arXiv subjects

Tymoteusz Chojecki

Publications and source records attributed to Tymoteusz Chojecki.

4 recordsLinked to original sources

arXiv2020

A probabilistic proof of apriori $l^p$ estimates for a class of divergence form elliptic operators

Suppose that ${\cal L}$ is a divergence form differential operator of the form ${\cal L}f:=(1/2) e^{U}\nabla_x\cdot\big[e^{-U}(I+H)\nabla_x f\big]$, where $U$ is scalar valued, $I$ identity matrix and $H$ an anti-symmetric matrix valued function. The coefficients are not assumed to be bounded, but are $C^2$ regular. We show that if $Z=\int_{\mathbb{R}^d}e^{-U(x) }dx<+\infty$ and the supremum of the numerical range of matrix $-\frac12\nabla^2_x U+\frac12\nabla_x\left\{\nabla_x\cdot H-[\nabla_x U]^TH\right\}$ satisfies some exponential integrability condition with respect to measure $dμ=Z^{-1}e^{-U}dx$, then for any $1 \le p 0$ such that $\left\| f\right\|_{W^{2,p}(μ)}\le C\Big(\left\|{\cal L}f\right\|_{L^q(μ)}+\left\|f\right\|_{L^q(μ)}\Big)$ for $f\in C_0^\infty(\mathbb{R}^d)$. Here $W^{2,p}(μ)$ is the Sobolev space of functions that are $L^p(μ)$ integrable with two derivatives. Our proof is probabilistic and relies on an application of the Malliavin calculus.

math.PR↗

arXiv2019

Homogenization of an advection equation with locally stationary random coefficients

In the paper we consider the solution of an advection equation with rapidly changing coefficients $\partial_t u_\eps+(1/\eps)V(t\eps^{-2},x/{\eps})\cdot\nabla_x u_\eps=0$ for $t 0$ is some small parameter and the drift term $\left(V(t,x)\right)_{(t,x)\in \bbR^{1+d}}$ is assumed to be a $d$-dimensional, vector valued random field with incompressible spatial realizations. We prove that when the field is Gaussian, locally stationary, quasi-periodic in the $x$ variable and strongly mixing in time the solutions $u_\eps(t,x)$ converge in law, as $\eps\to0$, to $ u_0(x(T;t,x))$, where $\left(x(s;t,x)\right)_{s\ge t}$ is a diffusion satisfying $x(t;t,x)=x$. The averages of $u_\eps(T,x)$ converge then to the solution of the corresponding Kolmogorov backward equation.

math.PR↗

arXiv2018

Passive tracer in non-Markovian, Gaussian velocity field

We consider the trajectory of a tracer that is the solution of an ordinary differential equation $\dot\bbX(t)=\bbV(t, \bbX(t)),\ X(0)=0$, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known, see [K-F;C-X;K-L-O], that $\bbX(t)/\sqrt{t}$ converges in law, as $t\to+\infty$, to a normal, zero mean vector, provided that the field $V(t,x)$ is Markovian and has the spectral gap property. We wish to extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernstein function.

math.PR↗

arXiv2011

On the central limit theorem for some birth and death process

Suppose that X_n, n>=0 is a stationary Markov chain and V is a certain function on a phase space of the chain, called an observanle. We say that the observable satisfies the central limit theorem (C.L.T.) if Y_n:=N^{-1/2}\sum_{n=0}^NV(X_n) converge in law to a normal random variable, as N goes to infinity. For a stationary Markov chain with the L^2 spectral gap the theorem holds for all V such that V(X_0) is centered and square integrable, see Gordin. The purpose of this article is to characterize a family of observables V for which the C.L.T. holds for a class of birth and death chains whose dynamics has no spectral gap, so that Gordin's result cannot be used and the result follows from an application of Kipnis-Varadhan theory.

math.PR↗