Narrow Results

In this paper we study the structure of square integrable functionals measurable with respect to coalescing stochastic flows. The case of the Wiener process stopped at the moment of hitting an irregular continuous level is considered. Relying on the change of measure technique, we present a new construction of multiple stochastic integrals with respect to stopped Wiener process. An intrinsic analogue of the Itô–Wiener expansion for the space of square integrable functionals measurable with respect to the stopped Wiener process is constructed.

An infection spreads in a binary tree of height n as follows: initially, each leaf is either infected by one of k states or it is not infected at all. The infection state of each leaf is independently distributed according to a probability vector p = (p₁, …, p_k+1). The remaining nodes become infected or not via annihilation and coalescence: nodes whose two children have the same state (infected or not) are infected (or not) by this state; nodes whose two children have different states are not infected; nodes such that only one of the children is infected are infected by this state. In this paper we characterize, for every p, the limiting distribution at the root node of as n goes to infinity.

We also consider a variant of the model when k = 2 and a mutation can happen, with a fixed probability q, at each infection step. We characterize, in terms of p and q, the limiting distribution at the root node of as n goes to infinity.

The distribution at the root node is driven by a dynamical system, and the proofs rely on the analysis of this dynamics.

Existence of random dynamical systems for a class of coalescing stochastic flows on $ℝ$ is proved. A new state space for coalescing flows is built. As particular cases coalescing flows of solutions to stochastic differential equations and coalescing Harris flows are considered.

We explain some results of [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.], discussed in our talk [G. Cotti, Monodromy of semisimple Frobenius coalescent structures, in Int. Workshop Asymptotic and Computational Aspects of Complex Differential Equations, CRM, Pisa, February 13–17, (2017).] in Pisa, February 2017. Consider an $n\times n$ linear system of ODEs with an irregular singularity of Poincaré rank 1 at $z=\infty$ and Fuchsian singularity at $z=0$, holomorphically depending on parameter $t$ within a polydisk in ${ℂ}^n$ centered at $t=0$. The eigenvalues of the leading matrix at $\infty$, which is diagonal, coalesce along a coalescence locus $\mathrm{\Delta }$ contained in the polydisk. Under minimal vanishing conditions on the residue matrix at $z=0$, we show in [G. Cotti, B. A. Dubrovin and D. Guzzetti, Isomonodromy deformations at an irregular singularity with coalescing eigenvalues, preprint (2017); arXiv:1706.04808.] that isomonodromic deformations can be extended to the whole polydisk, including $\mathrm{\Delta }$, in such a way that the fundamental matrix solutions and the constant monodromy data are well defined in the whole polydisk. These data can be computed just by considering the system at point of $\mathrm{\Delta }$, where it simplifies. Conversely, if the $t$-dependent system is isomonodromic in a small domain contained in the polydisk not intersecting $\mathrm{\Delta }$, and if suitable entries of the Stokes matrices vanish, then $\mathrm{\Delta }$ is not a branching locus for the fundamental matrix solutions. The results have applications to Frobenius manifolds and Painlevé equations.