Kernel estimates for nonautonomous Kolmogorov equations with potential term

Using time dependent Lyapunov functions, we prove pointwise upper bounds for the heat kernels of some nonautonomous Kolmogorov operators with possibly unbounded drift and diffusion coefficients and a possibly unbounded potential term

Cannon for Neutral Particles

Dynamics of spin-polarized neutral particles, such as neutrons or neutral atoms and molecules, in magnetic fields is studied. A new regime of motion is found where particles move mainly in one direction forming a well-collimated beam. This regime suggests a mechanism for creating devices emitting directed beams of neutral particles.Comment: 1 file, 7 pages, RevTex, no figures, final versio

Vibrations and fractional vibrations of rods, plates and Fresnel pseudo-processes

Different initial and boundary value problems for the equation of vibrations of rods (also called Fresnel equation) are solved by exploiting the connection with Brownian motion and the heat equation. The analysis of the fractional version (of order $\nu$) of the Fresnel equation is also performed and, in detail, some specific cases, like $\nu=1/2$, 1/3, 2/3, are analyzed. By means of the fundamental solution of the Fresnel equation, a pseudo-process $F(t)$, $t>0$ with real sign-varying density is constructed and some of its properties examined. The equation of vibrations of plates is considered and the case of circular vibrating disks $C_R$ is investigated by applying the methods of planar orthogonally reflecting Brownian motion within $C_R$. The composition of F with reflecting Brownian motion $B$ yields the law of biquadratic heat equation while the composition of $F$ with the first passage time $T_t$ of $B$ produces a genuine probability law strictly connected with the Cauchy process.Comment: 33 pages,8 figure

Conservative interacting particles system with anomalous rate of ergodicity

We analyze certain conservative interacting particle system and establish ergodicity of the system for a family of invariant measures. Furthermore, we show that convergence rate to equilibrium is exponential. This result is of interest because it presents counterexample to the standard assumption of physicists that conservative system implies polynomial rate of convergence.Comment: 16 pages; In the previous version there was a mistake in the proof of uniqueness of weak Leray solution. Uniqueness had been claimed in a space of solutions which was too large (see remark 2.6 for more details). Now the mistake is corrected by introducing a new class of moderate solutions (see definition 2.10) where we have both existence and uniquenes