degree dispersion increases the rate of rare events in population networks there is great interest in predicting rare and extreme events in complex systems and in particular understanding the role of network topology in facilitating such events in this letter we show that degree dispersion the fact that the number of local connections in networks varies broadly increases the probability of large rare fluctuations in population networks generically we perform explicit calculations for two canonical and distinct classes of rare events network extinction and switching when the distance to threshold is held constant and hence stochastic effects are fairly compared among networks we show that there is a universal exponential increase in the rate of rare events proportional to the variance of a network s degree distribution over its mean squared

generalized voterlike model on activity driven networks with attractiveness we study the behavior of a generalized consensus dynamics on a temporal network of interactions the activity driven network with attractiveness in this temporal network model agents are endowed with an intrinsic activity a ruling the rate at which they generate connections and an intrinsic attractiveness b modulating the rate at which they receive connections the consensus dynamics considered is a mixed voter and moran dynamics each agent either in state 0 or 1 modifies his or her state when connecting with a peer thus an active agent copies his or her state from the peer with probability p or imposes his or her state to him or her with the complementary probability 1 ensuremath p applying a heterogeneous mean field approach we derive a differential equation for the average density of voters with activity a and attractiveness b in state 1 which we use to evaluate the average time to reach consensus and the exit probability defined as the probability that a single agent with activity a and attractiveness b eventually imposes his or her state to a pool of initially unanimous population in the opposite state we study a number of particular cases finding an excellent agreement with numerical simulations of the model interestingly we observe a symmetry between voter and moran dynamics in pure activity driven networks and their static integrated counterparts that exemplifies the strong differences that a time varying network can impose on dynamical processes

asymptotic behaviour of the bessel heat kernels we consider dirichlet heat kernel p a mu t x y for the bessel differential operator l mu frac d 2 dx 2 frac 2 mu 1 2x mu in mathbf r in half line a infty a 0 and provide its asymptotic expansions for xy t rightarrow infty

fast fixation with a generic network structure we investigate the dynamics of a broad class of stochastic copying processes on a network that includes examples from population genetics spatially structured wright fisher models ecology hubbell type models linguistics the utterance selection model and opinion dynamics the voter model as special cases these models all have absorbing states of fixation where all the nodes are in the same state earlier studies of these models showed that the mean time when this occurs can be made to grow as different powers of the network size by varying the degree distribution of the network here we demonstrate that this effect can also arise if one varies the asymmetry of the copying dynamics while holding the degree distribution constant in particular we show that the mean time to fixation can be accelerated even on homogeneous networks when certain nodes are very much more likely to be copied from than copied to we further show that there is a complex interplay between degree distribution and asymmetry when they may covary and that the results are robust to correlations in the network or the initial condition

on the martingale property of certain local martingales the stochastic exponential z t rm exp m t m 0 1 2 langle m m rangle t of a continuous local martingale m is itself a continuous local martingale we give a necessary and sufficient condition for the process z to be a true martingale in the case where m t int 0 t b y u dw u and y is a one dimensional diffusion driven by a brownian motion w furthermore we provide a necessary and sufficient condition for z to be a uniformly integrable martingale in the same setting these conditions are deterministic and expressed only in terms of the function b and the drift and diffusion coefficients of y as an application we provide a deterministic criterion for the absence of bubbles in a one dimensional setting

evolutionary dynamics on degree heterogeneous graphs the evolution of two species with different fitness is investigated on degree heterogeneous graphs the population evolves either by one individual dying and being replaced by the offspring of a random neighbor voter model dynamics or by an individual giving birth to an offspring that takes over a random neighbor node invasion process dynamics the fixation probability for one species to take over a population of n individuals depends crucially on the dynamics and on the local environment starting with a single fitter mutant at a node of degree k the fixation probability is proportional to k for voter model dynamics and to 1 k for invasion process dynamics

phase transitions in the q voter model with noise on a duplex clique we study a nonlinear q voter model with stochastic noise interpreted in the social context as independence on a duplex network to study the role of the multilevelness in this model we propose three methods of transferring the model from a mono to a multiplex network they take into account two criteria one related to the status of independence local vs global and one related to peer pressure and vs or in order to examine the influence of the presence of more than one level in the social network we perform simulations on a particularly simple multiplex a duplex clique which consists of two fully overlapped complete graphs cliques solving numerically the rate equation and simultaneously conducting monte carlo simulations we provide evidence that even a simple rearrangement into a duplex topology may lead to significant changes in the observed behavior however qualitative changes in the phase transitions can be observed for only one of the considered rules local and for this rule the phase transition becomes discontinuous for q 5 whereas for a monoplex such behavior is observed for q 6 interestingly only this rule admits construction of realistic variants of the model in line with recent social experiments

on the fixation probability of superstars the moran process models the spread of genetic mutations through populations a mutant with relative fitness r is introduced and the system evolves either reaching fixation an all mutant population or extinction no mutants in a widely cited paper lieberman et al 2005 evolutionary dynamics on graphs nature 433 312 316 generalize the model to populations on the vertices of graphs they describe a class of graphs superstars with a parameter k and state that the fixation probability tends to 1 r k as the graphs get larger we show that this is untrue as stated specifically for k 5 we show that the fixation probability in the limit as graphs get larger cannot exceed 1 1 j r where j r r 4 contrary to the claimed result our proof is fully rigorous though we use a computer algebra package to invert a 31 31 symbolic matrix we do believe the qualitative claim of lieberman et al that superstar fixation probability tends to 1 as k increases and that it can probably be proved similarly to their sketch we were able to run larger simulations than the ones they presented simulations on graphs of around 40 000 vertices do not support their claim but these graphs might be too small to exhibit the limiting behaviour

cohomogeneity one topological manifolds revisited we prove a structure theorem for closed topological manifolds of cohomogeneity one this result corrects an oversight in the literature we complete the equivariant classification of closed simply connected cohomogeneity one topological manifolds in dimensions 5 6 and 7 and obtain topological characterizations of these spaces in these dimensions these manifolds are homeomorphic to smooth manifolds

finiteness and fluctuations in growing networks we study the role of finiteness and fluctuations about average quantities for basic structural properties of growing networks we first determine the exact degree distribution of finite networks by generating function approaches the resulting distributions exhibit an unusual finite size scaling behaviour and they are also sensitive to the initial conditions we argue that fluctuations in the number of nodes of degree k become gaussian for fixed degree as the size of the network diverges we also characterize the fluctuations between different realizations of the network in terms of higher moments of the degree distribution

convexity of the optimal stopping boundary for the american put option we show that the optimal stopping boundary for the american put option is convex in the standard black scholes model the methods are adapted from ice melting problems and rely upon studying the behavior of level curves of solutions to certain parabolic differential equations

effects of strategy migration direction and noise in the evolutionary spatial prisoner s dilemma spatial games are crucial for understanding patterns of cooperation in nature and to some extent society they are known to be more sensitive to local symmetries than e g spin models this paper concerns the evolution of the prisoner s dilemma game on regular lattices with three different types of neighborhoods char22 the von neumann moore and kagom e types we investigate two kinds of dynamics for the players to update their strategies that can be unconditional cooperator or defector depending on the payoff difference an individual can adopt the strategy of a random neighbor a voter model like dynamics vmld or impose its strategy on a random neighbor i e invasion process like dynamics ipld in particular we focus on the effects of noise in combination with the strategy dynamics on the evolution of cooperation we find that vmld compared to ipld better supports the spreading and sustaining of cooperation we see that noise has nontrivial effects on the evolution of cooperation maximum cooperation density can be realized either at a medium noise level in the limit of zero noise or in both these regions the temptation to defect and the local interaction structure determine the outcome especially in the low noise limit the local interaction plays a crucial role in determining the fate of cooperators we elucidate these both by numerical simulations and mean field cluster approximation methods

mapping the q voter model from a single chain to complex networks abstract we propose and compare six different ways of mapping the modified q voter model to complex networks considering square lattices barabasi albert watts strogatz and real twitter networks we ask the question if always a particular choice of the group of influence of a fixed size q leads to different behavior at the macroscopic level using monte carlo simulations we show that the answer depends on the relative average path length of the network and for real life topologies the differences between the considered mappings may be negligible

voter models on weighted networks we study the dynamics of the voter and moran processes running on top of complex network substrates where each edge has a weight depending on the degree of the nodes it connects for each elementary dynamical step the first node is chosen at random and the second is selected with probability proportional to the weight of the connecting edge we present a heterogeneous mean field approach allowing to identify conservation laws and to calculate exit probabilities along with consensus times in the specific case when the weight is given by the product of nodes degree raised to a power we derive a rich phase diagram with the consensus time exhibiting various scaling laws depending on and on the exponent of the degree distribution numerical simulations give very good agreement for small values of an additional analytical treatment heterogeneous pair approximation improves the agreement with numerics but the theoretical understanding of the behavior in the limit of large remains an open challenge

handbook of brownian motion facts and formulae i theory i stochastic processes in general ii linear diffusions iii stochastic calculus iv brownian motion v local time as a markov process vi differential systems associated to brownian motion appendix 1 briefly on some diffusions ii tables of distributions of functionals of brownian motion and related processes 1 brownian motion 2 brownian motion with drift 3 reflecting brownian motion 4 bessel process of order 5 bessel process of order 1 2 6 bessel process of order zero 7 ornstein uhlenbeck process 8 radial ornstein uhlenbeck process 9 geometric brownian motion appendix 2 special functions appendix 3 inverse laplace transforms appendix 4 differential equations and their solutions appendix 5 formulae for n fold differentiation

the law of the supremum of a stable levy process with no negative jumps let x xt t 0 be a stable levy process of index fi 2 1 2 with no negative jumps and let st sup0 st xs denote its running supremum for t 0 we show that the density function ft of st can be characterized as the unique solution to a weakly singular volterra integral equation of the flrst kind or equivalently as the unique solution to a flrst order riemann liouville fractional difierential equation satisfying a boundary condition at zero this yields an explicit series representation for ft recalling the familiar relation between st and the flrst entry time x of x into x 1 this further translates into an explicit series representation for the density function of x

nonlinear q voter model we introduce a nonlinear variant of the voter model the q voter model in which q neighbors with possible repetition are consulted for a voter to change opinion if the q neighbors agree the voter takes their opinion if they do not have a unanimous opinion still a voter can flip its state with probability epsilon we solve the model on a fully connected network i e in mean field and compute the exit probability as well as the average time to reach consensus by employing the backward fokker planck formalism and scaling arguments we analyze the results in the perspective of a recently proposed langevin equation aimed at describing generic phase transitions in systems with two z2 symmetric absorbing states in particular by deriving explicitly the coefficients of such a langevin equation as a function of the microscopic flipping probabilities we find that in mean field the q voter model exhibits a disordered phase for high epsilon and an ordered one for low epsilon with three possible ways to go from one to the other i a unique generalized voter like transition ii a series of two consecutive transitions one ising like in which the z2 symmetry is broken and a separate one in the directed percolation class in which the system falls into an absorbing state and iii a series of two transitions including an intermediate regime in which the final state depends on initial conditions this third so far unexplored scenario in which a type of ordering dynamics emerges is rationalized and found to be specific of mean field i e fluctuations are explicitly shown to wash it out in spatially extended systems

spectral decomposition of fractional operators and a reflected stable semigroup abstract in this paper we provide the spectral decomposition in hilbert space of the c 0 semigroup p and its adjoint p having as generator respectively the caputo and the right sided riemann liouville fractional derivatives of index 1 2 these linear operators which are non local and non self adjoint appear in many recent studies in applied mathematics and also arise as the infinitesimal generators of some substantial processes such as the reflected spectrally negative stable process our approach relies on intertwining relations that we establish between these semigroups and the semigroup of a bessel type process whose generator is a self adjoint second order differential operator in particular from this commutation relation we characterize the positive real axis as the continuous point spectrum of p and provide a power series representation of the corresponding eigenfunctions we also identify the positive real axis as the residual spectrum of the adjoint operator p and elucidate its role in the spectral decomposition of these operators by resorting to the concept of continuous frames we proceed by investigating the domain of the spectral operators and derive two representations for the heat kernels of these semigroups as a by product we also obtain regularity properties for these latter and also for the solution of the associated cauchy problem

mean field analysis of the q voter model on networks we present a detailed investigation of the behavior of the nonlinear q voter model for opinion dynamics at the mean field level we derive analytically for any value of the number q of agents involved in the elementary update the phase diagram the exit probability and the consensus time at the transition point the mean field formalism is extended to the case that the interaction pattern is given by generic heterogeneous networks we finally discuss the case of random regular networks and compare analytical results with simulations

asymptotically optimal amplifiers for the moran process abstract we study the moran process as adapted by lieberman hauert and nowak this is a model of an evolving population on a graph or digraph where certain individuals called mutants have fitness r and other individuals called non mutants have fitness 1 we focus on the situation where the mutation is advantageous in the sense that r 1 a family of digraphs is said to be strongly amplifying if the extinction probability tends to 0 when the moran process is run on digraphs in this family the most amplifying known family of digraphs is the family of megastars of galanis et al we show that this family is optimal up to logarithmic factors since every strongly connected n vertex digraph has extinction probability n 1 2 next we show that there is an infinite family of undirected graphs called dense incubators whose extinction probability is o n 1 3 we show that this is optimal up to constant factors finally we introduce sparse incubators for varying edge density and show that the extinction probability of these graphs is o n m where m is the number of edges again we show that this is optimal up to constant factors

sharp holder continuity of tangent cones for spaces with a lower ricci curvature bound and applications we prove a new estimate on manifolds with a lower ricci bound which asserts that the geometry of balls centered on a minimizing geodesic can change in at most a holder continuous way along the geodesic we give examples that show that the holder exponent along with essentially all the other consequences that follow from this estimate are sharp among the applications is that the regular set is convex for any non collapsed limit of einstein metrics in the general case of a potentially collapsed limit of manifolds with just a lower ricci curvature bound we show that the regular set is weakly convex and a e convex we also show two conjectures of cheeger colding one of these asserts that the isometry group of any even collapsed limit of manifolds with a uniform lower ricci curvature bound is a lie group the other asserts that the dimension of any limit space is the same everywhere

a note on the continuity of free boundaries in finite horizon optimal stopping problems for one dimensional diffusions we provide sufficient conditions for the continuity of the free boundary in a general class of finite horizon optimal stopping problems arising for instance in finance and economics the underlying process is a strong solution of a one dimensional time homogeneous stochastic differential equation sde the proof relies on both analytic and probabilistic arguments and is based on a contradiction scheme inspired by the maximum principle in partial differential equations theory mild local regularity of the coefficients of the sde and smoothness of the gain function locally at the boundary are required

distribution of the time to explosion for one dimensional diffusions we study the distribution of the time to explosion for one dimensional diffusions we relate this question to the computation of expectations of suitable nonnegative local martingales moreover we characterize the distribution function of the time to explosion as the minimal solution to a certain cauchy problem for an appropriate parabolic differential equation this leads to alternative characterizations of feller s criterion for explosions we discuss in detail several examples for which it is possible to obtain analytic expressions for the corresponding distribution of the time to explosion using the methodologies developed in the paper

a boundary of the set of the riemannian manifolds with bounded curvatures and diameters on etudie la structure de la fermeture de l ensemble constitue par des classes d isometrie de varietes de riemann a courbures et diametres bornes

detecting the maximum of a scalar diffusion with negative drift let x be a scalar diffusion process with drift coefficient pointing towards the origin i e x is mean reverting we denote by x the corresponding running maximum t 0 the first time x hits the level zero given an increasing and convex loss function ell we consider the following optimal stopping problem inf 0 leq theta leq t 0 mathbb e ell x t 0 x theta over all stopping times theta with values in 0 t 0 for the quadratic loss function and under mild conditions we prove that an optimal stopping time exists and is defined by theta t 0 wedge inf t geq 0 x t geq gamma x t where the boundary gamma is explicitly characterized as the concatenation of the solutions of two equations we investigate some examples such as the ornstein uhlenbeck process the cir feller process as well as the standard and drifted brownian motions

voter model on networks and the multivariate beta distribution in elections the vote shares or turnout rates show a strong spatial correlation the logarithmic decay with distance suggests that a two dimensional 2d noisy diffusive equation describes the system based on the study of u s presidential elections data it was determined that the fluctuations of vote shares also exhibit a strong and long range spatial correlation previously it was considered difficult to induce strong and long range spatial correlation of the vote shares without breaking the empirically observed narrow distribution we demonstrate that a voter model on networks shows such a behavior in the model there are many voters in a node who are affected by the agents in the node and by the agents in the linked nodes a multivariate wright fisher diffusion equation for the joint probability density of the vote shares is derived the stationary distribution is a multivariate generalization of the beta distribution in addition we also estimate the equilibrium values and the covariance matrix of the vote shares and obtain a correspondence with a multivariate normal distribution this approach largely simplifies the calibration of the parameters in the modeling of elections

optimal prediction of the last passage time of a transient diffusion we identify the integrable stopping time tau with minimal l 1 distance from the last passage time gamma z associated with a given level z 0 for an arbitrary nonnegative time homogeneous transient diffusion x we demonstrate that tau is in fact the first time that x assumes a value outside a half open interval 0 r the upper boundary r z of this interval is characterized either as the solution for a one dimensional optimization problem or as part of the solution for a free boundary problem a number of concrete examples illustrate the result

organization of growing random networks the organizational development of growing random networks is investigated these growing networks are built by adding nodes successively and linking each to an earlier node of degree k with an attachment probability a k when a k grows more slowly than linearly with k the number of nodes with k links n k t decays faster than a power law in k while for a k growing faster than linearly in k a single node emerges which connects to nearly all other nodes when a k is asymptotically linear n k t ensuremath sim mathrm tk ensuremath ensuremath nu with ensuremath nu dependent on details of the attachment probability but in the range 2l ensuremath nu l ensuremath infty the combined age and degree distribution of nodes shows that old nodes typically have a large degree there is also a significant correlation in the degrees of neighboring nodes so that nodes of similar degree are more likely to be connected the size distributions of the in and out components of the network with respect to a given node namely its descendants and ancestors are also determined the in component exhibits a robust s ensuremath 2 power law tail where s is the component size the out component has a typical size of order mathrm ln t and it provides basic insights into the genealogy of the network

on the optimal exercise boundaries of swing put options we use probabilistic methods to characterise time dependent optimal stopping boundaries in a problem of multiple optimal stopping on a finite time horizon motivated by financial applications we consider a payoff of immediate stopping of put type and the underlying dynamics follows a geometric brownian motion the optimal stopping region relative to each optimal stopping time is described in terms of two boundaries which are continuous monotonic functions of time and uniquely solve a system of coupled integral equations of volterra type finally we provide a formula for the value function of the problem

finite horizon optimal investment and consumption with transaction costs this paper concerns continuous time optimal investment and the consumption decision of a constant relative risk aversion crra investor who faces proportional transaction costs and a finite time horizon in the no consumption case it has been studied by liu and loewenstein review of financial studies 15 2002 pp 805 835 and dai and yi j differential equations 246 2009 pp 1445 1469 mathematically it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraints the problem gives rise to two free boundaries which stand for the optimal buying and selling strategies respectively we present an analytical approach to analyze the behaviors of free boundaries the regularity of the value function is studied as well our approach is essentially based on the connection between singular control and optimal stopping which is first revealed in the present problem

voter model with non poissonian interevent intervals recent analysis of social communications among humans has revealed that the interval between interactions for a pair of individuals and for an individual often follows a long tail distribution we investigate the effect of such a non poissonian nature of human behavior on dynamics of opinion formation we use a variant of the voter model and numerically compare the time to consensus of all the voters with different distributions of interevent intervals and different networks compared with the exponential distribution of interevent intervals i e the standard voter model the power law distribution of interevent intervals slows down consensus on the ring this is because of the memory effect in the power law case the expected time until the next update event on a link is large if the link has not had an update event for a long time on the complete graph the consensus time in the power law case is close to that in the exponential case regular graphs bridge these two results such that the slowing down of the consensus in the power law case as compared to the exponential case is less pronounced as the degree increases

epidemic dynamics and endemic states in complex networks we study by analytical methods and large scale simulations a dynamical model for the spreading of epidemics in complex networks in networks with exponentially bounded connectivity we recover the usual epidemic behavior with a threshold defining a critical point below that the infection prevalence is null on the contrary on a wide range of scale free networks we observe the absence of an epidemic threshold and its associated critical behavior this implies that scale free networks are prone to the spreading and the persistence of infections whatever spreading rate the epidemic agents might possess these results can help understanding computer virus epidemics and other spreading phenomena on communication and social networks

commitment versus persuasion in the three party constrained voter model in the framework of the three party constrained voter model where voters of two radical parties a and b interact with centrists c and c we study the competition between a persuasive majority and a committed minority in this model a s and b s are incompatible voters that can convince centrists or be swayed by them here radical voters are more persuasive than centrists whose sub population comprises susceptible agents c and a fraction of centrist zealots c whereas c s may adopt the opinions a and b with respective rates 1 a and 1 b with a b 0 c s are committed individuals that always remain centrists furthermore a and b voters can become susceptible centrists c with a rate 1 the resulting competition between commitment and persuasion is studied in the mean field limit and for a finite population on a complete graph at mean field level there is a continuous transition from a coexistence phase when b whereas all species coexist when a b when c and the initial density of centrists is low one finds lnn when n 1 our analytical findings are corroborated by stochastic simulations

fixation time in evolutionary graphs a mean field approach using an analytical method we calculate average conditional fixation time of mutants in a general graph structured population of two types of species the method is based on markov chains and uses a mean field approximation to calculate the corresponding transition matrix analytical results are compared with the results of simulation of the moran process on a number of network structures

optimal stopping of the maximum process the maximality principle the solution is found to the optimal stopping problem with payoff sup e s 0 t c x i dt where s s t t 0 is the maximum process associated with the one dimensional time homogeneous diffusion x x t t 0 the function x c x is positive and continuous and the supremum is taken over all stopping times of x for which the integral has finite expectation it is proved under no extra conditions that this problem has a solution that is the payoff is finite and there is an optimal stopping time if and only if the following maximality principle holds the first order nonlinear differential equation g s 2 g s l g s 2c g s l r l g s admits a maximal solution s g s which stays strictly below the diagonal in r 2 in this equation x x is the diffusion coefficient and x l x the scale function of x in this case the stopping time inf t 0 x t g s t is proved optimal and explicit formulas for the payoff are given the result has a large number of applications and may be viewed as the cornerstone in a general treatment of the maximum process

ising model on networks with an arbitrary distribution of connections we find the exact critical temperature t c of the nearest neighbor ferromagnetic ising model on an equilibrium random graph with an arbitrary degree distribution p k we observe an anomalous behavior of the magnetization magnetic susceptibility and specific heat when p k is fat tailed or loosely speaking when the fourth moment of the distribution diverges in infinite networks when the second moment becomes divergent t c approaches infinity the phase transition is of infinite order and size effect is anomalously strong

analytic treatment of consensus achievement in the single type zealotry voter model abstract we introduce zealots of one opinion in the voter model on a complete graph and examine how they affect consensus achievement using first step analysis for markov chains to obtain an exact solution we find that the mean consensus time scales with the population size n increasing the number of zealots z will decrease the consensus time in a power law fashion for large z the mean magnetization was also analytically obtained and was found to contain an exponential dependence on z the dynamics for the complete graph are qualitatively similar to those obtained in another study for the barabasi albert network in general the existence of zealots serves to hasten consensus except in the case where only a few zealots oppose the vast majority

natural models for evolution on networks evolutionary dynamics has been traditionally studied in the context of homogeneous populations mainly described by the moran process p moran random processes in genetics proceedings of the cambridge philosophical society 54 1 1958 60 71 recently this approach has been generalized in e lieberman c hauert m a nowak evolutionary dynamics on graphs nature 433 2005 312 316 by arranging individuals on the nodes of a network in general directed in this setting the existence of directed arcs enables the simulation of extreme phenomena where the fixation probability of a randomly placed mutant i e the probability that the offspring of the mutant eventually spread over the whole population is arbitrarily small or large on the other hand undirected networks i e undirected graphs seem to have a smoother behavior and thus it is more challenging to find suppressors amplifiers of selection that is graphs with smaller greater fixation probability than the complete graph i e the homogeneous population in this paper we focus on undirected graphs we present the first class of undirected graphs which act as suppressors of selection by achieving a fixation probability that is at most one half of that of the complete graph as the number of vertices increases moreover we provide some generic upper and lower bounds for the fixation probability of general undirected graphs as our main contribution we introduce the natural alternative of the model proposed in e lieberman c hauert m a nowak evolutionary dynamics on graphs nature 433 2005 312 316 in our new evolutionary model all individuals interact simultaneously and the result is a compromise between aggressive and non aggressive individuals we prove that our new model of mutual influences admits a potential function which guarantees the convergence of the system for any graph topology and any initial fitness vector of the individuals furthermore we prove fast convergence to the stable state for the case of the complete graph as well as we provide almost tight bounds on the limit fitness of the individuals apart from being important on its own this new evolutionary model appears to be useful also in the abstract modeling of control mechanisms over invading populations in networks we demonstrate this by introducing and analyzing two alternative control approaches for which we bound the time needed to stabilize to the healthy state of the system

voter model on the two clique graph i examine the mean consensus time i e exit time of the voter model in the so called two clique graph the two clique graph is composed of two cliques interconnected by some links and considered as a toy model of networks with community structure or multilayer networks i analytically show that as the number of interclique links per node is varied the mean consensus time experiences a crossover between a fast consensus regime i e o n and a slow consensus regime i e o n 2 where n is the number of nodes the fast regime is consistent with the result for homogeneous well mixed graphs such as the complete graph the slow regime appears only when the entire network has o 1 interclique links the present results suggest that the effect of community structure on the consensus time of the voter model is fairly limited

optimal stock selling based on the global maximum we aim to determine an optimal stock selling time to minimize the expectation of the square error between the selling price and the global maximum price over a given period assuming that stock price follows the geometric brownian motion we formulate the problem as an optimal stopping time problem or equivalently a variational inequality problem we provide a partial differential equation pde approach to characterize the resulting free boundary that corresponds to the optimal selling strategy the monotonicity and smoothness of the free boundary are addressed as well

reflected spectrally negative stable processes and their governing equations this paper explicitly computes the transition densities of a spectrally negative stable process with index greater than one reflected at its infimum first we derive the forward equation using the theory of sun dual semigroups the resulting forward equation is a boundary value problem on the positive half line that involves a negative riemann liouville fractional derivative in space and a fractional reflecting boundary condition at the origin then we apply numerical methods to explicitly compute the transition density of this space inhomogeneous markov process for any starting point to any desired degree of accuracy finally we discuss an application to fractional cauchy problems which involve a positive caputo fractional derivative in time

the trap of complacency in predicting the maximum where t 2 0 t and the functions t 7 b1 t and t 7 b2 t are continuous on t t with b1 t 0 and b2 t 1 2 if 0 then b1 is decreasing and b2 is increasing on t t with b1 t b2 t when t 6 0 using local time space calculus we derive a coupled system of nonlinear volterra integral equations of the second kind and show that the pair of optimal boundaries b1 and b2 can be characterized as the unique solution to this system this also leads to an explicit formula for v in terms of b1 and b2 if 0 then t 0 and b2 1 so that is expressed in terms of b1 only in this case b1 is decreasing on z t and increasing on 0 z for some z 2 0 t with z 0 if 0 and the system of two volterra equations reduces to one volterra equation if 0 then there is a closed form expression for b1 this problem was solved in 6 using the method of time change i e change of variables the method of time change cannot be extended to the case when 6 0 and the present paper settles the remaining cases using a difierent approach

margulis lemma for compact lie groups we improve margulis lemma for a compact connected lie group g there is a neighborhood u of the identity such that for any finite subgroup gamma subset g u cap gamma generates an abelian group we show that for each n there exists an integer w n 0 such that if h is a closed subgroup of a compact connected lie group g of dimension n then the quotient group h h0 has an abelian subgroup of index le w n where h0 is the identity component of h as an application we show that the fundamental group of the homogeneous space g h has an abelian subgroup of index le w n we show this same property for the fundamental groups of almost non negatively curved n manifolds whose universal coverings are not collapsed

time randomized stopping problems for a family of utility functions this paper studies stopping problems of the form v inf 0 leq tau leq t mathbb e u frac max 0 le s le t z s z tau for strictly concave or convex utility functions u in a family of increasing functions satisfying certain conditions where z is a geometric brownian motion and t is the time of the nth jump of a poisson process independent of z we obtain some properties of v and offer solutions for the optimal strategies to follow this provides us with a technique to build numerical approximations of stopping boundaries for the fixed terminal time optimal stopping problem presented in j du toit and g peskir ann appl probab 19 2009 pp 983 1014

stochastic classical solutions for space time fractional evolution equations on a bounded domain abstract space time fractional evolution equations are a powerful tool to model diffusion displaying space time heterogeneity we prove existence uniqueness and stochastic representation of classical solutions for an extension of caputo evolution equations featuring time nonlocal initial conditions we discuss the interpretation of the new stochastic representation as part of the proof a new result about inhomogeneous caputo evolution equations is proven

voter models on heterogeneous networks we study simple interacting particle systems on heterogeneous networks including the voter model and the invasion process these are both two state models in which in an update event an individual changes state to agree with a neighbor for the voter model an individual imports its state from a randomly chosen neighbor here the average time t n to reach consensus for a network of n nodes with an uncorrelated degree distribution scales as n ensuremath mu 1 2 ensuremath mu 2 where ensuremath mu k is the k text th moment of the degree distribution quick consensus thus arises on networks with broad degree distributions we also identify the conservation law that characterizes the route by which consensus is reached parallel results are derived for the invasion process in which the state of an agent is exported to a random neighbor we further generalize to biased dynamics in which one state is favored the probability for a single fitter mutant located at a node of degree k to overspread the population char22 the fixation probability char22 is proportional to k for the voter model and to 1 k for the invasion process

collapsing and pinching under a lower curvature bound in this paper we are concerned with collapsing phenomena and pinching problems of riemannian manifolds whose sectional curvatures are uniformly bounded from below for a positive integer n and for d 0 let x be the set of compact riemannian n manifolds m with sectional curvatures km 1 and diameters diam m d by the weak compactness theorem of gromov glp x is relatively compact in the set of compact metric spaces with respect to the hausdorff distance furthermore gromov proved in g2 that the sum of betti numbers of any element in x is uniformly bounded in terms of the given constants these results suggest that x is an object of study on which we could develop some geometry and topology in fact when volumes are uniformly bounded away from zero the following results are known let v be the subset of x with volume vol m v grove and petersen gp1 proved that the set of homotopy types of elements in 4 v is finite related results are in y2 for pinching problems otsu shiohama and the author osy y3 obtained some differential sphere theorems in the class x v according to a recent paper gpw if n 3 4 then v contains at most finitely many diffeomorphism types let mi i 1 2 be a convergent sequence in a and x be the limit of them we say that mi collapses to x if the hausdorff dimension of x is less than n by the volume condition no collapsing occurs in 4 v from this point of view it is quite natural to ask what collapsing phenomena occur and what types of pinching theorems hold in the class x in this situation it seems difficult to determine the singularities of x some examples are given in section 1 in this paper we study collapsing phenomena in the case when x is a riemannian manifold and establish a pinching theorem for a riemannian manifold m we denote by inj m the injectivity radius of m when n is written in place of x and the metric of n is normalized our

the critical contact process dies out by producing a finite box criterion for survival in a slab we show that the critical contact process dies out and that in two and more dimensions the critical parameter value coincides with the limit of slab thresholds using the techniques developed in this paper one may obtain the complete convergence and shape theorems for contact processes in all dimensions and for all infection rates our results apply also to the discrete time analogue of the contact process viz oriented percolation

supermartingales as radon nikodym densities and related measure extensions certain countably and finitely additive measures can be associated to a given nonnegative supermartingale under weak assumptions on the underlying probability space existence and non uniqueness results for such measures are proven

early fragmentation in the adaptive voter model on directed networks we consider voter dynamics on a directed adaptive network with fixed out degree distribution a transition between an active phase and a fragmented phase is observed this transition is similar to the undirected case if the networks are sufficiently dense and have a narrow out degree distribution however if a significant number of nodes with low out degree is present then fragmentation can occur even far below the estimated critical point due to the formation of self stabilizing structures that nucleate fragmentation this process may be relevant for fragmentation in current political opinion formation processes

predicting the ultimate supremum of a stable levy process with no negative jumps given a stable levy process x x t 0 t t of index alpha is an element of 1 2 with no negative jumps and letting s t sup 0 s t x s denote its running supremum for t is an element of 0 t we consider the optimal prediction problem