Carne--Varopoulos bounds for centered random walks

Discrete Hilbert Transform a la Gundy-Varopoulos

We show that the centered discrete Hilbert transform on integers applied to a function can be written as the conditional expectation of a transform of stochastic integrals, where the stochastic processes considered have jump components. The stochastic representation of the function and that of its Hilbert transform are under differential subordination and orthogonality relation with respect to the sharp bracket of quadratic covariation. This illustrates the Cauchy Riemann relations of analytic functions in this setting. This result is inspired by the seminal work of Gundy and Varopoulos on sto…

Sharp lower bounds for the asymptotic entropy of symmetric random walks

The entropy, the spectral radius and the drift are important numerical quantities associated to random walks on countable groups. We prove sharp inequalities relating those quantities for walks with a finite second moment, improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also deduce inequalities between these quantities and the volume growth of the group. Finally, we show that the equality case in our inequality is rather rigid.

Heat kernel upper bounds for interacting particle systems

We show a diffusive upper bound on the transition probability of a tagged particle in the symmetric simple exclusion process. The proof relies on optimal spectral gap estimates for the dynamics in finite volume, which are of independent interest. We also show off-diagonal estimates of Carne-Varopoulos type.

Properties of centered random walks on locally compact groups and Lie groups

The basic aim of this paper is to study asymptotic properties of the convolution powers K^(n) = K * K * ... * K of a possibly non-symmetric probability density K on a locally compact, compactly generated group G. If K is centered, we show that the Markov operator T associated with K is analytic in L^p(G) for 1<p<\infty, and establish Davies-Gaffney estimates in L^2 for the iterated operators T^n. These results enable us to obtain various Gaussian bounds on K^(n). In particular, when G is a Lie group we recover and extend some estimates of Alexopoulos and of Varopoulos for convolution powers of…

量子游走、离散波动方程与Chebyshev多项式

Quantum walks, the discrete wave equation and Chebyshev polynomials

本文受Terence Tao启发，提出从图上离散波动方程推导量子游走，将其描述为对随机游走转移矩阵应用Chebyshev多项式，从而解耦量子起源并证明格上扩散行为的二次改进。