Strong well-posedness for cyclic catalytic reaction--diffusion SPDEs with mutation and immigration

Jie Xiong; Wen Xu

Home/Archive/arXiv

arXiv · 2610.41400

Strong well-posedness for cyclic catalytic reaction--diffusion SPDEs with mutation and immigration

Jie Xiong·Wen Xu

View original source ↗Read source document ↗Export metadata ↓

Abstract

We study cyclic catalytic stochastic reaction-diffusion systems with mutation and immigration. The equations have square-root catalytic coefficients and spatially colored noise. Under a condition requiring the cyclic inward drift to dominate the corresponding noise covariance, we establish global strong well-posedness for nonnegative initial states. Our approach combines positivity-preserving approximations, uniform energy estimates, and a concave power entropy that controls the occupation of states near zero. These boundary estimates allow us to derive a Hellinger stability inequality for two synchronized weak variational solutions, yielding pathwise uniqueness directly in this solution class. We then use joint compactness of the approximating solutions and their driving noise, together with martingale identification and the Gyongy-Krylov criterion, to construct a probabilistically strong solution on the prescribed Wiener space.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jie Xiong, Wen Xu. 2026-10-30. Strong well-posedness for cyclic catalytic reaction--diffusion SPDEs with mutation and immigration. https://arxiv.org/abs/2610.41400

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Quantitative Propagation of Chaos for Singular Interacting Particle Systems Driven by Fractional Brownian Motion

We consider interacting systems particle driven by i.i.d. fractional Brownian motions, subject to irregular, possibly distributional, pairwise interactions. We show propagation of chaos and mean field convergence to the law of the associated McKean--Vlasov equation, as the number of particles $N\to\infty$, with quantitative sharp rates of order $N^{-1/2}$. Our results hold for a wide class of possibly time-dependent interactions, which are only assumed to satisfy a Besov-type regularity, related to the Hurst parameter $H\in (0,+\infty)\setminus \mathbb{N}$ of the driving noises. In particular, as $H$ decreases to $0$, interaction kernels of arbitrary singularity can be considered, a phenomenon frequently observed in regularization by noise results. Our proofs rely on a combinations of Sznitman's direct comparison argument with stochastic sewing techniques.

math.PR↗

arXiv2026

Power rate of convergence of discrete curves: framework and applications

We develop a framework for power-law rates of convergence of random discrete curves to Schramm Loewner Evolution (SLE). We show that a power-law rate of convergence of a suitable martingale observable gives a power-law rate of convergence of the interface to SLE, provided the discrete curves satisfy the Kemppainen-Smirnov crossing condition; the proof uses known estimates on the derivative of the SLE map. We apply the framework to the exploration process of critical site percolation on the triangular lattice, obtaining a power-law rate of convergence to SLE$_6$.

math.PR↗

arXiv2026

Refining Concentration Bounds for Gaussian Quadratic Chaos with Applications

This paper studies concentration of measure for Gaussian quadratic chaos in the non-asymptotic regime. We begin by tightening the classic Hanson-Wright inequality, where we increase its absolute constant $κ$ from 0.125 to at least 0.1457 in the symmetric case. We present a sharper version of Laurent-Massart inequality (LMI), which increases $κ$ from the largest available value of $1-\frac{\sqrt{3}}{2}\approx 0.134$ to $\frac{9-\sqrt{17}}{32}\approx 0.152$ in the positive-semidefinite case. We develop a sequence of inequalities indexed by $m\ge1$ that involve Schatten norms of the underlying matrix. The case $m=1$ recovers HWI. The bounds undergo a phase transition in the sense that if the tail parameter is smaller than a critical threshold $τ_c$, then $m=1$ is the tightest, whereas if it is larger than $τ_c$, then $m=\infty$ is the tightest. This leads to a novel bound called the $m_\infty$ bound. All HWI, LMI and the $m_\infty$ inequality rely on Markov's inequality. By avoiding Markov's inequality, we introduce the strong and weak $χ^2$ inequalities. We provide a list of concentration bounds that depend only on the operator norm in the positive-definite case, namely, the $m_\infty$ bound, the weak $χ^2$ bound, relaxed versions of HWI and LMI, and the so-called large-deviations bound. The sharpest among these is shown to be always either the $m_\infty$ bound or the weak $χ^2$ bound. By utilizing conditioning, modified versions of $m_\infty$, $χ^2$, and HW inequalities of various orders are proposed, all of which can be significantly tighter than the original inequalities. By leveraging the Ruben-Kotz expansion for the distribution of a positive-definite quadratic form, we introduce the Ruben-Kotz inequalities (RKIs) for general symmetric quadratic forms. A key advantage of RKIs is that they provide an explicit upper bound on the approximation gap.

math.PR↗