Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative $α$-stable processes: Existence, uniqueness and numerical approximation

Jiaquan Lu; Huaizhong Zhao

Home/Archive/arXiv

arXiv · 2610.35519

Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative $α$-stable processes: Existence, uniqueness and numerical approximation

Jiaquan Lu·Huaizhong Zhao

View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we study the periodic measures within the context of time-inhomogeneous Markov processes. By extending the Hairer--Mattingly method to the time-inhomogeneous case, we first establish the existence, uniqueness and exponential ergodicity of periodic measures for general time-periodic Markov processes. These results are subsequently applied to time-inhomogeneous stochastic differential equations under the assumption that the drift is weakly dissipative and locally Lipschitz. We further employ the truncated Euler--Maruyama scheme to study numerical approximations of the periodic measures and derive the convergence rate between the exact and discrete periodic measures. These numerical results are new even in the time-homogeneous case under the assumptions considered in this paper.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Jiaquan Lu, Huaizhong Zhao. 2026-10-26. Periodic measures for time-inhomogeneous stochastic differential equations driven by multiplicative $α$-stable processes: Existence, uniqueness and numerical approximation. https://arxiv.org/abs/2610.35519

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

KEEP EXPLORING

Related papers

arXiv2026

An Abstract Framework for Nonlinear Stochastic Parabolic Systems with Application to a Chemotaxis-Fluid Model

The purpose of the paper is twofold. Firstly, we want to present an abstract existence Theorem to show the existence of a martingale solution for coupled systems of non-linear stochastic partial differential equations. The idea is first to split the system by rewriting the nonlinear part in a linear part acting on a given process. This is done in such a way that the fixed point with respect to the given process would be the solution of the original Stochastic Partial Differential Equation (SPDE). However, it may be necessary to show the well-posedness of the linearized system using a cut-off argument, which may impose additional constraints. We work out under which conditions on the underlying spaces and the cut-off argument one gets a martingale solution of the original SPDE. A similar strategy was applied by Hausenblas and Tolle in [E. Hausenblas and J. Tolle. The stochastic Klausmeier system and a schauder tychonoff type theorem. Potential Analusis, 310:506-554, 2022] to investigate the stochastic Klausmeier system. Secondly, we want to verify the full applicability of the existence Theorem by showing the existence of a martingale solution of a highly nonlinear chemotaxis system with underlying fluid dynamics.

math.PR↗

arXiv2026

Hitting probability for Reflected Brownian Motion at Small Target

We derive the asymptotic behavior of hitting probability at small target of size $O(ε)$ for reflected Brownian motion in domains with suitable smooth boundary conditions, where the boundary of domain contains both reflecting part, absorbing part and target. In this case the domain could be localized near the target and explicit computations are possible. The asymptotic behavior is only related to $ε$ up to some multiplicative constants that depends on the domain and boundary conditions.

math.PR↗

arXiv2026

The Integrable Snake Model

A pure snake configuration is a bijection $σ:\mathbb{Z}^2 \to \mathbb{Z}^2$ containing no two-cycles and such that for each $x \in \mathbb{Z}^2$ we have $σ(x) \in \{ x , x+ \mathbf{e}^1, x+\mathbf{e}^2 , x- \mathbf{e}^2 \}.$ The non-trivial cycles of a pure snake configuration may be regarded as a collection of non-intersecting paths in $\mathbb{Z}^2$ that may travel right, up, or down (but not left) from a given vertex. Pure snake configurations are a generalisation of lozenge tilings, which are in natural correspondence with paths that only travel right or up. We introduce a partition function on a finite version of this model and study the probabilistic properties of random pure snake configurations chosen according to their contribution to this partition function. Under a suitable weighting, the model is integrable in the sense that we have access to explicit formulas for its partition function and correlation function. We utilise the integrable structure of this model in several applications through its various scaling limits, such as to prove a traffic representation of ASEP on the ring, generalising the analogous result for TASEP by the first author.

math.PR↗