Related papers: Diffusion transformations, Black-Scholes equation …
This article studies a dirichlet boundary value problem for singularly perturbed time delay convection diffusion equation with degenerate coefficient. A priori explicit bounds are established on the solution and its derivatives. For…
We consider the Cauchy problem for one-dimensional p-system with damping of space-dependent coefficient. This system models the compressible flow through porous media in the Lagrangean coordinate. Our concern is an asymptotic behavior of…
We provide an explicit rigorous derivation of a diffusion limit - a stochastic differential equation with additive noise - from a deterministic skew-product flow. This flow is assumed to exhibit time-scale separation and has the form of a…
We consider the problem of optimal stopping for a one-dimensional diffusion process. Two classes of admissible stopping times are considered. The first class consists of all nonanticipating stopping times that take values in [0,\infty],…
We prove the uniqueness in determining a spatially varying zeroth-order coefficient of a one-dimensional time-fractional diffusion equation by initial value and Cauchy data at one end point of the spatial interval.
Well-posedness and a number of qualitative properties for solutions to the Cauchy problem for the following nonlinear diffusion equation with a spatially inhomogeneous source $$ \partial_tu=\Delta u^m+|x|^{\sigma}u^p, $$ posed for…
The theory of Schroedinger bridges for diffusion processes is extended to classical and quantum discrete-time Markovian evolutions. The solution of the path space maximum entropy problems is obtained from the a priori model in both cases…
We study the Cauchy problem for a nonlocal heat equation, which is of fractional order both in space and time. We prove four main theorems: (i) a representation formula for classical solutions, (ii) a quantitative decay rate at which the…
It is shown that if the initial condition of the Cauchy problem for the diffusion equation on a general infinite countable ultrametric space is spherically symmetric with respect to some point, then this problem has an exact analytical…
We study the persistence and propagation (or blocking) phenomena for a species in periodically hostile environments. The problem is described by a reaction-diffusion equation with zero Dirichlet boundary condition. We first derive the…
Mathematically modelling diffusive and advective transport of particles in heterogeneous layered media is important to many applications in computational, biological and medical physics. While deterministic continuum models of such…
It is studied the Cauchy problem for the equations of Burgers' type but with bounded dissipation flux. Such equations degenerate to hyperbolic ones as the velocity gradient tends to infinity. Thus the discontinuous solutions are permitted.…
We study the Cauchy problem in the hyperbolic space for the heat equation with a Fisher-KPP type forcing term. Depending on the relative strength of diffusion, measured by the infimum of the spectrum of the Laplace-Beltrami operator, as…
For $n\ge 3$, $0<m<\frac{n-2}{n}$, $\beta<0$ and $\alpha=\frac{2\beta}{1-m}$, we prove the existence, uniqueness and asymptotics near the origin of the singular eternal self-similar solutions of the fast diffusion equation in…
We study the distribution of the time to explosion for one-dimensional diffusions. We relate this question to computing the expectations of suitable nonnegative local martingales, and to the distributions of related diffusions with unit…
We study solution techniques for an evolution equation involving second order derivative in time and the spectral fractional powers, of order $s \in (0,1)$, of symmetric, coercive, linear, elliptic, second-order operators in bounded domains…
Strang splitting is a widely used second-order method for solving diffusion-reaction problems. However, its convergence order is often reduced to order $1$ for Dirichlet boundary conditions and to order $1.5$ for Neumann and Robin boundary…
The connection between forward backward doubly stochastic differential equations and the optimal filtering problem is established without using the Zakai's equation. The solutions of forward backward doubly stochastic differential equations…
The fast diffusion equation is analyzed on a bounded domain with Dirichlet boundary conditions, for which solutions are known to extinct in finite time. We construct invariant manifolds that provide a finite-dimensional approximation near…
We consider the problem of "wall-to-wall optimal transport" in which we attempt to maximize the transport of a passive temperature field between hot and cold plates. Specifically, we optimize the choice of the divergence-free velocity field…