Tiered Moving Boundary Problem

• Tiered moving boundary problems are recursive systems where evolving interfaces in each tier depend on lower-tier solutions.
• They are analyzed via a hierarchical cascade of PDEs and forward-backward SDEs, capturing dynamic interactions and conditional absorption events.
• The framework ensures well-posedness and uniqueness using recursive boundary conditions, offering practical insights for models in finance, physics, and chemical reactions.

A tiered moving boundary problem involves a system where multiple interdependent moving interfaces or boundaries partition space or state into “tiers” (layers, regimes, or configurations), each with its own dynamics, and where the evolution of both the boundaries and the state in one tier depends recursively on the state and boundaries in other tiers. These problems arise in applications where hierarchical, cascade, or configuration-dependent coupling is intrinsic, including stochastic particle systems with feedback through absorption probabilities, phase-change kinetics with multi-layer structures, and multi-level interacting diffusions.

1. Definition and Mathematical Structure

A tiered moving boundary problem is characterized by a hierarchy of PDEs or stochastic models, each defined on a domain determined by the evolving position(s) of moving boundaries, with the domain and/or the boundary conditions of level $k$ specified by the solution at level $k-1$. Concretely, in a system of $N$ Brownian particles each absorbed when crossing a time-dependent boundary, the domain for each partial differential equation is defined by the survival of subsets $I \subset \{1,\dots,N\}$, with the boundary for the $i$th particle at time $t$ given by a functional $B^I_i(t,x)$, often expressed as a sum of expectation-dependent terms (e.g., $$B^I_i(t,x) = \sum_j D_{ij} v^{I\setminus\{i\},j}(t,x^{-i})$$) (Jettkant et al., 3 Oct 2025). This tiered coupling leads to a recursive definition where the solution at each level both influences and is influenced by solutions at neighboring tiers.

In canonical form, the system at tier $I$ consists of:

• A PDE for $v^{I,i}(t,x)$ (e.g., a heat equation or a Kolmogorov backward equation) on the domain $\mathcal{D}_T^I(v)$ determined by

$$\mathcal{D}_T^I(v) = \left\{ (t,x)\in[0,T]\times\mathbb{R}^I:\; x_i > \sum_{j} D_{ij} v^{I\setminus\{i\},j}(t, x^{-i})\;\forall i\in I \right\}$$

• Boundary condition (at the killing/absorption boundary) given by the solution at the next-lower tier:

$$v^{I,i}(t,x) = v^{I\setminus\{i_0\},i}(t, x^{-i_0})\quad\text{when}\;x_{i_0} = \sum_{j} D_{i_0 j}\,v^{I\setminus\{i_0\},j}(t, x^{-i_0})$$

• Terminal condition (e.g., $v^{I,i}_T(x) = 0$).

This recursive structure can be interpreted as a “cascade” of moving boundary problems, each conditioned on the survival or absorption events at higher tiers, leading to complex interdependence (Jettkant et al., 3 Oct 2025).

2. Forward–Backward Stochastic Representation

In the stochastic particle formulation, the system is encoded as a collection of forward SDEs for particle positions and backward SDEs for their conditional absorption probabilities. Specifically, for each particle $i$,

$$\begin{aligned} dX^i_t &= \sigma\, dW^i_t, \qquad X^i_0 = \xi_i, \ Y^i_t &= \mathbb{E}\left[\,\mathbf{1}_{\{\tau_i\leq T\}} ~|~ \mathcal{F}_t\,\right], \ \tau_i &= \inf\{t\in[0,T]\,:\, X^i_t \leq \sum_j D_{ij} Y^j_t\,\} \end{aligned}$$

Here, the hitting time $\tau_i$ is defined recursively via the $Y^j_t$, which themselves are backward processes depending on future (and thus, all) absorption events. This closed-loop, singular system is a forward–backward SDE with coupling through conditional probabilities and hitting times, exhibiting singular interaction at absorption events.

Classical methods for FBSDEs do not apply directly due to (i) discontinuity in the terminal conditions, (ii) degeneracy of the diffusion term at absorption, and (iii) recursive, nonlocal coupling through hitting times (Jettkant et al., 3 Oct 2025). Analysis proceeds by unraveling the FBSDE into a hierarchy of associated PDEs (decoupling fields) structured as described in Section 1.

3. Hierarchical Cascade and Domain Partitioning

The solution space for a tiered moving boundary problem is partitioned according to the set of surviving (unabsorbed/unmelted/unreacted) indices. Each subset $I$ corresponds to a tier with its own moving boundary problem, and the evolution of the system can be viewed as a cascade: as boundaries are reached (e.g., particle absorption), the tier drops to $I\setminus\{i_0\}$, and the PDE at that level prescribes the ongoing dynamics.

The boundaries between tiers are themselves functions of the lower-tier solutions, so that each tier’s domain is a non-cylindrical, history-dependent set:

$$\mathcal{D}_T^{I}(v) = \left\{ (t,x)\,:\, x_i > B^I_i(t,x),\,\forall i\in I \right\},\qquad B^I_i(t,x) = \sum_{j} D_{ij} v^{I\setminus\{i\},j}(t, x^{-i})$$

The coupling implies that the boundary in any tier is determined by the lower-tier’s value function, leading to recursive boundary determination throughout the hierarchy (Jettkant et al., 3 Oct 2025).

4. Recursive Boundary Conditions and Well-Posedness

Each tier’s PDE is endowed with boundary conditions set by projections of the lower-tier solution:

• On $\partial\mathcal{D}_T^I(v)$, when $x_{i_0} = B^I_{i_0}(t,x^{-i_0})$, set

$$v^{I,i}(t,x) = v^{I\setminus\{i_0\}, i}(t, x^{-i_0})$$

for all $i\in I\setminus\{i_0\}$. Lemmas in (Jettkant et al., 3 Oct 2025) show that this is well defined: the value-fields in lower tiers agree at their mutual boundaries. This structure ensures that as the system “descends” through tiers (by absorption events), the cascade closes and the entire hierarchy yields a unique, continuous solution profile.

Well-posedness and regularity of the classical solution are achieved by induction on $|I|$, using the smoothness of the solution in the interior and the compatibility of the cascade at the boundaries. Uniqueness is established by contradiction: any two distinct solutions to the FBSDE yield distinct hitting-time profiles and absorption events, contradicting properties of Brownian motion and the structure of the cascade.

5. Mathematical Formulation and Main Results

The general tiered moving boundary problem for $N$ particles with weights $D_{ij}\geq 0$ is summarized by:

• For each $I\subset\{1,\dots,N\}$ (the set of surviving indices), solve for $v^{I,i}: [0,T]\times\mathbb{R}^I\to\mathbb{R}$,

$$\partial_t v^{I,i}(t,x) + \frac{\sigma^2}{2} \Delta v^{I,i}(t,x) = 0, \quad (t,x)\in \mathcal{D}_T^{I}(v)$$

with

$$\begin{align*} \mathcal{D}_T^{I}(v) &= \{(t,x):\ x_k > \sum_j D_{kj} v^{I\setminus\{k\},j}(t, x^{-k}),\,\forall k\in I\} \ v^{I,i}_T(x) &= 0 \;\;(\textrm{terminal})\ v^{I,i}(t,x) &= v^{I\setminus\{i_0\},i}(t, x^{-i_0})\;\; \textrm{on}\;\; x_{i_0} = \sum_j D_{i_0 j} v^{I\setminus\{i_0\},j}(t, x^{-i_0}) \end{align*}$$

• The initial FBSDE solution is assembled from these fields as

$$Y^i_t = v^{I_t,i}(t, X_t), \quad Z^i_t = -\sigma \nabla_x v^{I_t,i}(t, X_t)$$

where $I_t$ is the (random) index set of alive particles at $t$.

The main result (Jettkant et al., 3 Oct 2025) affirms existence, uniqueness, and regularity for this system, and provides the first rigorous treatment of a singular, feed-back-coupled, tiered moving boundary problem of this form.

6. Broader Context and Significance

Tiered moving boundary problems generalize classical moving boundary problems (e.g., Stefan or one-phase model) to settings with multiple, interacting layers or regimes, each conditioned recursively on the others. This structural recursion is natural in systems with configuration-dependent interactions:

• Credit/economic contagion models, where the risk evolution of each institution depends recursively on the default probabilities of all others.
• Interacting particle systems with absorption, where the effective “absorbing boundary” for each particle is a function of the (time-evolving) absorption probability of the rest.
• Multiphase flows, wetted fronts, or chemical reactions with multistage interfaces.
• Multi-tiered order book models, where liquidity at one price level depends on executions or depletion at adjacent levels.

The recursive, cascade structure requires novel analytical and numerical techniques to handle the discontinuous transitions between tiers, and to ensure well-posedness and regularity in the presence of singular, feedback-driven boundary evolution.

7. Analytical Challenges and Mathematical Features

Key analytical features of tiered moving boundary problems include:

• The domains of definition are moving, non-cylindrical, and recursively defined.
• The boundary conditions at each tier are “fed” by the lower-tier solutions, leading to interlocking, nested PDEs or stochastic systems.
• Standard FBSDE theory does not apply due to lack of continuity, degenerate diffusion at killing, and recursion at the boundaries.
• Well-posedness is ensured by recursive compatibility of the boundary conditions, continuity of the decoupling fields up to $\partial\mathcal{D}_T^{I}$, and comparison lemmas tailored to the structure of hitting times in Brownian motion.
• An explicit Feynman–Kac formulation is available in this setting via the decoupling fields, which rigorously connect the hierarchy of solutions to the underlying FBSDE system.

In summary, tiered moving boundary problems present a mathematically rigorous approach for the analysis of hierarchical, feedback-driven moving interface systems where the evolution and position of each boundary—and the state within each domain—depends recursively on the configuration and solutions of all lower tiers. The resulting framework enables well-posedness and uniqueness results in complex, configuration-coupled stochastic and deterministic systems (Jettkant et al., 3 Oct 2025).