Related papers: Harnack Inequalities and Applications for Multival…
We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold $M$, with the sectional curvature bounded from below by $-\kappa$ for $\kappa\geq 0$. In the elliptic case, Wang and…
Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence…
A discrete analysis of the phase and dissipation errors of an explicit, semi-Lagrangian spectral element method is performed. The semi-Lagrangian method advects the Lagrange interpolant according the Lagrangian form of the transport…
In this paper we consider a new kind of inequality related to fractional integration, motivated by Gressman's paper. Based on it we investigate its multilinear analogue inequalities. Combining with the Gressman's work on multilinear…
We study universal spatial features of certain non-equilibrium steady states corresponding to flows of strongly correlated fluids over obstacles. This allows us to predict universal spatial features of far-from-equilibrium systems, which in…
We present an application of the theory of stochastic processes to model and categorize non-equilibrium physical phenomena. The concepts of uniformly continuous probability measures and modular evolution lead to a systematic hierarchical…
A system of a first order history-dependent evolutionary variational-hemivariational inequality with unilateral constraints coupled with a nonlinear ordinary differential equation in a Banach space is studied. Based on a fixed point theorem…
We study the convergence analysis for general degenerate and non-reversible stochastic differential equations (SDEs). We apply the Lyapunov method to analyze the Fokker-Planck equation, in which the Lyapunov functional is chosen as a…
This paper presents a novel method to synthesize stochastic control Lyapunov functions for a class of nonlinear, stochastic control systems. In this work, the classical nonlinear Hamilton-Jacobi-Bellman partial differential equation is…
In this thesis, we explore several related topics broadly regarding the symmetry and geometric properties of nonlocal partial differential equations (PDE). This thesis is split into three parts. In the first part, we study two…
We strengthen the Carleson-Hunt theorem by proving $L^p$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $p>\max\{r',2\}$. Four appendices are concerned with transference, a variation norm…
In this paper, by utilizing Wang's Harnack inequality with power and the Banach fixed point theorem, the weak well-posedness for McKean-Vlasov SDEs with integrable drift is investigated. In addition, using the decoupled method, some…
This introduction surveys a renormalisation group perspective on log-Sobolev inequalities and related properties of stochastic dynamics. We also explain the relationship of this approach to related recent and less recent developments such…
We consider non-conservative positive semigroups and obtain necessary and sufficient conditions for uniform exponential contraction in weighted total variation norm. This ensures the existence of Perron eigenelements and provides…
We establish the Krylov--Safonov theory for a large class of nonlocal operators of order $2s \in (0,2)$ on hyperbolic spaces $\mathbb{H}^{n}_{\kappa}$ with curvature $-\kappa<0$. We prove the Alexandrov--Bakelman--Pucci (ABP) estimates,…
We study the many body Schr\"odinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles.…
In this paper we study a family of nonlinear (conditional) expectations that can be understood as a diffusion with uncertain local characteristics. Here, the differential characteristics are prescribed by a set-valued function. We establish…
In this paper, we study the elliptic Harnack inequality and its applications on forward complete Finsler metric measure spaces under the conditions that the weighted Ricci curvature ${\rm Ric}_{\infty}$ has non-positive lower bound and the…
This article studies a Fokker-Planck type equation of fractional diffusion with conservative drift $\partial$f/$\partial$t = $\Delta$^($\alpha$/2) f + div(Ef), where $\Delta$^($\alpha$/2) denotes the fractional Laplacian and E is a…
We investigate the well-posedness and long-time behavior of a general continuum neural field model with Gaussian noise on possibly unbounded domains. In particular, we give conditions for the existence of invariant probability measures by…