Approximation of stochastic processes by non-expansive flows and coming down from infinity

Vincent Bansaye

Home/Archive/arXiv

arXiv · 1511.07396

Approximation of stochastic processes by non-expansive flows and coming down from infinity

Vincent Bansaye

View original source ↗Read source document ↗Export metadata ↓

Abstract

We approximate stochastic processes in finite dimension by dynamical systems. We provide trajectorial estimates which are uniform with respect to the initial condition for a well chosen distance. This relies on some non-expansivity property of the flow, which allows to deal with non-Lipschitz vector fields. We use the stochastic calculus and follow the martingale technics initiated in Berestycki and al [5] to control the fluctuations. Our main applications deal with the short time behavior of stochastic processes starting from large initial values. We state general properties on the coming down from infinity of one-dimensional SDEs, with a focus on stochastically monotone processes. In particular, we recover and complement known results on Lambda-coalescent and birth and death processes. Moreover, using Poincar{é}'s compactification technicsfor dynamical systems close to infinity, we develop this approach in two dimensions for competitive stochastic models. We classify the coming down from infinity of Lotka-Volterra diffusions and provide uniform estimates for the scaling limits of competitive birth and death processes.

Explore related subjects

math.PR

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Vincent Bansaye. 2017-01-10. Approximation of stochastic processes by non-expansive flows and coming down from infinity. https://arxiv.org/abs/1511.07396

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↗