Let iX, 03, m) be a finite measure space. We shall denote by L*im) (1 èp< °°) the Banach space of all real-valued (B-measurable functions/ defined on X such that |/[ p is m-integrable, and by L°°(w) the Banach space of all real-valued, (B-measurable, w-essentially bounded functions defined on X; as usual, the norm in Lpim) is given by 11/11,.= {fx\fix)\pdm}1'*, and the norm in Lxim) by \\g\\x =...

We study the homogenization of 2D linear transport equations, ut + ~a(~x/ε) · ∇~xu = 0, where ~a is a non-vanishing vector field with integral invariance on the torus T . When the underlying flow on T 2 is ergodic, we derive the efficient equation which is a linear transport equation with constant coefficients and quantify the pointwise convergence rate. This result unifies and illuminates the ...

Szemerédi’s Theorem asserts that any positive-density subset of the integers must contain arbitrarily long arithmetic progressions. It is one of the central results of additive combinatorics. After Szemeredi’s original combinatorial proof, Furstenberg noticed the equivalence of this result to a new phenomenon in ergodic theory that he called ‘multiple recurrence’. Furstenberg then developed som...

We study a stability property of probability laws with respect to small violations of algorithmic randomness. A sufficient condition of stability is presented in terms of Schnorr tests of algorithmic randomness. Most probability laws, like the strong law of large numbers, the law of iterated logarithm, and even Birkhoff’s pointwise ergodic theorem for ergodic transformations, are stable in this...