On Skorokhod Problems for Reflected and Singular Stochastic Heat Equations

Abstract: We prove a Skorokhod decomposition for the Markov processes $X<sup>a$ and $X$ associated to the gradient Dirichlet forms with respect to the measures $ρ<sup>aμ<sup>β$ and $ρμ<sup>β$, respectively. Here, $μ<sup>β$ is the law of the standard Brownian bridge $β$, while $ρ<sup>a$ and $ρ$ denote densities which are given by $$ρ^a(z) := \mathbf{1}<em>{[0,\infty)}(\bar{z}_a)$$ and $$ρ(z) := \int_0¹ \mathbf{1}</em>{[0,\infty)}(\bar{z}_x) \, dx$$, respectively, for all $z\in L<sup>2(0,1)$ which have a (unique) continuous representative $\bar{z}$ which vanishes at zero and one. To this end, we derive infinite-dimensional integration by parts formulas (IbPFs) w.r.t. $ρ<sup>aμ<sup>β$ and $ρμ<sup>β$, which contain Hida distributions alongside the usual drift terms. We represent these Hida distributions by integration w.r.t. vector measures of bounded variation. The vector measures in question are constructed via an approximation argument, making use of a generalization of Prokhorov's theorem for vector measures. We further prove that, almost surely, the sample paths of $X<sup>a$ and $X$ take values in the equivalence class of continuous functions vanishing at zero and one for all and $dt$-almost all times, respectively. The main motivation for studying $ρ<sup>aμ<sup>β$ and $ρμ<sup>β$ lies in the fact that the distributional terms in their IbPFs are simplifications of the distributional term in the IbPF w.r.t. the law of the reflected Brownian bridge on the unit interval $μ<sup>{|β|}$. Representing the latter by integration w.r.t. a vector measure of bounded variation is still an open problem.

• The paper establishes closable, quasi-regular Dirichlet forms for Brownian bridge laws weighted by non-negativity at one point or across all points, yielding associated distorted Ornstein–Uhlenbeck processes.
• It represents the resulting Hida-distribution boundary terms through bounded-variation vector measures in $H_0^{-1}(0,1)$ and proves that the corresponding densities belong to $mathrm{BV}(L,H_0^1)$, with contact supports identified in the one-point case.
• The paper derives rigorous Fukushima–Skorokhod decompositions with local-time drifts concentrated at a fixed spatial point or distributed over the interval, while leaving the full reflected Brownian bridge problem and several uniqueness questions open.

This paper by Grothaus and Renner (2606.11951) studies two gradient Dirichlet forms on $L^2(0,1)$ whose reference measures are weighted laws of the standard Brownian bridge, with weights given by indicators of non-negativity at a fixed point $a\in(0,1)$ or integrated over all points. The authors derive integration by parts formulas (IbPFs) for these measures containing Hida distributions, represent those distributional terms via $H_0^{-1}(0,1)$-valued vector measures of bounded variation, and thereby obtain Skorokhod decompositions of the associated distorted Ornstein–Uhlenbeck processes. The work is positioned as a tractable simplification of the IbPF for the law of the reflected Brownian bridge obtained in Grothaus–Vosshall, where a corresponding vector-measure representation remains open.

Setting and motivation

The underlying path space is $L = L^2(0,1)$, equipped with the law $\gamma$ of the standard Brownian bridge from 0 to 0, realized as the image of the white noise measure under the evaluation map. Two densities are considered: $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$ for a fixed $a \in (0,1)$, and $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$, both extended to $L$ by zero, where $\bar z$ denotes the continuous representative vanishing at the endpoints. These measures weight the Gaussian bridge law by the probability that the path is non-negative at one point, respectively at every point.

The motivation comes from the IbPF on the law of the reflected Brownian bridge established in GV18, whose right-hand side contains the Hida distribution $a\in(0,1)$0 involving Wick-renormalized squared white noise. Whether this distribution admits representation by integration against an $a\in(0,1)$1-valued vector measure of bounded variation is an open problem; prior work of Altman–Zambotti approximated the analogous term for Bessel bridges but did not resolve it. The present paper studies simplified versions of this term — dropping the renormalized square and, in one case, the spatial integral — as model problems that can be fully solved.

Quasi-regular Dirichlet forms

The symmetric bilinear forms

$a\in(0,1)$2

are shown to be closable on $a\in(0,1)$3 and $a\in(0,1)$4, with closures being quasi-regular local Dirichlet forms. The proof verifies the hypotheses of Röckner–Zhu–Zhu (RZZ12): the densities belong to $a\in(0,1)$5 and satisfy the ray Hamza condition in every eigenfunction direction of the Dirichlet Laplacian — here monotonicity of $a\in(0,1)$6 along rays is the key observation — and both measures have full topological support, established by explicit small-modification constructions around the points $a\in(0,1)$7, $a\in(0,1)$8, and $a\in(0,1)$9. By quasi-regularity, each form is associated via Revuz correspondence with a diffusion process (distorted Ornstein–Uhlenbeck process) $H_0^{-1}(0,1)$0 and $H_0^{-1}(0,1)$1 on $H_0^{-1}(0,1)$2.

Integration by parts formulas

A central technical ingredient is a version of the Cameron–Martin formula valid for displacements $H_0^{-1}(0,1)$3 in $H_0^{-1}(0,1)$4:

$H_0^{-1}(0,1)$5

It is proved by matching characteristic functionals on a separating class, using the white-noise Cameron–Martin formula together with the identity $H_0^{-1}(0,1)$6 for $H_0^{-1}(0,1)$7-a.e. $H_0^{-1}(0,1)$8, which itself follows from properties of the operator $H_0^{-1}(0,1)$9 mapping test functions to their bridge coordinates.

With smoothed densities $L = L^2(0,1)$0, $L = L^2(0,1)$1 obtained by mollifying the indicator with Gaussian kernels, differentiation under the integral sign yields smoothed IbPFs whose right-hand sides involve $L = L^2(0,1)$2 and $L = L^2(0,1)$3, interpreted as pairings with Donsker's delta in the Hida space. Passing to the limit $L = L^2(0,1)$4 using the convergence of mollified Donsker deltas in $L = L^2(0,1)$5 and density of $L = L^2(0,1)$6 in $L = L^2(0,1)$7, the main IbPF theorem states:

$L = L^2(0,1)$8

$L = L^2(0,1)$9

first for $\gamma$0 and then, by unique operator extension to $\gamma$1, for all $\gamma$2. These are precisely the announced simplifications of the reflected-Brownian-bridge IbPF: the singular term is now a local time-type functional without the Wick-renormalized square.

Vector-measure representation

The representation problem is solved by constructing approximating families of vector measures. For fixed $\gamma$3, one defines $\gamma$4 and $\gamma$5; analogously, $\gamma$6. Well-definedness of $\gamma$7 as an element of $\gamma$8 requires a norm estimate exploiting $\gamma$9-type bounds and Pettis measurability for the Bochner integral over $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$0.

Weak sequential compactness follows from a generalized Prokhorov theorem for vector measures (März–Shortt): uniform boundedness holds since $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$1 uniformly in $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$2, and uniform tightness w.r.t. the semivariation is proved using Hölder sets $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$3 (compact in $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$4 by Arzelà–Ascoli, of full $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$5-measure) combined with a "roughness of pinning" lemma showing that adding a pinned linear perturbation cannot smooth a rough path beyond controlled Hölder classes, together with the independence of $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$6 from the pinned bridge $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$7.

Fixing weakly convergent subsequences yields limit objects $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$8 and $\rho^a(z) = 1_{[0,\infty)}(\bar z_a)$9, with $a \in (0,1)$0 converging weakly to the natural candidate built from $a \in (0,1)$1. The main representation theorem then gives, for all $a \in (0,1)$2 and $a \in (0,1)$3,

$a \in (0,1)$4

where $a \in (0,1)$5 and $a \in (0,1)$6 are polar decompositions. A direct consequence via RZZ12 is that $a \in (0,1)$7 with $a \in (0,1)$8 and $a \in (0,1)$9. The support computation shows that, restricted to $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$0 with the supremum norm, $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$1, proved via Portmanteau arguments and the pinning-independence lemma.

Skorokhod decompositions and path regularity

Applying RZZ12's stochastic reflection theory, the paper obtains the Skorokhod decompositions: for $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$2-quasi-every starting point there exists a cylindrical Wiener process such that

$\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$3

with $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$4 the PCAF in Revuz correspondence with $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$5; analogously for $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$6 with $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$7. Heuristically, $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$8 solves an SPDE with drift $\rho(z) = \int_0^1 1_{[0,\infty)}(\bar z_x)\,dx$9 — a localized upward force at $L$0 acting when the process hits zero there — while $L$1 carries a spatially distributed local-time drift $L$2. The paper stresses that these SPDE interpretations are heuristic only; the Fukushima decompositions above are the last rigorous statements.

Path regularity is addressed with a transfer-method argument: $L$3 for every $L$4, and the same holds for $L$5 only for quasi-every starting point. The improvement for $L$6 uses log-concavity of $L$7 (inherited from $L$8 and convexity of the constraint set), which implies strong Feller semigroups via Ambrosio–Savaré–Zambotti; the paper notes explicitly that $L$9 is not log-concave, so this extension argument breaks down for $\bar z$0. Additionally, $\bar z$1 almost surely for a.e. time, and the local time measure satisfies $\bar z$2 quasi-surely, confirming that the singular drift acts exactly on the contact set.

Limitations and open questions

Several restrictions are acknowledged. The weak limits $\bar z$3 and $\bar z$4 may depend a priori on the choice of weakly convergent subsequence, and uniqueness of the limits is not established. Determining the support of $\bar z$5 remains open, as does a finer analysis of the unit field $\bar z$6. Whether $\bar z$7 — stronger than membership in $\bar z$8 — is left open, though the authors note indications in its favor. Most significantly, the original motivating problem persists: representing the full Hida distribution in the reflected-Brownian-bridge IbPF of GV18 by a vector measure of bounded variation is still unsolved, and the extension of the continuity statement for $\bar z$9 from quasi-every to every starting point is blocked by the failure of log-concavity of $a\in(0,1)$00.

Conclusion

The paper provides a complete treatment — closability, IbPFs, BV membership, vector-measure representations, and Skorokhod decompositions — for two singular reflected stochastic heat equations whose invariant measures are one-point and all-point non-negativity weightings of the Brownian bridge law. The combination of white noise analysis, generalized Prokhorov compactness for vector measures, and Dirichlet form methods yields rigorous meaning to otherwise ill-defined local-time drifts, and isolates precisely which analytic obstacles (subsequence dependence, log-concavity, the renormalized square term) separate these solved cases from the full reflected Brownian bridge problem.