Splitting Schemes for Rough Stochastic Heat Equations

Definitions and mathematical statement

Let $u$ denote the intended solution of the formal stochastic heat equation \[ \partial_t u(t,x)=\partial_{xx}u(t,x)+g(u(t,x))\dot W(t,x),\qquad t>0,\quad x\in(0,1), \] with homogeneous Dirichlet boundary conditions $u(t,0)=u(t,1)=0$ and initial datum denoted by $u_0$. The supplied record does not specify the law and covariance of $W$, the interpretation of the stochastic term, the solution concept, or the admissible initial data.

For $T>0$ and positive integers $M,N$, write \[ \tau=T/M,\qquad h=1/N,\qquad t_m=m\tau,\qquad x_n=nh. \] Let $u^{\mathrm{LT}}_{m,n}$ denote the fully discrete Lie--Trotter approximation referred to as scheme (4) in the source. Its update rule is absent from the supplied record. The excerpt identifies noise increments only through \[ \Delta_{m,n}W=W_n^N(t_{m+1})-W_n^N(t_m), \] without defining the processes $W_n^N$.

Source-reported baseline. Under unspecified technical assumptions, the source attributes to its reference [2] the properties \[ \mathbb P\!\left(u^{\mathrm{LT}}_{m,n}\geq0\text{ for every }0\leq m\leq M,\ 1\leq n\leq N-1\right)=1 \] and the following root-mean-square error bound: for every $\gamma>0$ and $T>0$, there exists $C=C_{\gamma,T}(u_0)>0$ such that, for every pair of positive integers $M,N$ satisfying $\tau\leq\gamma h^2$, \[ \sup_{0\leq m\leq M}\sup_{0\leq n\leq N} \left(\mathbb E\left[\left|u^{\mathrm{LT}}_{m,n}-u(t_m,x_n)\right|^2\right]\right)^{1/2} \leq C\tau^{1/4}. \] This display reconstructs the damaged mathematical typography of the supplied excerpt using its explicit description of mean-square order $1/4$; it requires checking against the source.

Requested extension. Establish these positivity and convergence properties for $g$ less regular than permitted in the source's reference [2], and determine a weak convergence rate for the same approximation. A precise theorem remains unresolved at the formulation level: neither the baseline regularity assumptions nor the intended weaker class of $g$ is supplied. The hypotheses ensuring positivity, including the restrictions on $u_0$ and the behavior of $g$ at zero, must also be recovered. The weak convergence request does not specify a class of test functions, an observable, or a topology. No particular choices for these missing data are asserted here.

Short English statement

The problem asks whether the source's fully discrete Lie--Trotter approximation remains nonnegative and retains a root-mean-square error bound of order $\tau^{1/4}$, under $\tau\leq\gamma h^2$, when the noise coefficient $g$ has weaker regularity. The curated statement also asks for a weak convergence rate. The numerical update, noise model, coefficient assumptions, initial-data assumptions, and weak-error criterion are missing, so this is a qualified statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30005936$, title: Splitting Schemes for Rough Stochastic Heat Equations; exact source problem code: catalogue entry. The record belongs to the Oberwolfach Reports Open Problems collection and gives proposed year 2024; no proposer is named.
• Supplied report citation: Constrained Dynamics, Stochastic Numerical Methods and the Modeling of Complex Systems (2024), Oberwolfach Reports, pp. 1445--1516, DOI 10.4171/owr/2024/26. Source URL and supplied literature source, labelled "Oberwolfach Report 26/2024": https://doi.org/10.4171/owr/2024/26.
• The excerpt attributes the baseline positivity and convergence results to its reference [2], whose bibliographic details are not supplied. It refers to SPDE (3) and scheme (4), whose complete definitions are not supplied.
• Literature checked at: 2026-08-22. This is supplied dated curation, not an independently verified assessment. It labels the problem open and states that the source leaves positivity and convergence for less regular $g$ open, with no authoritative subsequent resolution found in that curation. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry marks the statement corrected_verified and reports that it was reconstructed and checked against the official report. These are supplied provenance claims; no independent source consultation or verification is claimed here.