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We study the effectiveness of regional lockdown strategies to mitigate the spread of a pathogen across regional units, in the following called cities, within a country or region for a single infection wave. The heterogeneity in the epidemically relevant connectivity is defined via a random network model with cities as nodes, where the city's sizes determine their connectivity via a gravity type kernel function. Isolation of a whole city is initiated when infection numbers surpass defined thresholds. We consider two basic strategies for the lockdowns. Strategy~$(P)$ isolates cities based on a proportional threshold of infections, while stra\-tegy~\((U)\) uses a uniform infection threshold for all cities. Given the heavy-tailed distribution of city sizes, strategy \((P)\) can potentially result in more secondary infections from larger cities than strategy \((U)\). As an efficiency measure we use the ratio of individuals under lockdown and the number of infected individuals. Additionally, we analytically derive formulas for the basic reproduction numbers and prevalences. Our model is fitted to mobility data from France, Japan, and Poland, and validated through simulations. The findings indicate that while the model aligns well with data from France and Poland, it deviates in Japan, highlighting the importance of geographical nuances in pathogen spread modeling. Furthermore, it suggests that for France (and Japan) both strategies perform equally well, while for Poland strategy \((U)\) outperforms strategy \((P)\)., Comment: main text: 48 pages, supplementary material: 15p, 109 figures (including the supplementary material)

We consider discrete-time branching random walks with a radially symmetric distribution. Independently of each other individuals generate offspring whose relative locations are given by a copy of a radially symmetric point process $\mathcal{L}$. The number of particles at time $t$ form a supercritical Galton-Watson process. We investigate the maximal distance to the origin of such branching random walks. Conditioned on survival, we show that, under some assumptions on $\mathcal{L}$, it grows in the same way as for branching Brownian motion or a broad class of one-dimensional branching random walks: the first term is linear in time and the second logarithmic. The constants in front of these terms are explicit and depend only on the mean measure of $\mathcal{L}$ and dimension. Our main tool in the proof is a ballot theorem with moving barrier which may be of independent interest.

We study a general class of interacting particle systems over a countable state space $V$ where on each site $x \in V$ the particle mass $\eta(x) \geq 0$ follows a stochastic differential equation. We construct the corresponding Markovian dynamics in terms of strong solutions to an infinite coupled system of stochastic differential equations and prove a comparison principle with respect to the initial configuration as well as the drift of the process. Using this comparison principle, we provide sufficient conditions for the existence and uniqueness of an invariant measure in the subcritical regime and prove convergence of the transition probabilities in the Wasserstein-1-distance. Finally, for sublinear drifts, we establish a linear growth theorem showing that the spatial spread is at most linear in time. Our results cover a large class of finite and infinite branching particle systems with interactions among different sites.

We consider the continuous-time frog model on $\mathbb{Z}$. At time $t = 0$, there are $\eta (x)$ particles at $x\in \mathbb{Z}$, each of which is represented by a random variable. In particular, $(\eta(x))_{x \in \mathbb{Z} }$ is a collection of independent random variables with a common distribution $\mu$, $\mu(\mathbb{Z}_+) = 1$. The particles at the origin are active, all other ones being assumed as dormant, or sleeping. Active particles perform a simple symmetric continuous-time random walk in $\mathbb{Z} $ (that is, a random walk with $\exp(1)$-distributed jump times and jumps $-1$ and $1$, each with probability $1/2$), independently of all other particles. Sleeping particles stay still until the first arrival of an active particle to their location; upon arrival they become active and start their own simple random walks. Different sets of conditions are given ensuring explosion, respectively non-explosion, of the continuous-time frog model. Our results show in particular that if $\mu$ is the distribution of $e^{Y \ln Y}$ with a non-negative random variable $Y$ satisfying $\mathbb{E} Y < \infty$, then a.s. no explosion occurs. On the other hand, if $a \in (0,1)$ and $\mu$ is the distribution of $e^X$, where $\mathbb{P} \{X \geq t \} = t^{-a}$, $t \geq 1$, then explosion occurs a.s. The proof relies on a certain type of comparison to a percolation model which we call totally asymmetric discrete inhomogeneous Boolean percolation., Comment: Further examples and discussion are added; the proofs in section 5 are expanded significantly; minor changes, fixes, corrections, and improvements

In this paper we estimate the minimal controllability time for a class of non-linear control systems with a bounded convex state constraint. An explicit expression is given for the controllability time if the image of the control matrix is of co-dimension one. A lower bound for the controllability time is given in the general case. The technique is based on finding a lower dimension system with the similar controllability properties as the original system. The controls corresponding to the minimal time, or time close to the minimal one, are discussed and computed analytically. The effectiveness of the proposed approach is illustrated by a few examples., Comment: Figure 2 is now displayed correctly

It is well known that there are two regimes in a standard one-dimensional Boolean percolation model: either the entire space is covered a.s., or the covered volume fraction is strictly less than one. The aim of this work is to demonstrate that there is a third possibility in a Boolean model with totally asymmetric grains on a half-line: a.s. there is no unbounded component, but the covered volume fraction is one. An explicit condition is given characterizing the existence of an unbounded occupied component.

Consider a stochastic growth model on $\mathbb{Z} ^d$. Start with some active particle at the origin and sleeping particles elsewhere. The initial number of particles at $x \in \mathbb{Z} ^d$ is $\eta(x)$, where $\eta (x)$ are independent random variables distributed according to $\mu$. Active particles perform a simple continuous-time random walk while sleeping particles stay put until the first arrival of an active particle to their location. Upon the arrival all sleeping particles at the site activate at once and start moving according to their own simple random walks. The aim of this paper is to give conditions on $\mu$ under which the spread of the process is linear or faster than linear. The proofs rely on comparison to various percolation models., Comment: Very minor changes compared to the previous version. Accepted to publication in Annales de l'Institut Henri Poincar\'e, Probabilit\'es et Statistiques

The aim of this work is to demonstrate that the continuous-time frog model can spread arbitrary fast. The set of sites visited by an active particle can become infinite in a finite time., Comment: 8 pages

We consider a one-dimensional discrete-space birth process with a bounded number of particle per site. Under the assumptions of the finite range of interaction, translation invariance, and non-degeneracy, we prove a shape theorem. We also derive a limiting estimate and an exponential estimate on the fluctuations of the position of the rightmost particle., Comment: 17 pages; a short section devoted to numerical simulations is added; other small changes and improvements

We study a Markov birth-and-death process on a space of locally finite configurations, which describes an ecological model with a density dependent fecundity regulation mechanism. We establish existence and uniqueness of this process and analyze its properties. In particular, we show global time-space boundedness of the population density and, using a constructed Foster-Lyapunov-type function, we study return times to certain level sets of tempered configurations. We find also sufficient conditions that the degenerate invariant distribution is unique for the considered process.

We consider the speed of propagation of a {continuous-time continuous-space} branching random walk with the additional restriction that the birth rate at any spatial point cannot exceed $1$. The dispersion kernel is taken to have density that decays polynomially as $|x|^{- 2\alpha}$, $x \to \infty$. We show that if $\alpha > 2$, then the system spreads at a linear speed, {while for $\alpha \in (\frac 12 ,2]$ the spread is faster than linear}. We also consider the mesoscopic equation corresponding to the microscopic stochastic system. We show that in contrast to the microscopic process, the solution to the mesoscopic equation spreads exponentially fast for every $\alpha > \frac 12$., Comment: v2 update: A new result is added covering the case $alpha < 2$ for the microscopic model. Further remarks and heuristic comments are added, including connections to other models. Many minor changes are made

A short proof of the quenched central limit theorem for the random walk in random environment introduced by Boldrighini, Minlos, and Pellegrinotti is given.

We prove that under certain general conditions on the birth and death rates the Lebesgue-Poisson measure is a maximal irreducibility measure for the spatial birth-and-death process.

We formulate and prove a shape theorem for a continuous-time continuous-space stochastic growth model under certain general conditions. Similarly to the classical lattice growth models the proof makes use of the subadditive ergodic theorem. A precise expression for the speed of propagation is given in the case of a truncated free branching birth rate., Comment: v2 update: having modified the technique, we now do not need former Condition 2.6; a proposition explicitly proving that the growth is not faster than linear is added (Proposition 3.1); some structural changes are made, as well as many minor changes, such as dealing with typos or adding details. Some of the results from arxiv.org/abs/1507.05804 are given with proofs

We consider the pricing problem related to payoffs that can have discontinuities of polynomial growth. The asset price dynamic is modeled within the Black and Scholes framework characterized by a stochastic volatility term driven by a fractional Ornstein-Uhlenbeck process. In order to solve the aforementioned problem, we consider three approaches. The first one consists in a suitable transformation of the initial value of the asset price, in order to eliminate possible discontinuities. Then we discretize both the Wiener process and the fractional Brownian motion and estimate the rate of convergence of the related discretized price to its real value, the latter one being impossible to be evaluated analytically. The second approach consists in considering the conditional expectation with respect to the entire trajectory of the fractional Brownian motion (fBm). Then we derive a closed formula which involves only integral functional depending on the fBm trajectory, to evaluate the price; finally we discretize the fBm and estimate the rate of convergence of the associated numerical scheme to the option price. In both cases the rate of convergence is the same and equals $n^{-rH}$, where $n$ is a number of the points of discretization, $H$ is the Hurst index of fBm, and $r$ is the H\"{o}lder exponent of volatility function. The third method consists in calculating the density of the integral functional depending on the trajectory of the fBm via Malliavin calculus also providing the option price in terms of the associated probability density., Comment: 31 pages, 5 tables

We prove the exponential estimate \begin{equation*} P \{ s < \tau < \infty \} \leq C e^{-q s}, \quad s \geq 0, \end{equation*} where $C, q >0$ are constants and $ \tau $ is the extinction time of the supercritical branching random walk (BRW) on a cube. We cover both the discrete-space and continuous-space BRWs., Comment: 11 pages; minor text modifications only in the updated version, no essential changes

Spatial birth-and-death processes with time dependent rates are obtained as solutions to certain stochastic equations. The existence, uniqueness, uniqueness in law and the strong Markov property of unique solutions are proven when the integral of the birth rate over $\mathbb{R} ^ \mathrm{d}$ grows not faster than linearly with the number of particles of the system. Martingale properties of the constructed process provide a rigorous connection to the heuristic generator. We also study pathwise behavior of an aggregation model. The probability of extinction and the growth rate of the number of particles conditioning on non-extinction are estimated., Comment: Minor changes and improvements; journal ref added. arXiv admin note: text overlap with arXiv:1502.06783

Lattice birth-and-death Markov dynamics of particle systems with spins from the set of non-negative integers are constructed as unique solutions to certain stochastic equations. Pathwise uniqueness, strong existence, Markov property and joint uniqueness in law are proven, and a martingale characterization of the process is given. Sufficient conditions for the existence of an invariant distribution are formulated in terms of Lyapunov functions. We apply obtained results to discrete analogs of the Bolker--Pacala--Dieckmann--Law model and an aggregation model.

Spatial birth-and-death processes with a finite number of particles are obtained as unique solutions to certain stochastic equations. Conditions are given for existence and uniqueness of such solutions, as well as for continuous dependence on the initial conditions. The possibility of an explosion and connection with the heuristic generator of the process are discussed.