Qiao Huang

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Verified

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• PhD
• Associate Researcher at Southeast University

Jacobi structures are known to generalize Poisson structures, encompassing symplectic, cosymplectic, and Lie-Poisson manifolds. Notably, other intriguing geometric structures -- such as contact and locally conformal symplectic manifolds -- also admit Jacobi structures but do not belong to the Poisson category. In this paper, we employ global stocha...

Stochastic branching algorithms provide a useful alternative to grid-based schemes for the numerical solution of partial differential equations, particularly in high-dimensional settings. However, they require a strict control of the integrability of random functionals of branching processes in order to ensure the non-explosion of solutions. In thi...

This paper considers the probabilistic representation of the solutions of ordinary differential equations (ODEs) by the generation of random trees. We present sufficient conditions on equation coefficients that ensure the integrability and uniform integrability of the functionals of random trees used in this representation, and yield quantitative e...

We consider a family of marked binary Galton-Watson trees that can model individual types in population genetics, by allowing for mutation and reversion in discrete and continuous time. We derive a recursive formula for the computation of the joint distribution of types conditional to the value of the total progeny. This allows us to compute the ev...

We study a rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against Euclidean space, where the rolling is coupled with random slipping and twisting. The system is modeled by a stochastic differential equation of Stratonovich-type driven by semimartingales on the orthonormal frame bundle. T...

We prove a large deviation principle for stochastic differential equations driven by semimartingales, with additive controls. Conditions are given in terms of characteristics of driven semimartingales, so that if the noise-control pairs satisfy a large deviation principle with some good rate function, so do the solution processes. There is no joint...

В работе доказывается принцип больших уклонений для стохастических дифференциальных уравнений по семимартингалам с аддитивными управлениями. В терминах характеристик семимартингалов приводятся условия, гарантирующие, что если пары "шум-управление" удовлетворяют принципу больших уклонений с некоторой хорошей функцией скорости, то это же будет верно...

We provide a probabilistic representation of the solutions of ordinary differential equations (ODEs) by random generation of Butcher trees. This approach complements and simplifies a recent probabilistic representation of ODE solutions, by removing the need to generate random branching times. The random sampling of trees allows one to improve numer...

We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable L\'evy noises, whose stable indexes may be different. The homogenizing index $r_0$ of slow components has a relation with the stable index $\alpha_1$ of the noise of fast componen...

The most probable transition paths (MPTPs) of a stochastic dynamical system are the global minimisers of the Onsager–Machlup action functional and can be described by a necessary but not sufficient condition, the Euler–Lagrange (EL) equation (a second-order differential equation with initial-terminal condi- tions) from a variational principle. This...

In the framework of the dynamical solution of Schrödinger’s 1931 problem, we compare key aspects of its Lagrangian and Hamiltonian formalisms. This theory is regarded as a stochastic regularization of classical mechanics, in analogy with Feynman’s (informal) path integral approach to quantum mechanics. The role of our counterpart of quantum gauge i...

Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton-Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by...

Classical geometric mechanics, including the study of symmetries, Lagrangian and Hamiltonian mechanics, and the Hamilton–Jacobi theory, are founded on geometric structures such as jets, symplectic and contact ones. In this paper, we shall use a partly forgotten framework of second-order (or stochastic) differential geometry, developed originally by...

We study homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different scales...

We are describing relations between Schrödinger’s variational problem and Onsager’s approach to nonequilibrium statistical mechanics. Although the second work on reciprocal relations and detailed balance has been published in the same year (1931) as the first one, the impact of Schrödinger’s idea has not yet been considered in the classical context...

We study the homogenization for a class of non-symmetric pure jump Feller processes. The jump intensity involves periodic and aperiodic constituents, as well as oscillating and non-oscillating constituents. This means that the noise can come both from the underlying periodic medium and from external environments, and is allowed to have different sc...

We are describing relations between Schr\"odinger's variational problem and Onsager's approach to nonequilibrium statistical mechanics. Although the second work on reciprocal relations and detailed balance has been published the same year (1931) as the first one, the impact of Schr\"odinger's idea has not yet been considered in the classical contex...

We study the ``periodic homogenization'' for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, related to stochastic differential equations driven by multiplicative isotropic $\alpha$-stable L\'evy noise ($1<\alpha<2$) which is nonlinear in the noise component. Our homogenization method is p...

We study the Cucker–Smale (CS) flocking systems involving both singularity and noise. We first show the local strong well-posedness for the stochastic singular CS systems before the first collision time, which is a well-defined stopping time. Then, for communication with higher order singularity at origin (corresponding to α ≥ 1 in the case of ψ(r)...

The most probable transition paths of a stochastic dynamical system are the global minimizers of the Onsager-Machlup action functional and can be described by a necessary but not sufficient condition, the Euler-Lagrange equation (a second-order differential equation with initial-terminal conditions) from a variational principle. This work is devote...

We obtain sample-path large deviations for a class of one-dimensional stochastic differential equations (SDEs) with bounded drifts and heavy-tailed driven L\'evy processes. These heavy-tailed L\'evy processes don't satisfy the exponentially integrability condition, which is a common restriction on the driven L\'evy processes in classical large devi...

We study the Cucker-Smale (C-S) flocking systems involving both singularity and noise. We first show the local strong well-posedness for the stochastic singular C-S systems in a general setting, in which the communication weight $\psi$ is merely locally Lipschitz on $(0,\infty)$ and has lower bound (could be negative). Then, for the special case th...

In this paper, we study the nonlocal Fokker-Planck equations (FPEs) associated with Lévy-driven scalar stochastic dynamical systems. We first derive the Fokker-Planck equation for the case of multiplicative symmetric α-stable noises, by the adjoint operator method. Then we construct a finite difference scheme to simulate the nonlocal FPE on either...

We study the averaging principle for a family of multiscale stochastic dynamical systems. The fast and slow components of the systems are driven by two independent stable L\'evy noises, whose stable indexes may be different. The homogenizing index $r_0$ of slow components has a relation with the stable index $\alpha_1$ of the noise of fast componen...

The Fokker-Planck equations (FPEs) for stochastic systems driven by additive symmetric α-stable noises may not adequately describe the time evolution for the probability densities of solution paths in some practical applications, such as hydrodynamical systems, porous media, and composite materials. As a continuation of previous works on additive c...

We revisit the rolling model from the perspective of probability. More precisely, we consider a Riemannian manifold rolling against the Euclidean space, where the rolling is combined with random slipping and twisting. The system is modelled as a stochastic differential equation of Stratonovich-type driven by semimartingales, on the orthonormal fram...

As a class of Lévy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under Lévy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for 'parabo...

As a class of L\'evy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under L\'evy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for `pa...

We study the "periodic homogenization" for a class of nonlocal partial differential equations of parabolic-type with rapidly oscillating coefficients, related to stochastic differential equations driven by multiplicative isotropic $\alpha$-stable L\'evy noise ($1<\alpha<2$) which is nonlinear in the noise component. Our homogenization method is pro...

As a class of L\'evy type Markov generators, nonlocal Waldenfels operators appear naturally in the context of investigating stochastic dynamics under L\'evy fluctuations and constructing Markov processes with boundary conditions (in particular the construction with jumps). This work is devoted to prove the weak and strong maximum principles for `pa...