R W

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 Name R W Member for 2 years Seen 5 hours ago Website Location Age
 May6 answered Modern Mathematical Achievements Accessible to Undergraduates Apr30 accepted Fundamental inequality of entropy in random walks Apr29 answered Fundamental inequality of entropy in random walks Apr25 comment An action or two of $SL_2(\Bbb Z)$?Misha, you are not quite right. The action of $SL(2,Z)$ on $R^2∖ 0$ (rather, on the quotient of this space by the central symmetry) has a clear geometrical interpretation. It is isomorphic to the action of $SL(2,Z)$ on the space of horocycles in the hyperbolic plane (i.e., to the extension of the boundary action by the Busemann cocycle). Apr18 accepted Obtaining conditional probabilities as pushforwards of [0,1] Apr18 answered Obtaining conditional probabilities as pushforwards of [0,1] Apr14 awarded ● Fanatic Apr12 accepted Absolute continuity of probabilities on Polish spaces and open sets. Apr12 comment Absolute continuity of probabilities on Polish spaces and open sets.There is a lot of pairwise singular measures with the same support. For instance, look at the distribution of the sum $$\sum_{k=1}^\infty x_k/2^k \;,$$ where $x_k$ are independent and take values 1 with probability $p$ and 0 with probability $1−p$. The resulting measures $m_p$ are pairwise singular (as it follows, for instance, from the law of large numbers) and have the same support $[0,1]$ (for $p=1/2$ this is the Lebesgue measure). Apr12 comment Mean value theorem for harmonic functions on ellipsoidOK - but if you formulate the claim in terms of harmonic measures, then there is no need for any assumptions like that - right? Apr12 answered Absolute continuity of probabilities on Polish spaces and open sets. Apr11 comment Mean value theorem for harmonic functions on ellipsoidAlexandre, why do you impose the centrality condition? Apr9 accepted The relations between conservative part and conservativity Apr6 answered $\Delta f \le - \lambda f$ then ${\lambda _1}\left( M \right) \ge \lambda$? Apr4 comment Methods for solving two variable recurrenceConcerning the variance it's just a simple calculation (since the variance of exponential distributions is explicitly known) - then one has to apply the standard Chebyshev inequality (in the same way as in the proof of the weak law of large numbers). Apr4 revised Methods for solving two variable recurrenceadded 305 characters in body Apr4 revised Methods for solving two variable recurrencedeleted 287 characters in body Apr4 answered Methods for solving two variable recurrence Apr3 comment The relations between conservative part and conservativityI see. I wouldn't immediately know any type III examples for smooth $\mathbb Z$ actions. The ones I know are either $\mathbb Z$ actions on pretty abstract spaces (e.g., shift on product spaces) or smooth actions of much bigger groups (e.g., boundary actions of Fuchsian or Kleinian groups). Apr3 answered The relations between conservative part and conservativity Apr2 comment Local concentration of measure on Erdos-Rényi graphAlmost sure convergence a priori does not make any sense when talking about weak convergence of a sequence of measures. For a.s. convergence one needs a measure in the space of sequences (in your case in the space of sequences of graphs whose size n goes to infinity) - what is the measure with respect to which you want to consider a.s. convergence? Apr2 comment Local concentration of measure on Erdos-Rényi graphDon't understand why you would need any concentration here. If you reverse your condition and talk about the probability of "not being a tree", then you have to deal with convergence of non-negative functions to 0. Apr2 answered Local concentration of measure on Erdos-Rényi graph Mar30 comment Finite vertex-transitive graphs that look like infinite vertex-transitive graphsOh - are you claiming that Schreier graphs are transitive? They are not! Mar30 comment Finite vertex-transitive graphs that look like infinite vertex-transitive graphsThere is something I don't understand in this argument. Any, say 4-regular graph, can be edge-labelled to become a Schreier graph of the free group $F_2$. Do you claim it's embeddable into a homogeneous graph? Mar30 answered Finite vertex-transitive graphs that look like infinite vertex-transitive graphs Mar27 answered Ergodicity for a Probabilistic Cellular Automaton on a finite space Mar19 accepted Weak convergence in measure for negligible sets. Mar19 answered Weak convergence in measure for negligible sets. Mar17 accepted Compactness of sigma-algebra for the $L^1$ metrics Mar17 comment Convergence of probability measure and the *-weak convergence ? Sigma-additive probability measures are dense in finitely additive probability measures with respect to the weak topology, whereas tightness is precisely the condition which guarantees that the limit of sigma-additive measures is also a sigma-additive measure. Mar17 revised Compactness of sigma-algebra for the $L^1$ metricsdeleted 1 characters in body Mar17 answered Compactness of sigma-algebra for the $L^1$ metrics Mar17 answered Convergence of probability measure and the *-weak convergence ? Mar16 answered Equivalence of two definitions of Lyapunov exponents Mar15 accepted The property of a Markov measure Mar15 answered The property of a Markov measure Mar15 comment The property of a Markov measureAnother question: are you assuming that $m$ is shift invariant? Mar15 comment The property of a Markov measureWhat are $a$ and $b$ in your cylinder sets? Are you talking about one-dimensional cylinders determined by first letters? Mar14 answered Is the first eigenvalue of a parabolic ends of a Riemanian manifold 0? Mar14 comment Integral of a harmonic function on a manifold with two non-parabolic endsIf only one end is non-parabolic, then you may have no harmonic functions to talk about (I presume you are interested in bounded or at least positive ones). Mar14 accepted Integral of a harmonic function on a manifold with two non-parabolic ends Mar14 answered Integral of a harmonic function on a manifold with two non-parabolic ends Mar1 comment How does a quasi-isometry affect Poisson or Martin boundaries?@Igor Khavkine - Your comment about comparison of Martin and Busemann compactifications is only valid for symmetric spaces. Mar1 comment Spanning subgaph with trivial Poisson boundariesThis is a theorem of Kaimanovich-Vershik-Rosenblatt: a group is amenable iff it carries a Liouville random walk. However, the step distribution of this random walk need not be finitely supported (for instance, for higher dimensional lamplighter groups). She proves in that paper that actually for these groups all measures with finite entropy have non-trivial Poisson boundary. Mar1 answered Spanning subgaph with trivial Poisson boundaries Feb10 comment Invariant measures for Cellular automataThank you, I see now - looks like an interesting question. Is there any known a priory reason why all known $F$-invariant measures happen to be shift invariant? Feb8 comment Invariant measures for Cellular automataCould you explain in more detail how the local rules are defined in terms of the matrix $A$? Feb4 accepted Metrization of weak convergence of signed measures Jan31 comment Is a typical path of a Brownian motion on a torus equidistributed?The Riemannian volume is stationary for the Brownian motion on any Riemannian manifold. For a compact manifold (more generally, for a finite volume one) this is the only stationary measure (which follows, for instance, from the fact that the transition probabilities are absolutely continuous with respect to the Riemannian volume), in particular, it is ergodic. Equidistribution of a.e. sample path is then essentially just the usual ergodic theorem for the Brownian motion.