Regular Flows for Diffusions with Rough Drifts

According to DiPerna-Lions theory, velocity fields with weak derivatives in $L^p$ spaces possess weakly regular flows. When a velocity field is perturbed by a white noise, the corresponding (stochastic) flow is far more regular in spatial variables; a $d$-dimensional diffusion with a drift in $L^{r,q}$ space ($r$ for the spatial variable and $q$ for the temporal variable) possesses weak derivatives with stretched exponential bounds, provided that $r/d+2/q<1$. As an application we show that a Hamiltonian system that is perturbed by a white noise produces a symplectic flow provided that the corresponding Hamiltonian function $H$ satisfies $\nabla H\in L^{r,q}$ with $r/d+2/q<1$. As our second application we derive a Constantin-Iyer type circulation formula for certain weak solutions of Navier-Stokes equation

Stochastically Symplectic Maps and Their Applications to Navier-Stokes Equation

Poincare's invariance principle for Hamiltonian flows implies Kelvin's principle for solution to Incompressible Euler Equation. Iyer-Constantin Circulation Theorem offers a stochastic analog of Kelvin's principle for Navier-Stokes Equation. Weakly symplectic diffusions are defined to produce stochastically symplectic flows in a systematic way. With the aid of symplectic diffusions, we produce a family of martigales associated with solutions to Navier-Stokes Equation that in turn can be used to prove Iyer-Constantin Circulation Theorem. We also review some basic facts in symplectic and contact geometry and their applications to Euler Equation

Kinetic description of scalar conservation laws with Markovian data

We derive a kinetic equation to describe the statistical structure of solutions $\rho$ to scalar conservation laws $\rho_t=H(x,t,\rho )_x$, with certain Markov initial conditions. When the Hamiltonian function is convex and increasing in $\rho$, we show that the solution $\rho(x,t)$ is a Markov process in $x$ (respectively $t$) with $t$ (respectively $x$) fixed. Two classes of Markov conditions are considered in this article. In the first class, the initial data is characterize by a drift $b$ which satisfies a linear PDE, and a jump density $f$ which satisfies a kinetic equation as time varies. In the second class, the initial data is a concatenation of fundamental solutions that are characterized by a parameter $y$, which is a Markov jump process with a jump density $g$ satisfying a kinetic equation. When $H$ is not increasing in $\rho$, the restriction of $\rho$ to a line in $(x,t)$ plane is a Markov process of the same type, provided that the slope of the line satisfies an inequality

Coagulation, diffusion and the continuous Smoluchowski equation

The Smoluchowski equation is a system of partial differential equations modelling the diffusion and binary coagulation of a large collection of tiny particles. The mass parameter may be indexed either by positive integers, or by positive reals, these corresponding to the discrete or the continuous form of the equations. In dimension at least 3, we derive the continuous Smoluchowski PDE as a kinetic limit of a microscopic model of Brownian particles liable to coalesce, using a similar method to that used to derive the discrete form of the equations in Hammond and Rezakhanlou [4]. The principal innovation is a correlation-type bound on particle locations that permits the derivation in the continuous context while simplifying the arguments of [4]. We also comment on the scaling satisfied by the continuous Smoluchowski PDE, and its potential implications for blow-up of solutions of the equations.Comment: 42 page

Central Limit Theorem for a Tagged Particle in Asymmetric Simple Exclusion

We prove a Functional Central Limit Theorem for the position of a Tagged Particle in the one-dimensional Asymmetric Simple Exclusion Process in the hyperbolic scaling, starting from a Bernoulli product measure conditioned to have a particle at the origin. We also prove that the position of the Tagged Particle at time $t$ depends on the initial configuration, by the number of empty sites in the interval $[0,(p-q)\alpha t]$ divided by $\alpha$ in the hyperbolic and in a longer time scale, namely $N^{4/3}$.Comment: 28 pages, no figure

The random Arnold Conjecture: a new probabilistic Conley-Zehnder Theory for symplectic maps

We take the first steps to develop Conley-Zehnder Theory, as conjectured by Arnold, in the world of probability. As far as we know, this paper provides the first probabilistic theorems about density of fixed points of symplectic twist maps in dimensions greater than $2$. In particular we will show that, when the analogue conditions to classical Conley-Zehnder theory hold, quasiperiodic symplectic twist maps have infinitely many fixed points almost surely. The paper contains also a number of theorems which go well beyond the quasiperiodic case.Comment: 42 page