Arnab Sen

We study the notion of reverse hypercontractivity. We show that reverse hypercontractive inequalities are implied by standard hypercontractive inequalities as well as by the modified log-Sobolev inequality. Our proof is based on a new comparison lemma for Dirichlet forms and an extension of the Strook-Varapolos inequality. A consequence of our analysis is that {\em all} simple operators $L=Id-\E$ as well as their tensors satisfy uniform reverse hypercontractive inequalities. That is, for all $q<p<1$ and every positive valued function $f$ for $t \geq \log \frac{1-q}{1-p}$ we have $\| e^{-tL}f\|_{q} \geq \| f\|_{p}$. This should be contrasted with the case of hypercontractive inequalities for simple operators where $t$ is known to depend not only on $p$ and $q$ but also on the underlying space. The new reverse hypercontractive inequalities established here imply new mixing and isoperimetric results for short random walks in product spaces, for certain card-shufflings, for Glauber dynamics in high-temperatures spin systems as well as for queueing processes. The inequalities further imply a quantitative Arrow impossibility theorem for general product distributions and inverse polynomial bounds in the number of players for the non-interactive correlation distillation problem with $m$-sided dice.

Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. For sedentary populations in a spatially homogeneous yet temporally variable environment, a simple model of population growth is a stochastic differential equation dZ _t = μ Z _t dt + σ Z _t dW _t , t ≥ 0, where the conditional law of Z _t+Δt − Z _t given Z _t = z has mean and variance approximately z μΔt and z ² σ ²Δt when the time increment Δt is small. The long-term stochastic growth rate \({\lim_{t \to \infty} t^{-1}\log Z_t}\) for such a population equals \({\mu -\frac{\sigma^2}{2}}\) . Most populations, however, experience spatial as well as temporal variability. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study an analogous model \({{\bf X}_t = (X_t^1, \ldots, X_t^n)}\) , t ≥ 0, for the population abundances in n patches: the conditional law of X _t+Δt given X _t = x is such that the conditional mean of \({X_{t+\Delta t}^i - X_t^i}\) is approximately \({[x^i \mu_i + \sum_j (x^j D_{ji} - x^i D_{ij})] \Delta t}\) where μ _i is the per capita growth rate in the ith patch and D _ij is the dispersal rate from the ith patch to the jth patch, and the conditional covariance of \({X_{t+\Delta t}^i - X_t^i}\) and \({X_{t + \Delta t}^j - X_t^j}\) is approximately x ⁱ x ^j σ _ij Δt for some covariance matrix Σ = (σ _ij ). We show for such a spatially extended population that if \({S_t = X_t^1 + \cdots + X_t^n}\) denotes the total population abundance, then Y _t = X _t /S _t , the vector of patch proportions, converges in law to a random vector Y _∞ as \({t \to \infty}\) , and the stochastic growth rate \({\lim_{t \to \infty} t^{-1}\log S_t}\) equals the space-time average per-capita growth rate \({\sum_i \mu_i \mathbb{E}[Y_\infty^i]}\) experienced by the population minus half of the space-time average temporal variation \({\mathbb{E}[\sum_{i,j}\sigma_{ij}Y_\infty^i Y_\infty^j]}\) experienced by the population. Using this characterization of the stochastic growth rate, we derive an explicit expression for the stochastic growth rate for populations living in two patches, determine which choices of the dispersal matrix D produce the maximal stochastic growth rate for a freely dispersing population, derive an analytic approximation of the stochastic growth rate for dispersal limited populations, and use group theoretic techniques to approximate the stochastic growth rate for populations living in multi-scale landscapes (e.g. insects on plants in meadows on islands). Our results provide fundamental insights into “ideal free” movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology. For example, our analysis implies that even in the absence of density-dependent feedbacks, ideal-free dispersers occupy multiple patches in spatially heterogeneous environments provided environmental fluctuations are sufficiently strong and sufficiently weakly correlated across space. In contrast, for diffusively dispersing populations living in similar environments, intermediate dispersal rates maximize their stochastic growth rate.

Classical ecological theory predicts that environmental stochasticity increases extinction risk by reducing the average per-capita growth rate of populations. To understand the interactive effects of environmental stochasticity, spatial heterogeneity, and dispersal on population growth, we study the following model for population abundances in $n$ patches: the conditional law of $X_{t+dt}$ given $X_t=x$ is such that when $dt$ is small the conditional mean of $X_{t+dt}^i-X_t^i$ is approximately $[x^i\mu_i+\sum_j(x^j D_{ji}-x^i D_{ij})]dt$, where $X_t^i$ and $\mu_i$ are the abundance and per capita growth rate in the $i$-th patch respectivly, and $D_{ij}$ is the dispersal rate from the $i$-th to the $j$-th patch, and the conditional covariance of $X_{t+dt}^i-X_t^i$ and $X_{t+dt}^j-X_t^j$ is approximately $x^i x^j \sigma_{ij}dt$. We show for such a spatially extended population that if $S_t=(X_t^1+...+X_t^n)$ is the total population abundance, then $Y_t=X_t/S_t$, the vector of patch proportions, converges in law to a random vector $Y_\infty$ as $t\to\infty$, and the stochastic growth rate $\lim_{t\to\infty}t^{-1}\log S_t$ equals the space-time average per-capita growth rate $\sum_i\mu_i\E[Y_\infty^i]$ experienced by the population minus half of the space-time average temporal variation $\E[\sum_{i,j}\sigma_{ij}Y_\infty^i Y_\infty^j]$ experienced by the population. We derive analytic results for the law of $Y_\infty$, find which choice of the dispersal mechanism $D$ produces an optimal stochastic growth rate for a freely dispersing population, and investigate the effect on the stochastic growth rate of constraints on dispersal rates. Our results provide fundamental insights into "ideal free" movement in the face of uncertainty, the persistence of coupled sink populations, the evolution of dispersal rates, and the single large or several small (SLOSS) debate in conservation biology.