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We study large deviations asymptotics for a class of unbounded additive functionals, interpreted as normalized accumulated areas, of one-dimensional Langevin diffusions with sub-linear gradient drifts. Our results provide parametric…
In the course of Darwinian evolution of a population, punctualism is an important phenomenon whereby long periods of genetic stasis alternate with short periods of rapid evolutionary change. This paper provides a mathematical interpretation…
We propose a finite difference scheme for the numerical solution of a two-dimensional singularly perturbed convection-diffusion partial differential equation whose solution features interacting boundary and interior layers, the latter due…
We study approximation of non-autonomous linear differential equations with variable delay over infinite intervals. We use piecewise constant argument to obtain a corresponding discrete difference equation. The study of numerical…
Reversing a diffusion process by learning its score forms the heart of diffusion-based generative modeling and for estimating properties of scientific systems. The diffusion processes that are tractable center on linear processes with a…
We investigate a one-dimensional nonlinear wave system which arises from a variational principle modeling a type of cholesteric liquid crystals. The problem treated here is the Cauchy problem for the same wave speed case with initial data…
Two-scale homogenization limits of parabolic cross-diffusion systems in a heterogeneous medium with no-flux boundary conditions are proved. The heterogeneity of the medium is reflected in the diffusion coefficients or by the perforated…
We investigate the fractional diffusion approximation of a kinetic equation in the upper-half plane with diffusive reflection conditions at the boundary. In an appropriate singular limit corresponding to small Knudsen number and long time…
This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which…
This work is devoted to the analysis of the backward problem for a viscous Hamilton-Jacobi equation with degenerate diffusion and a general Hamiltonian that is not necessarily quadratic. First, we focus on linear degenerate parabolic…
In this paper, we are concerned with the dynamical behavior of the stochastic nonclassical parabolic equation, more precisely, it is shown that the inviscid limits of the stochastic nonclassical diffusion equations reduces to the stochastic…
The problem of imaging of a moving target is formulated as a Coefficient Inverse Problem for a hyperbolic equation with its coefficient depending on all three spatial variables and time. As the initial condition, the point source running…
We establish symmetrization results for the solutions of the linear fractional diffusion equation $\partial_t u +(-\Delta)^{\sigma/2}u=f$ and itselliptic counterpart $h v +(-\Delta)^{\sigma/2}v=f$, $h>0$, using the concept of comparison of…
We study a nonlinear evolutionary partial differential equation that can be viewed as a generalization of the heat equation where the temperature gradient is a~priori bounded but the heat flux provides merely \mbox{$L^1$-coercivity}.…
We provide a stochastic representation for a general class of viscous Hamilton-Jacobi (HJ) equations, which has convexity and superlinear nonlinearity in its gradient term, via a type of backward stochastic differential equation (BSDE) with…
In this paper, we study both the direct and inverse random source problems associated with the multi-term time-fractional diffusion-wave equation driven by a fractional Brownian motion. Regarding the direct problem, the well-posedness is…
In this paper we are interested in a degenerate parabolic system of reaction-diffusion equations arising in biology when studying cell adhesion at the protein level. In this modeling the unknown is the couple of the distribution laws of the…
In this paper we prove that solutions to a transmission problem degenerating on the interface are H\"older differentiable up to the interface with universal estimates. Furthermore, we obtain a sharper pointwise $C^{1,\alpha(\cdot)}$ with…
Analytic solutions to the nonlinear radiation diffusion equation with an instantaneous point source for a non-homogeneous medium with a power law spatial density profile, are presented. The solutions are a generalization of the well known…
Consider a set of discounted optimal stopping problems for a one-parameter family of objective functions and a fixed diffusion process, started at a fixed point. A standard problem in stochastic control/optimal stopping is to solve for the…