Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0803.0600

Superposition rules and stochastic Lie-Scheffers systems

Joan-Andreu Lázaro-Camí·Juan-Pablo Ortega
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution properties of the distribution generated by the vector fields that define it. When stated in the particular case of standard deterministic systems, our main theorem improves various aspects of the classical Lie-Scheffers result. We show that the stochastic analog of the classical Lie-Scheffers systems can be reduced to the study of Lie group valued stochastic Lie-Scheffers systems; those systems, as well as those taking values in homogeneous spaces are studied in detail. The developments of the paper are illustrated with several examples.

Explore related subjects

math.PRmath.DG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Joan-Andreu Lázaro-Camí, Juan-Pablo Ortega. 2008-03-05. Superposition rules and stochastic Lie-Scheffers systems. https://arxiv.org/abs/0803.0600

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

KEEP EXPLORING

Related papers

arXiv2026

Bounded weak solutions to cross-diffusion semiconductor model with electron-hole scattering

Semiconductor model is a system of parabolic partial differential equations with cross-diffusion phenomenon. Previous results showed that a weak solution exists and is not bounded in general. So semiconductor model was categorized as a cross-diffusion system without bounded weak solutions. In this work, we show that once the initial value is bounded, there exists a weak solution that is also bounded. The entropy method is a major tool in global existence analysis of cross-diffusion systems. We notice that traditional entropies in volume-filling cases may not provide required positive semi-definiteness result for the existence proof. In this situation, a transformation of variables technique has been applied. The product between Hessian matrix of the entropy and replacement diffusion matrix is positive semi-definite, then we apply the entropy method to show semiconductor model has a bounded weak solution.

math.PR↗
arXiv2026

Global existence and uniqueness analysis of cross-diffusion multispecies chemotaxis system with volume-filling

The system of multispecies chemotaxis equations is a cross-diffusion system with volume-filling. In this work, we show that a weak solution of the two species chemotaxis system exists. The entropy method is a major tool in existence analysis of cross-diffusion systems. Previous investigations indicate that traditional entropies in volume-filling cases may not be able to provide required gradient estimates. In this situation, we upgrade existing matrix computation methods to derive gradient estimates. Due to the cross-diffusion phenomenon, the uniqueness of the weak solution to a cross-diffusion system is very difficult to prove in general. In this work, we apply the distance functional to show that when parameters of the chemotaxis system are identical, the weak solution is unique.

math.PR↗
arXiv2026

Self-normalized scaled quadratic variation

The concept of a scaled quadratic variation was originally introduced by E. Gladyshev in 1961 for processes with Gaussian increments. Using certain deterministic scaling, arrived at from the covariance of the process, Gladyshev showed that the sum of scaled square increments along the dyadic partition sequence converges almost surely to a finite limit. In this paper, we propose a pathwise counterpart in which the deterministic normalization is replaced by a self-normalizing factor built from the $p$-th variation of the path along a given sequence of partitions. The resulting quantity requires no probabilistic assumption and no knowledge of a covariance structure, and its scale is both path-dependent and sensitive to the partition sequence. Under a mild regularity condition on the limiting $p$-th variation, we show that the self-normalized and the classical deterministic normalizations are comparable, and for fractional Brownian motion the two agree up to a multiplicative constant. We establish a switching behaviour in the index, and prove that for $p \ge 2$ the self-normalized scaled quadratic variation obeys a smooth-transformation formula under $C^2$ maps; at $p=2$ this recovers the known transformation rule for quadratic variation. Since only squared increments are scaled, the construction polarizes, yielding a matrix-valued scaled quadratic variation for every $p \geq 1$ for $\mathbb R^d$ valued paths. We conclude with examples beyond the Gaussian setting.

math.PR↗