Symmetric rearrangements around infinity with applications to Lévy processes

Abstract

We prove a new rearrangement inequality for multiple integrals, which partly generalizes a result of Friedberg and Luttinger [FL76] and can be interpreted as involving symmetric rearrangements of domains around ∞. As applications, we prove two comparison results for general Lévy processes and their symmetric rearrangements. The first application concerns the survival probability of a point particle in a Poisson field of moving traps following independent Lévy motions. We show that the survival probability can only increase if the point particle does not move, and the traps and the Lévy motions are symmetrically rearranged. This essentially generalizes an isoperimetric inequality of Peres and Sousi [PS11] for the Wiener sausage. In the second application, we show that the q-capacity of a Borel measurable set for a Lévy process can only decrease if the set and the Lévy process are symmetrically rearranged. This result generalizes an inequality obtained by Watanabe [W83] for symmetric Lévy processes.AMS 2010 subject classification: Primary 26D15, 60J65. Secondary 60D05, 60G55, 60G50.

Beyond the Entropy Power Inequality, via Rearrangements

“…where we have used Fubini's Theorem implicitly. The first term on the right side of the above display is precisely the right side of the inequality (26), while the fact that the second term is zero follows from (30).…”

“…One application that rearrangement inequalities have found in probability is in the area of isoperimetric inequalities for stochastic processes. Representative works in this area include Watanabe [51] on capacities associated to Lévy processes, Burchard and Schmuckenschläger [18] on exit times of Brownian motions on the sphere or hyperbolic space, Bañuelos and Méndez-Hernández on exit times and more for general Lévy processes [2], and Drewitz, Sousi and Sun [26] on survival probabilities in a field of Lévy-moving traps. Our results also have implications for stochastic processes and these are developed in Section X.…”

Annealed Asymptotics for Brownian Motion of Renormalized Potential in Mobile Random Medium

“…Proof: Let K(x) ≥ 0 be bounded, locally supported and radially symmetric. As a corollary of Theorem 1.5, [17],…”

A Monotonicity Result for the Range of a Perturbed Random Walk

“…2.1 for p(·), which satisfies condition Theorem 1.1 (i), is based on induction and a suitable notion of symmetric domination of measures on Z, which a priori may seem mysterious. A variant of this argument was recently applied by the second author with coauthors in [DSS11] to prove a rearrangement inequality for Lévy processes in R d , which can be regarded as a generalized version of the Pascal principle for a trapping problem, where traps follow independent Lévy motions. The argument in [DSS11] involves symmetric decreasing rearrangement of increment distributions in R d , which in the discrete setting we consider here would require p(·) to be symmetric and satisfy p(0) ≥ p(1) ≥ p(2) ≥ · · · .…”

“…This however does not cover the simple symmetric random walk. Our key observation here is that by using a weaker notion of symmetric domination than in [DSS11] and careful manipulations, we can deal with p(·) that satisfies condition Theorem 1.1 (i), which in particular includes the simple symmetric random walk.…”