A compact support property for infinite-dimensional SDEs with Hölder continuous coefficients

Abstract: We consider non-negative solutions to some infinite-dimensional SDEs on $\mathbb{Z}^d$ with Hölder continuous noise coefficients. We prove that if the Hölder exponent is less than $1/2$, solutions are compactly supported for almost all times, a variant of the classical compact support property for SPDEs. Our results imply that the instantaneous propagation of supports for superprocesses associated to discontinuous spatial motions is effectively sharp. We also show in a special case that the set of times when the support is arbitrarily large is dense. The proof uses a general approach which we expect can be applied to prove similar results for non-local SPDEs. It is based on an analysis of the excursions and zero sets of semimartingales whose quadratic variation satisfies a certain lower bound. As a corollary of our method, we show that the zero sets of non-negative solutions to some simple one-dimensional SDEs have positive Lebesgue measure, despite the absence of "sticky" dynamics.

• The paper establishes a compact support property for infinite-dimensional SDEs with γ in (0,1/2), showing that solutions remain compactly supported for almost all times.
• It employs martingale-based excursion theory and pseudo-strong Markov techniques to control non-Markovian propagation in discrete stochastic systems.
• The results reveal a sharp phase transition at γ=1/2, where the support becomes noncompact, thereby extending classical SPDE support properties to nonlocal models.

Compact Support Phenomena for Infinite-Dimensional SDEs with Hölder Continuous Coefficients

Introduction and Context

The paper "A compact support property for infinite-dimensional SDEs with Hölder continuous coefficients" (2603.29442) addresses the propagation of supports in spatially structured interacting stochastic systems, focusing on infinite-dimensional SDEs on $\mathbb{Z}^d$ whose multiplicative noise coefficients are strictly Hölder continuous with exponent $\gamma\in(0,1)$. The work's primary motivation is to analyze to what extent localized initial conditions remain localized under the dynamics driven by non-Lipschitz, subcritical noise, extending the classical compact support property (CSP) well-known from stochastic partial differential equations (SPDEs) with continuous spatial motion and additive or locally Lipschitz multiplicative noise.

The context is set by recalling several canonical stochastic models where propagation of support is a significant qualitative property: for instance, for the stochastic heat equation with multiplicative (possibly nonlinear) noise, and for measure-valued branching processes such as superprocesses driven by Brownian motion or more general Lévy processes. In the continuous case ($\alpha=2$), for nonlinearity $u^\gamma$ with $\gamma\in (0,1)$, the CSP holds—the solution has compact support for all times if started compactly supported. This fails for certain nonlocal (e.g., stable) spatial motions or for $\gamma\geq 1$, as typified by instantaneous propagation of support in super-stable processes.

Main Results

The authors rigorously establish a compact support property for a class of infinite-dimensional SDEs when the noise coefficient is Hölder with exponent $\gamma \in (0,1/2)$. Specifically, solutions on $\mathbb{Z}^d$ with compactly supported initial data remain compactly supported for almost all times, in Lebesgue measure. This structure is shown to be sharp: for $\gamma=1/2$, corresponding to the critical superprocess (super-Brownian motion), instantaneous propagation of support holds, so the support is noncompact at all but a null set of times.

Strong claims and technical highlights include:

• Measure-theoretic CSP: For all $T>0$, with probability one, the set of times $\gamma\in(0,1)$0 for which the solution has compact support has full Lebesgue measure. This generalizes the CSP from continuous-space SPDEs to discrete, possibly nonlocal systems.
• Sharp phase transition at $\gamma\in(0,1)$1: The results exhibit that $\gamma\in(0,1)$2 is the critical value below which compact support for almost all times is retained, matching the threshold for propagation phenomena in superprocesses driven by discontinuous spatial motion.
• Unboundedness of total support: While solutions are compactly supported for almost all times, the union of supports over any nontrivial time interval is almost surely unbounded. Exceptional times with unbounded support are dense.
• Qualitative analysis of zero sets: For one-dimensional SDEs with nonnegative solutions and coefficients behaving like $\gamma\in(0,1)$3, the set of times the solution spends identically at zero has positive Lebesgue measure, producing a 'fat' Cantor set, despite the absence of sticky behavior.

Analytical Framework and Methods

The principal technical contribution consists in leveraging the interplay between the local extinction mechanism induced by subcritical noise (small $\gamma\in(0,1)$4), and discontinuous spatial motion captured by the generator $\gamma\in(0,1)$5 of a random walk or jump process on $\gamma\in(0,1)$6.

The analysis proceeds by:

• Half-space aggregation: The support propagation is reduced to understanding the process $\gamma\in(0,1)$7 (for $\gamma\in(0,1)$8), which counts total mass to the right of position $\gamma\in(0,1)$9. The process $\alpha=2$0 is not Markov; however, its quadratic variation can be bounded below in terms of $\alpha=2$1 itself, leading to powerful control over excursions to zero.
• Martingale-based excursion theory: The core is a coupling of $\alpha=2$2 to a class of continuous nonnegative semimartingales with drift $\alpha=2$3 and quadratic variation bounded below by $\alpha=2$4, denoted as $\alpha=2$5. The authors demonstrate that, for these semimartingales, the set of times spent at zero has positive (indeed, full) density as $\alpha=2$6.
• Pseudo-strong Markov property: Despite non-Markovianity, the quadratic variation bound permits strong recurrence estimates and pseudo-strong Markov techniques, crucial for the inductive and downcrossing-based estimates of zero-set times.
• Sharp tail bounds and scaling relations: The exit times from small intervals near zero are quantitatively controlled by exponential moment estimates and scaling properties, ensuring the validity of the Lebesgue measure lower bounds on the zero set.

Theoretical Implications

The results establish that subcritical regimes ($\alpha=2$7) in high-dimensional, infinite stochastic systems can preserve near-complete localization—despite spatial jumps and in the absence of strong (Lipschitz) noise regularity. This shows a substantial qualitative difference from the critical and supercritical cases, and extends compact support phenomena to a broad discrete and nonlocal context.

The techniques employed, particularly the reduction to semimartingale excursion behavior via quadratic variation bounds, are general and can, with moderate adaptation, be transferred to nonlocal SPDEs (e.g., stable operator-driven SPDEs and other jump models).

The explicit construction and analysis of exceptional times (times of unbounded support despite compact support at almost every other time) also open up connections to questions in random fractal geometry and potential theory for stochastic flows.

Practical Implications and Extension to SPDEs

The machinery developed enables a systematic study of support phenomena even when the system is non-Markovian or lacks weak uniqueness, a situation prevalent in high-dimensional and SPDE limits. The results imply that, in modeling spatial populations, chemical reaction systems, or genetic dynamics with sufficiently subcritical noise and spatial transport, spatial mass remains essentially confined for nearly all times, permitting a more tractable description of macroscale behavior.

The conjectured extension of these properties to nonlocal SPDEs with general stable-like generators and $\alpha=2$8 is well-supported by the present framework, contingent primarily on transferring certain uniform a priori estimates. This would fill a recognized gap in the theory of SPDEs with non-Lipschitz coefficients and long-range operators.

Highlighted Numerical and Pathwise Results

• For SDEs $\alpha=2$9 with $u^\gamma$0, nonnegative initial data, and positive drift, the set of times $u^\gamma$1 for which $u^\gamma$2 has positive Lebesgue measure almost surely.
• For the infinite-dimensional SDE system on $u^\gamma$3 with compactly supported initial data and $u^\gamma$4, $u^\gamma$5, the measure of times with compact support is full, while times of unbounded support are dense.

Speculations for Further AI and Mathematical Developments

The framework of analyzing support propagation in terms of quadratic variation and excursion counting is potentially applicable to neural field models, interacting particle systems with non-Lipschitz interactions, and extensions involving random environments or interactions. Non-Markovian coupling techniques and pseudo-strong Markov properties may also inspire advances in the control of rare-event and metastable behavior in high-dimensional stochastic models relevant to stochastic numerics, inference, and spatially extended machine learning architectures.

Additionally, the explicit characterization of zero sets in non-sticky, non-Lipschitz SDEs may find applications in the probabilistic analysis of random interface models and the geometric analysis of path-support in random fields.

Conclusion

The paper establishes measure-theoretic compact support properties for infinite-dimensional SDEs with Hölder continuous coefficients with exponent $u^\gamma$6. These results display a phase transition at $u^\gamma$7 and are the first to extend compact support behavior to infinite systems with discontinuous spatial motion. The analysis introduces methodology for non-Markovian semimartingale classes, obtaining sharp quantitative and qualitative control over zero sets and support propagation. The implications are substantial for the theory of stochastic spatial systems, and the framework is poised for significant extension to nonlocal SPDEs and other infinite-dimensional models (2603.29442).