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In this work we consider the asymptotic behavior of the nonlinear semigroup defined by a semilinear parabolic problem with homogeneous Neumann boundary conditions posed in a bounded region of the plane that degenerates into a line segment…
We show that the dimension of the exit distribution of planar partially reflected Brownian motion can be arbitrarily close to 2.
For every bounded planar domain $D$ with a smooth boundary, we define a `Lyapunov exponent' $\Lambda(D)$ using a fairly explicit formula. We consider two reflected Brownian motions in $D$, driven by the same Brownian motion (i.e., a…
An inhomogeneous Kaluza-Klein compactification of a higher dimensional spacetime may give rise to an effective 4d spacetime with distinct domains having different sizes of the extra dimensions. The domains are separated by domain walls…
We study the long time behavior of a Brownian particle moving in an anomalously diffusing field, the evolution of which depends on the particle position. We prove that the process describing the asymptotic behaviour of the Brownian particle…
We study a class of reflected McKean-Vlasov diffusions over a convex domain with self-stabilizing coefficients. This includes coefficients that do not satisfy the classical Wasserstein Lipschitz condition. Further, the process is…
We give a comprehensive study of the analytic properties and long-time behavior of solutions of a reaction-diffusion system in a bounded domain in the case where the nonlinearity satisfies the standard monotonicity assumption. We pay the…
We formulate the theory for a self-propelled domain in an excitable reaction-diffusion system in two dimensions where the domain deforms from a circular shape when the propagation velocity is increased. In the singular limit where the width…
Second initial boundary problem in narrow domains of width $\epsilon\ll 1$ for linear second order differential equations with nonlinear boundary conditions is considered in this paper. Using probabilistic methods we show that the solution…
We discuss the diffusion phenomenon in the parabolic and hyperbolic regimes. New effects related to the finite velocity of the diffusion process are predicted, that can partially explain the strange behavior associated to adsorption…
We study the asymptotic behavior of a diffusion process with small diffusion in a domain $D$. This process is reflected at $\partial D$ with respect to a co-normal direction pointing inside $D$. Our asymptotic result is used to study the…
We study trajectories of d-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyper-plane. Our Poissonian potential V can be associated to a field of traps whose centers location is given by a Poisson…
We predict a novel nonlinear electromagnetic phenomenon in layered superconducting slabs irradiated from one side by an electromagnetic plane wave. We show that the reflectance and transmittance of the slab can vary over a wide range, from…
A solution to the ultra-relativistic strong explosion problem with a non-power law density gradient is delineated. We consider a blast wave expanding into a density profile falling off as a steep radial power-law with small, spherically…
We develop an effective theory of pulse propagation in a nonlinear {\it and} disordered medium. The theory is formulated in terms of a nonlinear diffusion equation. Despite its apparent simplicity this equation describes novel phenomena…
In this paper we study a family of semilinear reaction-diffusion equations on thin spatial domains, lying close to a lower dimensional submanifold $M$. As the thickness tends to zero, the domains collapse onto (a subset of) $M$. As it was…
We consider a semi-infinite spatially dispersive dielectric with unequal transverse and longitudinal susceptibilities. The effect of the boundary is characterized by arbitrary reflection coefficients for polarization waves in the material…
This paper provides information about the asymptotic behavior of a one-dimensional Brownian polymer in random medium represented by a Gaussian field $W$ on ${\mathbb{R}}_+\times{\mathbb{R}}$ which is white noise in time and function-valued…
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…
The justification of hydrodynamic limits in non-convex domains has long been an open problem due to the singularity at the grazing set. In this paper, we investigate the unsteady neutron transport equation in a general bounded domain with…