Reflected Stochastic Heat Equation

• Reflected stochastic heat equations are SPDEs that incorporate constraints via sticky reflections, geometric quantile adjustments, or singular boundary drifts.
• In the sticky-reflected model, noise is deactivated at zero while a deterministic drift pushes the solution positive, ensuring the non-negativity of the process.
• The RSHE and Dirichlet-form approaches treat reflection as a structural constraint that preserves quantile ordering, with each method employing distinct analytical and approximation techniques.

A reflected stochastic heat equation is a stochastic partial differential equation in which the heat flow is constrained by a mechanism that prevents the solution from leaving a prescribed region or violating an ordering constraint. In the materials considered here, this label encompasses at least three distinct constructions: a sticky-reflected stochastic heat equation on $[0,1]$ driven by colored noise, where the noise is switched off at the zero set and replaced by a positive drift; the Rearranged Stochastic Heat Equation (RSHE) on the circle, where a reflection term keeps the solution in the cone of symmetric quantile functions; and Dirichlet-form-based Markov processes whose Fukushima–Skorokhod decomposition yields a rigorous reflected or singular stochastic heat dynamics associated with distorted Brownian-bridge measures (Konarovskyi, 2020, Delarue et al., 2024, Grothaus et al., 10 Jun 2026).

1. Core formulations of reflection in stochastic heat dynamics

In the sticky-reflected model, the unknown is a weak solution $X_t(u)\ge 0$, continuous in $(t,u)$, on the spatial interval $[0,1]$. The equation is the usual stochastic heat equation away from the zero level set, but at points where $X_t(u)=0$ the noise vanishes and a positive drift $\lambda \mathbf 1_{\{X_t(u)=0\}}$ pushes the solution away from zero. The noise is colored because it is obtained from a space-time white noise $W$ by a non-negative definite self-adjoint Hilbert-Schmidt operator $Q$ on $L^2[0,1]$, and the drift nonlinearity $f:[0,\infty)\to[0,\infty)$ is continuous, has linear growth, and satisfies $X_t(u)\ge 0$0 (Konarovskyi, 2020).

In the RSHE, the state variable $X_t(u)\ge 0$1 evolves on the circle $X_t(u)\ge 0$2 according to

$X_t(u)\ge 0$3

where $X_t(u)\ge 0$4 is a colored noise in the symmetric $X_t(u)\ge 0$5-space $X_t(u)\ge 0$6, and $X_t(u)\ge 0$7 is a reflection term. Here the constraint is not non-negativity pointwise, but membership in

$X_t(u)\ge 0$8

the set of symmetric, non-increasing functions with the appropriate semicontinuity conditions. These are precisely the symmetric quantile functions (Delarue et al., 2024).

A third formulation arises from gradient Dirichlet forms on $X_t(u)\ge 0$9 with respect to the distorted measures $(t,u)$0 and $(t,u)$1, where $(t,u)$2 is the law of the standard Brownian bridge. The associated Markov processes $(t,u)$3 and $(t,u)$4 are interpreted heuristically as stochastic heat flows with either a single-point reflection or repulsion at a spatial point $(t,u)$5, or with reflection spread over the whole spatial interval. The formal SPDEs involve local-time-type terms, but the rigorous object is the Fukushima–Skorokhod decomposition derived from the Dirichlet-form construction (Grothaus et al., 10 Jun 2026).

These examples show that “reflection” in stochastic heat equations is not a single mechanism. It may be implemented by discontinuous coefficients, by a monotone force keeping the solution in a convex cone, or by a singular additive functional extracted from an integration-by-parts formula.

2. Sticky reflection and the zero-level set

The sticky-reflected stochastic heat equation is presented as an infinite-dimensional analogue of sticky-reflected Brownian motion on the real line. Its defining feature is the pair of discontinuous coefficients: $(t,u)$6 When $(t,u)$7, the dynamics coincide with the usual stochastic heat equation. When $(t,u)$8, there is no stochastic forcing at that point, and a deterministic drift pushes the value positive. The process can therefore “stick” at zero only in the sense that the dynamics at zero are altered by removing the noise and adding a drift that keeps it from spending too much time there (Konarovskyi, 2020).

The weak solution is formulated through a martingale problem. For every admissible test function $(t,u)$9, the process

$[0,1]$0

is required to be a martingale, and its quadratic variation is

$[0,1]$1

This formulation makes explicit that the noise acts only on the strictly positive part of the profile. In this sense, the equation is an infinite-dimensional sticky diffusion whose covariance structure is supported on $[0,1]$2 (Konarovskyi, 2020).

The main existence result proves the existence of a weak solution under the compatibility condition stated in Theorem 1.2. The paper describes this as the requirement that the drift parameter $[0,1]$3 must vanish on the set where the noise is active; it also notes that a solution may exist even when this condition fails, for instance if the solution stays strictly positive where the issue would arise (Konarovskyi, 2020).

The limiting process obtained from the approximation is tight in

$[0,1]$4

and any limit point is continuous in time and space, non-negative, locally Hölder continuous in $[0,1]$5 with exponent $[0,1]$6 in the sense inherited from the discrete approximation argument, adapted, and semimartingale-valued in $[0,1]$7. The paper treats either Neumann or Dirichlet boundary behavior through the discrete Laplacian and passes the chosen boundary condition to the limit (Konarovskyi, 2020).

3. Reflection as geometric constraint: the rearranged stochastic heat equation

The RSHE replaces pointwise reflection at the zero set by reflection into a geometric constraint set. The noise admits the expansion

$[0,1]$8

where $[0,1]$9, $X_t(u)=0$0, $X_t(u)=0$1 are independent Brownian motions, and $X_t(u)=0$2 for large $X_t(u)=0$3, with $X_t(u)=0$4. The reflection term $X_t(u)=0$5 is part of the solution and lives in $X_t(u)=0$6 (Delarue et al., 2024).

The state space $X_t(u)=0$7 is isometric to $X_t(u)=0$8 through the law map

$X_t(u)=0$9

This identifies the RSHE as a diffusion on probability measures built by evolving a quantile function and reflecting it whenever it tries to leave the monotone symmetric cone. The reflection is therefore not an external boundary local time in the classical finite-dimensional sense, but a monotone force preserving quantile structure (Delarue et al., 2024).

A central result is an Itô formula for smooth functionals

$\lambda \mathbf 1_{\{X_t(u)=0\}}$0

that are smooth in Lions’ sense. If

$\lambda \mathbf 1_{\{X_t(u)=0\}}$1

then the resulting Itô expansion contains the heat contribution, stochastic integral, and second-order corrections through

$\lambda \mathbf 1_{\{X_t(u)=0\}}$2

but no reflection term. Equivalently, the reflection does not contribute to the generator of the induced Markov process $\lambda \mathbf 1_{\{X_t(u)=0\}}$3 on $\lambda \mathbf 1_{\{X_t(u)=0\}}$4 (Delarue et al., 2024).

The paper states the orthogonality principle explicitly: the reflection term vanishes when tested against smooth functionals of the law, and the induced generator $\lambda \mathbf 1_{\{X_t(u)=0\}}$5 on Wasserstein space contains a drift-like term from the heat operator, a diffusion correction through $\lambda \mathbf 1_{\{X_t(u)=0\}}$6, and a measure-valued second derivative term through $\lambda \mathbf 1_{\{X_t(u)=0\}}$7, but not the reflection. This gives a precise sense in which the reflection is built into the geometry of the quantile representation rather than appearing in the generator acting on smooth mean-field observables (Delarue et al., 2024).

4. Dirichlet forms, integration by parts, and Skorokhod decomposition

In the Dirichlet-form framework, the reference Gaussian measure is the law $\lambda \mathbf 1_{\{X_t(u)=0\}}$8 of the standard Brownian bridge on

$\lambda \mathbf 1_{\{X_t(u)=0\}}$9

with covariance operator

$W$0

Equivalently, $W$1 with Dirichlet boundary conditions satisfies $W$2. For $W$3, the distorted densities are

$W$4

extended by $W$5 outside $W$6 (Grothaus et al., 10 Jun 2026).

The corresponding closable gradient forms

$W$7

generate quasi-regular local Dirichlet forms and hence Markov diffusion processes $W$8 and $W$9. Heuristically, $Q$0 behaves like a stochastic heat flow with a single-point reflection or repulsion at $Q$1, whereas $Q$2 behaves like a heat flow with reflection spread over the whole spatial interval, the drift being generated by local times at all spatial points (Grothaus et al., 10 Jun 2026).

The rigorous result is a Skorokhod decomposition. For $Q$3-quasi-every starting point $Q$4, there exists an $Q$5-cylindrical Wiener process $Q$6 such that for all $Q$7 and all $Q$8,

$Q$9

and analogously for $L^2[0,1]$0. Here $L^2[0,1]$1 is the unit field in the polar decomposition of the vector measure associated with the boundary term, and $L^2[0,1]$2 is the positive continuous additive functional in Revuz correspondence with the boundary measure. The decomposition isolates the martingale term, the heat drift, and the reflection or local-time push (Grothaus et al., 10 Jun 2026).

This framework is tied to infinite-dimensional integration-by-parts formulas. For $L^2[0,1]$3 and $L^2[0,1]$4, the limiting identities are

$L^2[0,1]$5

and

$L^2[0,1]$6

The right-hand sides are Hida-distribution-valued and provide the singular terms from which the boundary measures and Skorokhod decompositions are derived (Grothaus et al., 10 Jun 2026).

5. Approximation and identification methods

A major theme across these models is that the main analytical difficulty lies in identifying reflection or singular terms after approximation.

For the sticky-reflected equation, the solution is constructed from a finite-dimensional particle system with discrete Laplacian $L^2[0,1]$7, correlated Brownian motions $L^2[0,1]$8, and discontinuous coefficients regularized by smooth approximations $L^2[0,1]$9. The hard part is to identify the limit of the terms supported on $f:[0,\infty)\to[0,\infty)$0 and $f:[0,\infty)\to[0,\infty)$1. The paper avoids direct pointwise passage to the limit and instead uses a quadratic-variation characterization of the limit semimartingale: from the approximations one obtains an $f:[0,\infty)\to[0,\infty)$2-valued semimartingale $f:[0,\infty)\to[0,\infty)$3, and a new theorem identifies its quadratic variation structure, yielding that the limiting covariance operator equals the one supported on $f:[0,\infty)\to[0,\infty)$4, while the drift term is exactly the one supported on $f:[0,\infty)\to[0,\infty)$5. A central theorem states that for an $f:[0,\infty)\to[0,\infty)$6-valued heat semimartingale $f:[0,\infty)\to[0,\infty)$7,

$f:[0,\infty)\to[0,\infty)$8

where $f:[0,\infty)\to[0,\infty)$9 is the operator governing the quadratic variation (Konarovskyi, 2020).

For the RSHE, the proof of the Itô formula uses a discrete rearrangement scheme rather than a direct SPDE argument: $X_t(u)\ge 0$00 Here $X_t(u)\ge 0$01 is the rearrangement map into $X_t(u)\ge 0$02. This scheme makes the reflected structure explicit: the process first evolves by heat plus noise, then is rearranged back into the monotone symmetric cone. The key analytic input is the gradient estimate

$X_t(u)\ge 0$03

which gives the $X_t(u)\ge 0$04-control needed for passage to the limit in terms involving $X_t(u)\ge 0$05 (Delarue et al., 2024).

For the Dirichlet-form models, the singular Hida-distribution terms are represented by integration with respect to $X_t(u)\ge 0$06-valued vector measures of bounded variation. The approximation proceeds through mollified densities and yields uniformly bounded and uniformly tight families of vector measures. Uniform tightness is proved using compact Hölder sets

$X_t(u)\ge 0$07

together with a pinning decomposition

$X_t(u)\ge 0$08

A generalized Prokhorov theorem for vector measures then provides weak sequential compactness, after which the limiting vector measures represent the Hida distributions and imply

$X_t(u)\ge 0$09

(Grothaus et al., 10 Jun 2026).

These approaches are methodologically distinct, but they address a common obstacle: the reflection term is either discontinuous, geometric, or distributional, so standard smooth-coefficient SPDE arguments do not directly apply.

6. Regularity, interpretation, and open questions

The available results support several distinct interpretations of reflected stochastic heat equations. In the sticky-reflected model, the solution is explicitly described as an infinite-dimensional sticky-reflected Brownian motion: the noise is turned off at zero, the heat operator couples spatial points, and the sticky behavior is propagated through the PDE rather than acting independently at each spatial site (Konarovskyi, 2020).

In the RSHE, the reflection term is orthogonal to the Lions derivative of smooth functionals on $X_t(u)\ge 0$10. A common misconception is that a reflected stochastic heat equation must display its reflection term directly in the generator. The RSHE shows otherwise: when the process is expressed through quantile functions and then projected to the induced law-valued process $X_t(u)\ge 0$11, the generator contains only the heat and noise contributions, while the reflection is invisible to smooth mean-field observables because it acts only to preserve the quantile ordering constraint (Delarue et al., 2024).

In the Dirichlet-form setting, path properties depend on the underlying distorted measure. For $X_t(u)\ge 0$12, the paper proves that for every starting point $X_t(u)\ge 0$13,

$X_t(u)\ge 0$14

and also

$X_t(u)\ge 0$15

For $X_t(u)\ge 0$16, the corresponding path-valued statement is established only for $X_t(u)\ge 0$17-quasi-every starting point. The paper attributes the stronger result for $X_t(u)\ge 0$18 to the fact that $X_t(u)\ge 0$19 is log-concave, which implies a strong Feller property, whereas $X_t(u)\ge 0$20 is not log-concave (Grothaus et al., 10 Jun 2026).

The same paper also identifies a boundary between tractable singular models and the “true” reflected stochastic heat equation. It states that the reflected Brownian bridge measure $X_t(u)\ge 0$21 is the canonical invariant measure for the “true” reflected stochastic heat equation on $X_t(u)\ge 0$22, but representing the distributional term in its integration-by-parts formula by a vector measure of bounded variation remains open. Also open are determining the support of the limiting vector measure for $X_t(u)\ge 0$23, clarifying the precise behavior of the associated positive continuous additive functional $X_t(u)\ge 0$24 for $X_t(u)\ge 0$25, and proving the plausible stronger statement that $X_t(u)\ge 0$26 (Grothaus et al., 10 Jun 2026).

Taken together, these works delineate a research area in which “reflection” may mean sticky deactivation of noise at the zero set, normal-cone correction in a quantile geometry, or singular boundary forcing recovered from infinite-dimensional integration by parts. The unifying feature is the heat operator under stochastic forcing together with a constraint mechanism that is rigorous, but model-dependent, at the level of weak solutions, generators, or Dirichlet forms.