encoding the interface's nonlocal behavior (Kopytko et al., 2019).

In the fractional stable setting,

$D_2(s) = (h(s), +\infty)$6

where $D_2(s) = (h(s), +\infty)$7 is a Lévy-type process with probabilistic reflection, transmission, or absorption at the interface, gives both a Duhamel (series) and a Feynman–Kac-type representation of solutions (Komorowski et al., 2019).

4. Interface and Boundary Condition Taxonomy

The general Feller–Wentzell interface condition encompasses:

• Partial Reflection: Parameters $D_2(s) = (h(s), +\infty)$8, $D_2(s) = (h(s), +\infty)$9, leading to matched fluxes on either side—standard reflection.
• Absorption: One of $D(s) = D_1(s) \cup D_2(s)$0 or $D(s) = D_1(s) \cup D_2(s)$1 is zero, the corresponding flux vanishes, modeling absorption at the interface.
• Jump-like Exit: $D(s) = D_1(s) \cup D_2(s)$2, $D(s) = D_1(s) \cup D_2(s)$3, the process jumps on contact, distributed according to $D(s) = D_1(s) \cup D_2(s)$4.
• Combination (General Case): All terms present; the process reflects, absorbs, or jumps according to specified weights.
• Nonlocal (Fractional) Transmission and Reflection: Three exit channels—reflection, transmission, absorption—via weights $D(s) = D_1(s) \cup D_2(s)$5, $D(s) = D_1(s) \cup D_2(s)$6, $D(s) = D_1(s) \cup D_2(s)$7, in nonlocal fractional models (Komorowski et al., 2019), paralleling the “jump-flux” balance.

These interface conditions interpolate between classical boundary conditions (Neumann, Dirichlet, Robin, Wentzell) and nonlocal stochastic behaviors.

Case	$D(s) = D_1(s) \cup D_2(s)$8, $D(s) = D_1(s) \cup D_2(s)$9	$D_i(s)$0	Interpretation
Reflecting	$D_i(s)$1	$D_i(s)$2	Common flux (reflection)
Absorbing	$D_i(s)$3 or $D_i(s)$4	$D_i(s)$5	Dirichlet at interface
Pure jump-exit	$D_i(s)$6	$D_i(s)$7	Jump distributed by $D_i(s)$8
Partial reflection + jump	$D_i(s)$9	$$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$0	General Feller–Wentzell

5. Green's Functions, Resolvents, and Spectral Structure

The resolvent operator,

$$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$1

satisfies $$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$2 in the bulk, together with interface conditions $$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$3 at the membrane. The Green's function $$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$4 is constructed via the free-space resolvent and parabolic single- and double-layer potentials at the interface; see Section V.2 in Friedman (1964) for analogous constructions (Kopytko et al., 2019).

Under stationary or time-homogeneous conditions (e.g., fixed membrane), spectral expansions in eigenmodes subject to Wentzell boundary conditions yield explicit representations for the transition semigroup. In non-stationary (moving interface) cases, such spectral decompositions are only available via “freezing” $$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$5 (Kopytko et al., 2019).

6. Existence and Uniqueness of Solutions

Under the hypotheses of uniform parabolicity, regularity (Hölder continuity) of data, and positivity for coefficients ($$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$6, $$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$7), existence and uniqueness of bounded classical solutions to the nonlocal interface problem are established via:

• Construction of a Poisson potential for initial data and a simple-layer potential (unknown density $$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$8).
• Use of interface conditions to derive a coupled $$L_i u(s, x) = b_i(s, x) \partial_x^2 u(s, x) + a_i(s, x) \partial_x u(s, x), \qquad i=1,2,$$9 Volterra system for $b_i(s, x) > 0$0.
• Transformation and solution of the Volterra equations by successive approximations and kernel estimates to ensure convergence and boundedness.
• Reconstruction of the solution and verification of compatibility with the PDE and interface conditions.
• Uniqueness follows by reduction to a homogeneous Volterra system and application of energy estimates or Grönwall-type inequalities (Kopytko et al., 2019); (Komorowski et al., 2019).

In probabilistic (fractional) frameworks, uniqueness in the finite-energy class is ensured by decay of the $b_i(s, x) > 0$1 energy and absence of a boundary layer due to the interface condition (Komorowski et al., 2019).

7. Connections to Microscopic Models and Macroscopic Limits

In kinetic models with reflection, transmission, and absorption at a microscopic interface, the macroscopic limit under heavy-tailed jump kernels leads to fractional diffusion equations with Feller–Wentzell interface conditions. The reflection, transmission, and absorption weights arise precisely as the low-frequency limits of the microscopic probabilities, and the fractional index $b_i(s, x) > 0$2 encodes the degree of superdiffusivity determined by microscopic scattering (Komorowski et al., 2019).

This demonstrates that Feller–Wentzell interface diffusions serve as universal scaling limits for a broad class of physical and stochastic systems with interfacial mechanisms—unifying standard and anomalous diffusion, local and nonlocal transitions, and connecting mesoscopic (kinetic) rules directly to macroscopic interface laws. The analytical machinery of boundary integrals, Volterra equations, and careful semigroup construction underpins this theory, situating Feller–Wentzell interface diffusions as a central model class for rigorous analysis of processes with nontrivial interface or boundary phenomena (Kopytko et al., 2019); (Komorowski et al., 2019).