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arxiv: 2310.18726 · v3 · pith:X4MGSNDJnew · submitted 2023-10-28 · 🧮 math.PR

Stochastic partial differential equations associated with Feller processes

Jian Song , Meng Wang , Wangjun Yuan This is my paper

Pith reviewed 2026-05-24 06:32 UTC · model grok-4.3

classification 🧮 math.PR
keywords stochastic partial differential equationsFeller processesFeynman-Kac representationsStratonovich solutionsSkorohod solutionscolored Gaussian noiseHölder continuityregularity of laws
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The pith

Feynman-Kac representations are derived for Stratonovich and Skorohod solutions of SPDEs with Feller process generators and time-colored Gaussian noise.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines the stochastic partial differential equation where the time derivative of the solution equals the action of the generator on the solution plus the solution multiplied by the noise. The noise is Gaussian and colored in time while the generator is that of a Feller process. Feynman-Kac type representations are obtained for the Stratonovich and Skorohod solutions as well as for their moments. The regularity properties of the law of the solution and its Hölder continuity are also investigated. A sympathetic reader would care because the representations tie the SPDE directly to path expectations over the Feller process and thereby supply a probabilistic route to analyzing existence, moments, and smoothness.

Core claim

For the SPDE ∂u/∂t = ℒu + uẆ where Ẇ denotes Gaussian noise colored in time and ℒ is the infinitesimal generator of a Feller process X, Feynman-Kac type representations are obtained for the Stratonovich and Skorohod solutions together with their moments. The regularity of the law and the Hölder continuity of the solutions are also studied.

What carries the argument

Feynman-Kac type representations that express the SPDE solutions and moments through expectations involving paths of the Feller process X and the driving noise.

Load-bearing premise

The SPDE admits well-defined Stratonovich and Skorohod solutions when the driving noise is Gaussian and colored in time and when the spatial operator is the generator of a Feller process.

What would settle it

An explicit closed-form solution computed for the SPDE when the Feller process is standard Brownian motion, checked directly against the proposed representation for mismatch in the Stratonovich or Skorohod case.

read the original abstract

For the stochastic partial differential equation $\frac{\partial u}{\partial t}=\mathcal L u +u\dot W$ where $\dot W$ is Gaussian noise colored in time and $\mathcal L$ is the infinitesimal generator of a Feller process $X$, we obtain Feynman-Kac type of representations for the Stratonovich and Skorohod solutions as well as for their moments. The regularity of the law and the H\"older continuity of the solutions are also studied.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript claims to obtain Feynman-Kac type representations for the Stratonovich and Skorohod solutions (and their moments) to the SPDE ∂u/∂t = ℒu + uẆ, where Ẇ is Gaussian noise colored in time and ℒ is the generator of a Feller process X; it further studies the regularity of the law and Hölder continuity of the solutions.

Significance. If the existence of the solutions can be established under explicit conditions and the representations are rigorously derived, the work would connect SPDE theory with Feller semigroups in a useful way, potentially enabling semigroup-based analysis of moments and path regularity for colored-noise driven equations.

major comments (2)
  1. [Introduction / Setup] The central claims presuppose that the SPDE admits well-defined Stratonovich and Skorohod solutions for general Feller generators ℒ and the given time-colored Gaussian noise, yet no conditions on the semigroup or covariance are derived or cited to guarantee existence; this renders the representations and all subsequent regularity results conditional (see the setup preceding the main theorems).
  2. [Main results section (representations)] The Feynman-Kac representations for the solutions and moments are stated without accompanying verification steps, error bounds, or explicit use of the Feller property in the derivations; the absence of these controls makes it impossible to assess whether the representations hold independently of the existence assumption.
minor comments (2)
  1. [Preliminaries] Notation for the noise Ẇ and the precise definition of the Stratonovich vs. Skorohod integrals should be introduced with explicit references to the underlying probability space early in the paper.
  2. [Regularity results] The statement of Hölder continuity results would benefit from a clear statement of the Hölder exponents in terms of the noise regularity parameters.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments. Below we address each major comment point by point. We agree that the assumptions on existence need to be stated more explicitly and that the derivations would benefit from additional detail; we will revise accordingly.

read point-by-point responses

  1. Referee: [Introduction / Setup] The central claims presuppose that the SPDE admits well-defined Stratonovich and Skorohod solutions for general Feller generators ℒ and the given time-colored Gaussian noise, yet no conditions on the semigroup or covariance are derived or cited to guarantee existence; this renders the representations and all subsequent regularity results conditional (see the setup preceding the main theorems).

    Authors: We acknowledge that the manuscript presupposes existence of the Stratonovich and Skorohod solutions and does not derive new existence criteria. The setup relies on standard conditions from the SPDE literature for equations driven by time-colored Gaussian noise (typically Hölder regularity of the covariance in time and suitable growth/boundedness assumptions on the Feller semigroup). We will revise the introduction and the setup section preceding the main theorems to cite the relevant existence results explicitly and to list the precise conditions on the covariance and on ℒ under which the solutions are assumed to exist. This will clarify that the representations and regularity statements are conditional on those hypotheses. revision: yes

  2. Referee: [Main results section (representations)] The Feynman-Kac representations for the solutions and moments are stated without accompanying verification steps, error bounds, or explicit use of the Feller property in the derivations; the absence of these controls makes it impossible to assess whether the representations hold independently of the existence assumption.

    Authors: The derivations invoke the Feller property via the Markov semigroup generated by ℒ and the pathwise properties of the underlying Feller process X to express the solutions as expectations. Nevertheless, we agree that the verification steps are presented too concisely and that explicit error bounds are missing. In the revised version we will expand the proofs in the main results section to include the intermediate approximation steps, the precise invocation of the Feller semigroup, and the error estimates needed to justify passing to the limit, thereby making the dependence on the existence assumption transparent. revision: yes

Circularity Check

0 steps flagged

No significant circularity; representations are conditional on assumed existence

full rationale

The paper assumes well-defined Stratonovich and Skorohod solutions exist for the given SPDE driven by time-colored Gaussian noise and Feller generator ℒ, then derives Feynman-Kac representations for those solutions and their moments along with regularity results. No step reduces a claimed prediction or representation to a fitted parameter or self-referential definition by construction, nor does any load-bearing premise rest solely on a self-citation chain. The derivation chain is self-contained once the existence assumption is granted; the conditional nature of the results is a limitation on scope rather than circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Ledger inferred from abstract only; full paper likely invokes standard properties of Feller semigroups and stochastic integrals.

axioms (2)
  • standard math Feller processes possess strongly continuous contraction semigroups on the space of continuous functions vanishing at infinity
    Standard background fact for the generator ℒ in the SPDE.
  • domain assumption The driving noise is a centered Gaussian process that is colored in time
    Explicitly stated in the SPDE setup.

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