34 research outputs found
Einstein's model of "the movement of small particles in a stationary liquid" revisited: finite propagation speed
The aforementioned celebrated model, though a breakthrough in Stochastic processes and a great step toward the construction of the Brownian motion, leads to a paradox: infinite propagation speed and violation of the 2nd law of thermodynamics. We adapt the model by assuming the diffusion matrix is dependent on the concentration of particles, rather than constant it was up to Einstein, and prove a finite propagation speed under the assumption of a qualified decrease of the diffusion for small concentrations. The method involves a nonlinear degenerated parabolic PDE in divergent form, a parabolic Sobolev-type inequality, and the Ladyzhenskaya-Ural’tseva iteration lemma
Kolmogorov equations in infinite dimensions: Well-posedness and regularity of solutions, with applications to stochastic generalized Burgers equations
We develop a new method to uniquely solve a large class of heat equations,
so-called Kolmogorov equations in infinitely many variables. The equations are
analyzed in spaces of sequentially weakly continuous functions weighted by
proper (Lyapunov type) functions. This way for the first time the solutions are
constructed everywhere without exceptional sets for equations with possibly
nonlocally Lipschitz drifts. Apart from general analytic interest, the main
motivation is to apply this to uniquely solve martingale problems in the sense
of Stroock--Varadhan given by stochastic partial differential equations from
hydrodynamics, such as the stochastic Navier--Stokes equations. In this paper
this is done in the case of the stochastic generalized Burgers equation.
Uniqueness is shown in the sense of Markov flows.Comment: Published at http://dx.doi.org/10.1214/009117905000000666 in the
Annals of Probability (http://www.imstat.org/aop/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Gradient estimates for degenerate quasi-linear parabolic equations
For a general class of divergence type quasi-linear degenerate parabolic
equations with differentiable structure and lower order coefficients form
bounded with respect to the Laplacian we obtain -estimates for the
gradients of solutions, and for the lower order coefficients from a Kato-type
class we show that the solutions are Lipschitz continuous with respect to the
space variable
A critical phenomenon for sublinear elliptic equations in cone-like domains
We study positive supersolutions to an elliptic equation : , in cone-like domains in (). We
prove that in the sublinear case there exists a critical exponent
such that equation has a positive supersolution if and only if
. The value of is determined explicitly by and the
geometry of the cone.Comment: 6 pages, 2 figure
On the Lp-theory of C0-semigroups associated with second-order elliptic operators with complex singular coefficients
A work in Perturbation Theory, with a purpose to consider well-posedness of elliptic and parabolic PDE with singular complex coefficient