arXiv subjects

Yawei Wei

Publications and source records attributed to Yawei Wei.

At least 19 recordsLinked to original sources

arXiv2026

Mean field games with Hormander diffusions

In this paper, we study a class of degenerate mean field games (MFG, for short) systems with Hormander diffusion, in which the typical agent can move only along admissible direction. We establish the well-posedness of the MFG systems in intrinsic Holder spaces, which describes the Nash equilibria for a differential game with infinitely many small players. The analysis builds upon degenerate parabolic equations defined on the tours induced by Hormander vector fields without any group structure, for which we develop a global regularity theory for the first time in this general setting. A central difficulty arises from the anisotropic geometry of the state space, which induces non-commutativity and inhomogeneity of the underlying vector fields. Instead of the classical fundamental solution framework, we present an alternative method to derive a priori Schauder estimates for general Hormander degenerate parabolic equations via Campanato spaces.

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arXiv2026

State-Dependent Delays in Optimal Control and Hamilton--Jacobi Equations

We develop a Hamilton--Jacobi theory for finite-horizon optimal control problems governed by state-dependent delay equations. In this setting, the delayed-time map depends on the controlled trajectory itself, so that both the state and the point at which the past state is evaluated vary simultaneously. The natural state variable is therefore the entire history, and the value functional is defined on a space of Lipschitz histories. Under suitable growth, Lipschitz, and monotonicity assumptions, we establish the dynamic programming principle and characterize the value functional as the unique viscosity solution of the associated Hamilton--Jacobi equation with co-invariant derivatives. To accommodate Lipschitz histories, we introduce a notion of viscosity solution based on finite-dimensional projections generated by polygonal extensions, and prove a comparison principle in the resulting class of functionals. Under additional regularity assumptions, we derive a Pontryagin minimum principle whose adjoint equation contains advanced terms induced by the state dependence of the delay. We also obtain a generalized transversality relation formulated through the delay superdifferential of the value functional. Finally, we establish semiconcavity in the history variable and a joint semiconcavity estimate on an appropriate solution manifold.

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arXiv2026

Stochastic Keller Segel System with Porous Medium Diffusion and Nonlinear Chemotactic Sensitivity

In this paper, we investigate a stochastic Keller--Segel system with porous medium diffusion and nonlinear chemotactic sensitivity on a bounded one-dimensional domain. The model describes cell aggregation in complex environments, where the dispersal of cells is governed by density-dependent diffusion $Δu^{[m]}$, reflecting the combined effects of porous media and population crowding, and the perception of chemoattractants follows Stevens' power law, leading to the nonlinear chemotactic sensitivity $\nabla\cdot(u\nabla v^{[a]})$. In addition, random environmental fluctuations are incorporated through multiplicative noise $u\,dW(t)$, which represents stochastic perturbations in population dynamics. For $a\geq1$ and $m\geq2a+1$, we establish the global existence of martingale solutions, uniform a priori estimates, and preservation of non-negativity. The condition $m\geq2a+1$ reveals a balance between nonlinear chemotactic aggregation and porous-medium diffusion: stronger sensing response requires stronger diffusion to prevent excessive aggregation. The proof combines a decoupled auxiliary system, energy estimates, and a stochastic Schauder--Tychonoff fixed point argument to overcome the difficulties caused by degenerate diffusion, nonlinear drift, and stochastic perturbations.

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arXiv2026

Stochastic Two-Species Chemotaxis Competition: Global Well Posedness and Continuous Dependence

In this paper, we investigate a stochastic two species competition Keller--Segel model. Chemotactic movement, interspecific competition, and environmental fluctuations may act simultaneously when two species respond to the same chemical signal. We consider a stochastic two-species chemotaxis--competition system on a bounded smooth domain $\mathcal O\subset\mathbb R^n$, $n\in\{1,2\}$. Both species contribute to the production of the signal and move along its gradient; their population equations also include nonlinear competition and multiplicative noise. If the self-damping exponents exceed two, we prove that the system has a unique global adapted nonnegative mild solution. We also establish continuous dependence in probability on admissible initial data over finite time intervals. The results prove the global well-posedness under the stated assumptions and we still show that small changes in the initial populations, in probability, to small changes in the solution. Due to the two coupled chemotactic fluxes and the stochastic terms, the $L^p$ estimates contain terms with no fixed sign. They are controlled by parabolic estimates for the signal equation, superquadratic self-damping, and stochastic convolution estimates.

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arXiv2026

Asymptotic Flocking Behavior in Particle and Kinetic Systems with biological Memory Term

We investigate a Cucker-Smale type flocking model with a biological memory term and conclude that memory makes flocking harder. The model introduces $h(t)$ to describe memory, with dynamics $\frac{dh}{dt}=x(t)+v(t)-λh(t)$ reflecting path, velocity dependence and exponential decay. In microscopic level, the flocking phenomenon becomes conditional, requires communication kernel $ψ(t)$ has polynomial lower bound and memory deviation $H(t)$ decays exponentially. In mesoscopic level, flocking rates exhibit exponential or algebraic decay and the range of communication strength $β$ become smaller.

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arXiv2026

Mean field and N-agent games for optimal relative consumption-investment with jump risk and common noise

This paper studies an optimal consumption--investment problem for competitive agents in an $N$-player game and its associated mean field game. Each agent invests in an individual risky asset subject to idiosyncratic noise, common noise and downward jump risk, and the interaction among agents is induced by relative performance concerns in both consumption and terminal wealth. In the mean field limit, we characterize a deterministic mean field equilibrium in analytical form by using the stochastic maximum principle. Numerical experiments are presented to illustrate the resulting equilibrium and its financial implications. Finally, based on the obtained mean field equilibrium, we construct an approximate Nash equilibrium for the $N$-player game. This model is motivated by \cite{Merton1971} and \cite{Lacker2020}.

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arXiv2026

Well-posedness of the mean field game master equation on Carnot tori

We study the master equation for a second-order mean field game on Carnot tori, which means the generic player can move periodically only along admissible trajectories given by the family of vector fields generating the Carnot group. As examples of sub-Riemannian manifolds, Carnot groups represent a type of non-commutative groups characterized by stratified Lie algebra structures. In order to obtain the well-posedness of the master equation, we analyze the properties of its solution by investigating a degenerate mean field game system for which there exists an equivalent characterization with the master equation. The main part of this paper lies in leveraging the regularity properties of solutions to two classes of linear degenerate parabolic equations and a class of linear degenerate coupled systems to derive the existence of solutions to the master equation. The research in this paper is motivated by \cite{19CDLL,24MMM}.

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arXiv2026

Existence results for nonlinear cone degenerate Laplace equations

This paper concerns a class of non-divergence nonlinear elliptic equations driven by the cone degenerate Laplacian, which is motivated by cone calculus. We establish the existence of viscosity solutions by proving the Alexandrov-Bakelman-Pucci and Hölder estimates. Furthermore, we obtain the existence of weak solutions by proving the equivalence between weak solutions and viscosity solutions.

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arXiv2026

Regularity results for linear parabolic equations on Carnot tori via mollifier kernel construction

This paper first proves the existence, uniqueness and regularity of the solution to a class of linear backward parabolic equations on Carnot tori, namely the periodic linear parabolic equation on Carnot groups. Such groups are non-commutative and typical examples of sub-Riemannian manifolds. Moreover, we apply the results for this equation to its dual equation (i.e., the Fokker-Planck-Kolmogorov equation in the general form), and derive the existence, uniqueness and regularity of its weak solution. To obtain the regularity results for solutions to the linear parabolic equation and its dual equation, firstly, we construct several families of mollifiers adapted respectively to the Hörmander vector fields generating Carnot groups, Carnot tori and dual spaces of non-isotropic Hölder spaces; secondly, we use the theory of singular integral operators to establish stronger a priori regularity for the solutions.

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arXiv2026

On a class of Nonlinear Grushin equations

In this paper, we study two kinds of nonlinear degenerate elliptic equations containing the Grushin operator. First, we prove radial symmetry and a decay rate at infinity of solutions to such a Grushin equation by using the moving plane method in combination with suitable integral inequalities. Applying similar methods, we obtain nonexistence results for solutions to a second type of Grushin equation in Euclidean half space. Finally, we derive a priori estimates and the existence for positive solutions to more general types of Grushin equations by employing blow up analysis and topological degree methods, respectively.

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arXiv2025

Mean-Field Game for Gene Expression of Beetles

In this paper, we investigate the probability of the expression of genes that control the size of beetles under competitive relationships. We use the mean field game (MFG) theory in multiple populations to characterize the different competitive pressures of large and small beetles in the population, and simulate the probability of gene expression in finite time $[0, T]$. Therefore, we prove the existence and uniqueness of the solution of the equation under some assumptions.

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arXiv2025

Removable singularities and Harnack inequality for nonlinear Hörmander degenerate subelliptic equations

This paper concerns the quasilinear subelliptic function derived from Hörmander vector fields. Based on the significant work of J. Serrin in \cite{SER}, M. Meier in \cite{MM1}, and L. Capogna, D. Danielli and N. Garofalo in \cite{LC1,LDN}, we obtain the removable singularities and Harnack inequality by a sharp Sobolev inequalities under weaker integrability of coefficients in structure conditions. Furthermore, we get the Hölder continuity when domain $Ω$ is equiregular.

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arXiv2025

Pohozaev identities for weak solutions of Grushin type p-sub-Laplacian equation via domain variations

In this paper, we study Pohozaev identities for weak solutions of degenerate elliptic equations involving Grushin type p-sub-Laplacian under only $C^1$-regularity assumption. By using domain variations, we obtain the local Pohozaev identities of translating type and scaling type. As an application, a global Pohozaev identity of scaling type in $\mathbb{R}^{N+l}$ is also derived.

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arXiv2024

The kernel space of linear operator on a class of Grushin equation

In this paper, we concern the kernel of linear operator for a class of Grushin equation. First, we study the kernel space of linear operator for a general Grushin equation. Then, we provide an exact expression for the kernel space of linear operator for a special Grushin equation. Finally, we prove the linear operator related to the singularly perturbed Grushin equation is invertible when restricted to the complement of its approximate kernel space.

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arXiv2024

Existence and uniqueness for cone degenerate p-Laplace equation

In this paper, we study the cone degenerate p-Laplace equation. We provide the existence of the viscosity solutions by proving Alexandrov-Bakelman-Pucci and Hölder estimates. Further more, we give the comparison principle by an equivalent transformation. Finally, we obtain the existence of weak solutions by analyzing the relationship between weak solutions and viscosity solutions.

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arXiv2024

Stackelberg games with the third party

In this paper, we introduce the third party to achieve the Stackelberg equilibrium with the time inconsistency in three different Stackelberg games, which are the discrete-time games, the dynamic games, and the mean field games. Here all followers are experiencing learning-by-doing. The role of a third party is similar to industry associations, they supervise the leader's implementation and impose penalties for the defection with the discount factor. Then we obtain different forms of discount factors in different models and effective conditions to prevent defection.These results are consistent and the third party intervention is effective and maneuverable in practice.

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arXiv2024

Fujita phenomena in nonlinear fractional Rayleigh-Stokes equations

This paper concerns the Cauchy problems for the nonlinear Rayleigh-Stokes equation and the corresponding system with time-fractional derivative of order $α\in(0,1)$, which can be used to simulate the anomalous diffusion in viscoelastic fluids. It is shown that there exists the critical Fujita exponent which separates systematic blow-up of the solutions from possible global existence, and the critical exponent is independent of the parameter $α$. Different from the general scaling argument for parabolic problems, the main ingredients of our proof are suitable decay estimates of the solution operator and the construction of the test function.

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arXiv2024

Pohozaev identities and Kelvin transformation of semilinear Grushin equation

In this paper, we study Pohozaev identities, Kelvin transformation and their applications of semilinear Grushin equation. First, we establish two Pohozaev identities generated from translations and determine the location of the concentration point for solution of a kind of Grushin equation by such identities. Next, we establish Pohozaev identity generated from scaling and prove the nonexistence of nontrivial solutions of another kind of Grushin equation by such identity. Finally, we provide the change of Grushin operator by Kelvin transformation and obtain the decay rate of solution at infinity for a critical Grushin equation by Kelvin transformation.

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