the perturbed compound poisson risk model with two sided jumps in this paper we consider a perturbed compound poisson risk model with two sided jumps the downward jumps represent the claims following an arbitrary distribution while the upward jumps are also allowed to represent the random gains assuming that the density function of the upward jumps has a rational laplace transform the laplace transforms and defective renewal equations for the discounted penalty functions are derived and the asymptotic estimate for the probability of ruin is also studied for heavy tailed downward jumps finally some explicit expressions for the discounted penalty functions as well as numerical examples are given

growth and decay of shock and acceleration waves in a traffic flow model with relaxation abstract the usual ficken based constitutive relation for traffic flux is replaced with one based on the maxwell cattaneo model the resulting flux law which now takes into account the reaction time of driver and vehicle results in a second order hyperbolic generalization of burgers equation as the pde governing the traffic density an analytical study of this equation is presented with an emphasis on shock and related kinematic wave phenomena specifically the exact traveling wave solution tws is derived and it is shown that shock formation is possible only if the diffusivity is non vanishing using singular surface theory exact amplitude expressions for both shock and acceleration waves are obtained and their temporal evolution determined the exact upper bound of the reaction time parameter is established and connections between singular surface and tws results are noted

the evolution laws of dilatational spherical and cylindrical weak nonlinear shock waves in elastic non conductors the theory of singular surfaces yields a set of coupled evolution equations for the shock amplitude and the amplitudes of the higher order discontinuities which accompany the shock to solve these equations we use perturbation methods with a perturbation parameter e characterising the initial shock amplitude it is shown that for decaying shock waves if the accompanying second order discontinuity is of order one the straightforward perturbation procedure yields uniformly valid solutions but if the accompanying second order discontinuity is of order e the method of multiple scales is needed in order to render the perturbation solutions uniformly valid with respect to the distance of travel we also construct shock wave solutions from modulated simple wave solutions which are obtained with the aid ofhunter keller s weakly nonlinear geometrical optics method the two approaches give exactly the same results within their common range of validity the explicit evolution laws thus obtained enable us to see clearly how weak nonlinear curved shock waves are attenuated because of the effects of geometry and material nonlinearity and on what length scale these effects are most pronounced

skewness and kurtosis in statistical kinetics we obtain lower and upper bounds on the skewness and kurtosis associated with the cycle completion time of unicyclic enzymatic reaction schemes analogous to a well known lower bound on the randomness parameter the lower bounds on skewness and kurtosis are related to the number of intermediate states in the underlying chemical reaction network our results demonstrate that evaluating these higher order moments with single molecule data can lead to information about the enzymatic scheme that is not contained in the randomness parameter

continuous dependence on the relaxation time and modeling and unbounded growth in theories of heat conduction with finite propagation speeds abstract the problem is studied of how the solution varies as the relaxation time is altered for hyperbolic heat conduction theories of maxwell cattaneo type we also investigate the behaviour of the maxwell cattaneo models as the relaxation time tends to zero for the improperly posed backward in time problem we show that the solution depends holder continuously on changes in the model finally a model of heat propagation is studied which allows for three relaxation times we show that under a large class of conditions for the coefficients this theory always allows choices of initial data for which solutions are unstable

monotonic continuous time random walks with drift and stochastic reset events in this paper we consider a stochastic process that may experience random reset events which suddenly bring the system to the starting value and analyze the relevant statistical magnitudes we focus our attention on monotonic continuous time random walks with a constant drift the process increases between the reset events either by the effect of the random jumps or by the action of the deterministic drift as a result of all these combined factors interesting properties emerge like the existence for any drift strength of a stationary transition probability density function or the faculty of the model to reproduce power law like behavior general formulas for two extreme statistics the survival probability and the mean exit time are also derived to corroborate in an independent way the results of the paper monte carlo methods were used these numerical estimations are in full agreement with the analytical predictions

restart could optimize the probability of success in a bernoulli trial the recently noticed ability of restart to reduce the expected completion time of first passage processes allows appealing opportunities for performance improvement in a variety of settings however complex stochastic processes often exhibit several possible scenarios of completion which are not equally desirable in terms of efficiency here we show that restart may have profound consequences on the splitting probabilities of a bernoulli like first passage process i e of a process which can end with one of two outcomes particularly intriguing in this respect is the class of problems where a carefully adjusted restart mechanism maximizes the probability that the process will complete in a desired way we reveal the universal aspects of this kind of optimal behavior by applying the general approach recently proposed for the problem of first passage under restart

beyond the poisson renewal process a tutorial survey after sketching the basic principles of renewal theory and recalling the classical poisson process we discuss two renewal processes characterized by waiting time laws with the same power asymptotics defined by special functions of mittag leffler and of wright type we compare these three processes with each other

search in unknown random environments n searchers are sent out by a source in order to locate a fixed object which is at a finite distance d but the search space is infinite and d would be in general unknown each of the searchers has a finite random lifetime and may be subject to destruction or failures and it moves independently of other searchers and at intermediate locations some partial random information may be available about which way to go when a searcher is destroyed or disabled or when it dies naturally after some time the source becomes aware of this and it sends out another searcher which proceeds similarly to the one that it replaces the search ends when one of the searchers finds the object being sought we use n coupled brownian motions to derive a closed form expression for the average search time as a function of d which will depend on the parameters of the problem the number of searchers the average lifetime of searchers the routing uncertainty and the failure or destruction rate of searchers we also examine the cost in terms of the total energy that is expended in the search

diffusion of epicenters of earthquake aftershocks omori s law and generalized continuous time random walk models the epidemic type aftershock sequence etas model is a simple stochastic process modeling seismicity based on the two best established empirical laws the omori law power law decay ensuremath sim 1 t 1 ensuremath theta of seismicity after an earthquake and gutenberg richter law power law distribution of earthquake energies in order to describe also the space distribution of seismicity we use in addition a power law distribution ensuremath sim 1 r 1 ensuremath mu of distances between triggered and triggering earthquakes the etas model has been studied for the last two decades to model real seismicity catalogs and to obtain short term probabilistic forecasts here we present a mapping between the etas model and a class of ctrw continuous time random walk models based on the identification of their corresponding master equations this mapping allows us to use the wealth of results previously obtained on anomalous diffusion of ctrw after translating into the relevant variable for the etas model we provide a classification of the different regimes of diffusion of seismic activity triggered by a mainshock specifically we derive the relation between the average distance between aftershocks and the mainshock as a function of the time from the mainshock and of the joint probability distribution of the times and locations of the aftershocks the different regimes are fully characterized by the two exponents ensuremath theta and ensuremath mu our predictions are checked by careful numerical simulations we stress the distinction between the bare omori law describing the seismic rate activated directly by a mainshock and the renormalized omori law taking into account all possible cascades from mainshocks to aftershocks of aftershock of aftershock and so on in particular we predict that seismic diffusion or subdiffusion occurs and should be observable only when the observed omori exponent is less than 1 because this signals the operation of the renormalization of the bare omori law also at the origin of seismic diffusion in the etas model we present predictions and insights provided by the etas to ctrw mapping which suggest different ways for studying seismic catalogs finally we discuss the present evidence for our predicted subdiffusion of seismicity triggered by a main shock stressing the caveats and limitations of previous empirical works

intermittent search strategies revisited effect of the jump length and biased motion we study the kinetics of a search of a single fixed target by a large number of searchers performing an intermittent biased random walk in a homogeneous medium our searchers carry out their walks in one of two states between which they switch randomly one of these states search phase is a nearest neighbor walk characterized by the probability of stepping in a given direction i e the walks in this state are not necessarily isotropic the other relocation phase is characterized by the length of the jumps i e when in this state a walker does not perform a nearest neighbor walk within such a framework we propose a model to describe the searchers dynamics generalizing results of our previous work we have obtained and numerically evaluated analytic results for the mean number of distinct sites visited up to a maximum evolution time we have studied the dependence of this quantity on both the transition probability between the states and the parameters that characterize each state in addition to our theoretical approach we have implemented monte carlo simulations finding excellent agreement between the theoretical numerical and simulations results

optimal random search for a single hidden target a single target is hidden at a location chosen from a predetermined probability distribution then a searcher must find a second probability distribution from which random search points are sampled such that the target is found in the minimum number of trials here it will be shown that if the searcher must get very close to the target to find it then the best search distribution is proportional to the square root of the target distribution regardless of dimension for a gaussian target distribution the optimum search distribution is approximately a gaussian with a standard deviation that varies inversely with how close the searcher must be to the target to find it for a network where the searcher randomly samples nodes and looks for the fixed target along edges the optimum is either to sample a node with probability proportional to the square root of the out degree plus 1 or not to do so at all

an algebraic approach to h output feedback control problems abstract we derive an output feedback controller which stabilizes a system and satisfies a prescribed h norm bound of the closed loop transfer function the proposed design method is available for any system i e there are no restrictions on d12 and d21 this approach utilizes only algebraic operations thus proofs are simple and clear

optimal first arrival times in levy flights with resetting we consider the diffusive motion of a particle performing a random walk with levy distributed jump lengths and subject to a resetting mechanism bringing the walker to an initial position at uniformly distributed times in the limit of an infinite number of steps and for long times the process converges to superdiffusive motion with replenishment we derive a formula for the mean first arrival time mfat to a predefined target position reached by a meandering particle and we analyze the efficiency of the proposed searching strategy by investigating criteria for an optimal a shortest possible mfat

front propagation into unstable states universal algebraic convergence towards uniformly translating pulled fronts fronts that start from a local perturbation and propagate into a linearly unstable state come in two classes pulled fronts and pushed fronts the term pulled front expresses that these fronts are pulled along by the spreading of linear perturbations about the unstable state accordingly their asymptotic speedv equals the spreading speed of perturbations whose dynamics is governed by the equations linearized about the unstable state the central result of this paper is the analysis of the convergence of asymptotically uniformly traveling pulled fronts towardsv we show that when such fronts evolve from sufficiently steep initial conditions which initially decay faster than e x forx 1 t hey have auniversal relaxation behavioras timet 1 the velocity of a pulled front always relaxes algebraically like v t d v 3 2t c 3 p d d 2 t 3 2 c o 1 t 2 the parametersv and d are determined through a saddle point analysis from the equation of motion linearized about the unstable invaded state this front velocity is independent of the precise value of the front amplitude which one tracks to measure the front position the interior of the front is essentially slaved to the leading edge and develops universally as x t d8v t x rt dt 0 v t 0 c o 1 t 2 where8v x vt is a uniformly translating front solution with velocity v v our result which can be viewed as a general center manifold result for pulled front propagation is derived in detail for the well known nonlinear diffusion equation of type t d 2 x c 3 where the invaded unstable state is d 0 even for this simple case the subdominant t 3 2 term extends an earlier result of bramson our analysis is then generalized to more general sets of partial differential equations with higher spatial or temporal derivatives to pdes with memory kernels and also to difference equations such as those that occur in numerical finite difference codes our universal result for pulled fronts thus implies independence i of the level curve which is used to track the front position ii of the precise nonlinearities iii of the precise form of the linear operators in the dynamical equation and iv of the precise initial conditions as long as they are sufficiently steep the only remainders of the explicit form of the dynamical equation are the nonlinear solutions 8v and the three saddle point parameters v and d as our simulations confirm all our analytical predictions in every detail it can be concluded that we have a complete analytical understanding of the propagation mechanism and relaxation behavior of pulled fronts if they are uniformly translating for t 1 an immediate consequence of the slow algebraic relaxation is that the standard moving boundary approximation breaks down for weakly curved pulled fronts in two or three dimensions in addition to our main result for pulled fronts we also discuss the propagation and convergence of fronts emerging from

asymmetric inclusion process we introduce and explore the asymmetric inclusion process asip an exactly solvable bosonic counterpart of the fermionic asymmetric exclusion process asep in both processes random events cause particles to propagate unidirectionally along a one dimensional lattice of n sites in the asep particles are subject to exclusion interactions whereas in the asip particles are subject to inclusion interactions that coalesce them into inseparable clusters we study the dynamics of the asip derive evolution equations for the mean and probability generating function pgf of the sites occupancy vector obtain explicit results for the above mean at steady state and describe an iterative scheme for the computation of the pgf at steady state we further obtain explicit results for the load distribution in steady state with the load being the total number of particles present in all lattice sites finally we address the problem of load optimization and solve it under various criteria the asip model establishes bridges between statistical physics and queueing theory as it represents a tandem array of queueing systems with unlimited batch service and a tandem array of growth collapse processes

spectral content of fractional brownian motion with stochastic reset we analyse the power spectral density psd s t f with t being the observation time and f the frequency of a fractional brownian motion fbm with an arbitrary hurst index undergoing a stochastic resetting to the origin at a constant rate r the resetting process introduced some time ago as an example of an efficient optimisable search algorithm to this end we first derive an exact expression for the covariance function of an arbitrary not necessarily a fbm process with a reset expressing it through the covariance function of the parental process without a reset which yields the desired result for the fbm in a particular case we then use this result to compute exactly the power spectral density for fbm for all frequency f the asymptotic large frequency f behaviour of the psd turns out to be distinctly different for sub and super diffusive fbms we show that for large f the psd has a power law tail where the exponent for sub diffusive fbm while for all thus somewhat unexpectedly the exponent in the superdiffusive case h 1 2 sticks to its brownian value and does not depend on h

disturbance attenuation and h sub infinity control via measurement feedback in nonlinear systems a solution to the problem of disturbance attenuation via measurement feedback with internal stability is presented for an affine nonlinear system it is shown that the concept of disturbance attenuation in the sense of truncated l sub 2 norms can be given an interpretation in terms of the response to periodic inputs in the sense of rms amplitude even in the nonlinear setup in the case of state feedback a family of controllers is also provided the proofs of all these results are simple and provide deeper insight even in the analysis of the corresponding linear control problem

l sub 2 gain analysis of nonlinear systems and nonlinear state feedback h sub infinity control previously obtained results on l2 gain analysis of smooth nonlinear systems are unified and extended using an approach based on hamilton jacobi equations and inequalities and their relation to invariant manifolds of an associated hamiltonian vector field on the basis of these results a nonlinear analog is obtained of the simplest part of a state space approach to linear h sub infinity control namely the state feedback h sub infinity optimal control problem furthermore the relation with h sub infinity control of the linearized system is dealt with

robust random search with scale free stochastic resetting a new model of search based on stochastic resetting is introduced wherein rate of resets depends explicitly on time elapsed since the beginning of the process it is shown that rate inversely proportional to time leads to paradoxical diffusion which mixes self similarity and linear growth of the mean square displacement with nonlocality and non gaussian propagator it is argued that such resetting protocol offers a general and efficient search boosting method that does not need to be optimized with respect to the scale of the underlying search problem e g distance to the goal and is not very sensitive to other search parameters both subdiffusive and superdiffusive regimes of the mean squared displacement scaling are demonstrated with more general rate functions

on the emergence of a power law in the distribution of covid 19 cases the first confirmed case of coronavirus disease 2019 covid 19 in the us was reported on january 21 2020 by the end of march 2020 there were more than 180 000 confirmed cases in the us distributed across more than 2000 counties we find that the right tail of this distribution exhibits a power law with pareto exponent close to one we investigate whether a simple model of the growth of covid 19 cases involving gibrat s law can explain the emergence of this power law the model is calibrated to match i the growth rates of confirmed cases and ii the varying lengths of time during which covid 19 had been present within each county thus calibrated the model generates a power law with pareto exponent nearly exactly equal to the exponent estimated directly from the distribution of confirmed cases across counties at the end of march

non gaussianity and dynamical trapping in locally activated random walks we propose a minimal model of locally activated diffusion in which the diffusion coefficient of a onedimensional brownian particle is modified in a prescribed way either increased or decreased upon each crossing of the origin such a local mobility decrease arises in the formation of atherosclerotic plaques due to diffusing macrophage cells accumulating lipid particles we show that spatially localized mobility perturbations have remarkable consequences on diffusion at all scales such as the emergence of a non gaussian multipeaked probability distribution and a dynamical transition to an absorbing static state in the context of atherosclerosis this dynamical transition can be viewed as a minimal mechanism that causes macrophages to aggregate in lipid enriched regions and thereby to the formation of atherosclerotic plaques

weakly nonlinear high frequency waves abstract in this paper we derive a method for finding small amplitude high frequency solutions to hyperbolic systems of quasilinear partial differential equations our solution is the sum of two parts i a superposition of small amplitude high frequency waves ii a slowly varying mean solution each high frequency wave displays nonlinear distortion of the wave profile and shocks may form shock conditions are derived for conservative systems different high frequency waves do not interact provided the frequencies and wave numbers of two waves are not linearly related to those of a third the mean solution is found by solving a linear partial differential equation this method generalizes whitham s nonlinearization technique 9 for single waves to problems where many waves are present we obtain these results by generalizing a scheme first proposed by choquet bruhat 1 which employs the method of multiple scales author

brief paper disturbance attenuation and trajectory tracking via a reduced order output feedback controller for robot manipulators this paper presents a solution to the problem of trajectory tracking with external disturbance attenuation in robotics systems via a reduced order output feedback controller without velocity measurement the proposed control law ensures both semi global asymptotic stability of the closed loop system and external disturbance attenuation the approach is based on the notion of the l 2 gain and requires to solve two algebraic riccati inequalities the proposed controller design is illustrated by a simulation example

a re examination of the basic postulates of thermomechanics this paper is mainly concerned with a re examination of the basic postulates and the consequent procedure for the construction of the constitutive equations of material behaviour in thermomechanics however the implication of the basic postulates and the significance of the related procedure for the development of the constitutive equations is also illustrated in some detail in the context of flow of heat in a rigid solid with particular reference to the propagation of thermal waves at finite speed more specifically after briefly examining the nature of the basic equations of motion for a system of particles within the scope of the classical newtonian mechanics the basic postulates of the purely mechanical theory for a continuum including its specialization for a rigid body is re examined this includes some differences from the usual procedure on the subject next thermal variables are introduced and after observing a useful analogy between the thermal and mechanical variables a discussion of a theory of heat or a purely thermal theory is provided which differs from the usual development in the classical thermodynamics a detailed application of the latter development is then made to the problem of heat flow in a stationary rigid solid using several different and well motivated constitutive equations special cases of these include linearized theories of the classical heat flow by conduction and of heat flow transmitted as thermal waves the remainder of the paper is concerned with thermal mechanical theory of deformable media along with discussions of a number of related issues on the subject

spatial behavior of anomalous transport we present a general derivation of one dimensional spatial concentration distributions for anomalous transport regimes such transport can be captured in the framework of a continuous time random walk with a broad transition time distribution this general theory includes a fokker planck equation as a particular limiting case all of the concentration profiles as well as the associated temporal first passage time distributions can be written in terms of a single special function that belongs to the class of fox functions in addition we consider the first two moments of the spatial concentration distributions and determine not only their scaling behavior with time but also the coefficients and correction terms

the evolution of travelling waves in the weakly hyperbolic generalized fisher model in this paper we consider an initial boundary value problem for a weakly hyperbolic generalized fisher equation of order p 0 we show that for each p 0 the large time structure of the solution involves the evolution of a propagating wavefront however we demonstrate that the case p 1 is a bifurcation point in the sense that for p 1 the wavefront is of reaction diffusion type while for 0 p

growth decay and bifurcation of shock amplitudes under the type ii flux law by replacing fick s diffusion law with green and nagdhi s type ii flux law a hyperbolic counterpart to the classical fisherkpp equation is obtained in this article an analytical study of this pa

optimal search behavior and classic foraging theory random walk methods and diffusion theory pervaded ecological sciences as methods to analyze and describe animal movement consequently statistical physics was mostly seen as a toolbox rather than as a conceptual framework that could contribute to theory on evolutionary biology and ecology however the existence of mechanistic relationships and feedbacks between behavioral processes and statistical patterns of movement suggests that beyond movement quantification statistical physics may prove to be an adequate framework to understand animal behavior across scales from an ecological and evolutionary perspective recently developed random search theory has served to critically re evaluate classic ecological questions on animal foraging for instance during the last few years there has been a growing debate on whether search behavior can include traits that improve success by optimizing random stochastic searches here we stress the need to bring together the general encounter problem within foraging theory as a mean for making progress in the biological understanding of random searching by sketching the assumptions of optimal foraging theory oft and by summarizing recent results on random search strategies we pinpoint ways to extend classic oft and integrate the study of search strategies and its main results into the more general theory of optimal foraging

continuous time random walk model for financial distributions we apply the formalism of the continuous time random walk to the study of financial data the entire distribution of prices can be obtained once two auxiliary densities are known these are the probability densities for the pausing time between successive jumps and the corresponding probability density for the magnitude of a jump we have applied the formalism to data on the u s dollar deutsche mark future exchange finding good agreement between theory and the observed data

multiplicative processes and power laws takayasu sato and takayasu phys rev lett 79 966 1997 revisited the question of stochastic processes with multiplicative noise which have been studied in several different contexts over the past decades we focus on the regime found for a generic set of control parameters in which stochastic processes with multiplicative noise produce intermittency of a special kind characterized by a power law probability density distribution we briefly explain the physical mechanism leading to a power law probability distribution function and provide a list of references for these results dating back from a quarter of century we explain how the formulation in terms of the characteristic function developed by takayasu sato and takayasu can be extended to exponents ensuremath mu g2 which explains the reason for the lucky coincidence the multidimensional generalization of the results of takayasu sato and takayasu and the present status of the problem are briefly summarized the discovery of stretched exponential tails in the presence of the cutoff introduced by takayasu sato and takayasu is explained theoretically we end by briefly listing applications

uncoupled continuous time random walks solution and limiting behavior of the master equation a detailed study is presented for a large class of uncoupled continuous time random walks ctrws the master equation is solved for the mittag leffler survival probability the properly scaled diffusive limit of the master equation is taken and its relation with the fractional diffusion equation is discussed finally some common objections found in the literature are thoroughly reviewed

comparison theorems and variable speed waves for a scalar reaction diffusion equation this paper concerns the reaction diffusion equation u t u xx u 2 1 u previous numerical solutions of this equation have demonstrated various different types of wave front solutions generated by different initial conditions in this paper the authors use a phase plane form of comparison theorems for partial differential equations pdes to confirm analytically these numerical results in particular they show that initial conditions with an exponentially decaying tail evolve to the unique exponentially decaying travelling wave while initial conditions with algebraically decaying tails evolve either to an algebraically decaying travelling wave or to the exponentially decaying wave or to a perpetually accelerating wave dependent upon the exact form of the decay of the initial conditions we then focus on the case of accelerating waves and investigate their form in more detail by approximating the full equation in this case with a hyperbolic pde which we solve using the method of characteristics we use this approximate solution to derive a leading order approximation to the wave speed

stochastic multiplicative processes with reset events we study a stochastic multiplicative process with reset events it is shown that the model develops a stationary power law probability distribution for the relevant variable whose exponent depends on the model parameters two qualitatively different regimes are observed corresponding to intermittent and regular behavior in the boundary between them the mean value of the relevant variable is time independent and the exponent of the stationary distribution equals ensuremath 2 the addition of diffusion to the system modifies in a nontrivial way the profile of the stationary distribution numerical and analytical results are presented

fractal random walks we consider a class of random walks on lattices and in continuous spaces having infinite mean squared displacement per step the probability distribution functions considered generate fractal self similar trajectories the characteristic functions structure functions of the walks are nonanalytic functions and satisfy scaling equations

first passages in bounded domains when is the mean first passage time meaningful we study the first passage statistics to adsorbing boundaries of a brownian motion in bounded two dimensional domains of different shapes and configurations of the adsorbing and reflecting boundaries from extensive numerical analysis we obtain the probability p ensuremath omega distribution of the random variable ensuremath omega ensuremath tau 1 ensuremath tau 1 ensuremath tau 2 which is a measure for how similar the first passage times ensuremath tau 1 and ensuremath tau 2 are of two independent realizations of a brownian walk starting at the same location we construct a chart for each domain determining whether p ensuremath omega represents a unimodal bell shaped form or a bimodal m shaped behavior while in the former case the mean first passage time mfpt is a valid characteristic of the first passage behavior in the latter case it is an insufficient measure for the process strikingly we find a distinct turnover between the two modes of p ensuremath omega characteristic for the domain shape and the respective location of absorbing and reflective boundaries our results demonstrate that large fluctuations of the first passage times may occur frequently in two dimensional domains rendering quite vague the general use of the mfpt as a robust measure of the actual behavior even in bounded domains in which all moments of the first passage distribution exist

lotka volterra systems with stochastic resetting we study the dynamics of predator prey systems where prey are confined to a single region of space and where predators move randomly according to a power law l vy dispersal kernel site fidelity an important feature of animal behavior is incorporated in the model through stochastic resetting dynamics of the predators to the prey patch we solve in the long time limit the lotka volterra rate equations that describe the evolution of the two species densities fixing the demographic parameters and the l vy exponent the total population of predators can be maximized for a certain value of the resetting rate this optimal value achieves a compromise between the overexploitation and underutilization of the habitat similarly at a fixed resetting rate there exists a l vy exponent that is optimal regarding predator abundance these findings are supported by 2d stochastic simulations and show that the combined effects of diffusion and resetting can broadly extend the region of species coexistence in ecosystems facing resource scarcity

h sub spl infin compensator design with minimal order observers a method based on a bounded real lemma is presented in the form of sufficient conditions for designing observer based proper reduced order h sub spl infin compensators the proposed compensators are shown to include the strictly proper full order central h sub spl infin compensators as a special class a compensator design algorithm is included to illustrate the use of the proposed compensator design equations

subdiffusive continuous time random walks with stochastic resetting we analyze two models of subdiffusion with stochastic resetting each of them consists of two parts subdiffusion based on the continuous time random walk scheme and independent resetting events generated uniformly in time according to the poisson point process in the first model the whole process is reset to the initial state whereas in the second model only the position is subject to resets the distinction between these two models arises from the non markovian character of the subdiffusive process we derive exact expressions for the two lowest moments of the full propagator stationary distributions and first hitting time statistics we also show with an example of a constant drift how these models can be generalized to include external forces possible applications to data analysis and modeling of biological systems are also discussed

continuous time random walks under power law resetting we study continuous time random walks ctrw with power law distribution of waiting times under resetting which brings the walker back to the origin with a power law distribution of times between the resetting events two situations are considered under complete resetting the ctrw after the resetting event starts anew with a new waiting time independent of the prehistory under incomplete resetting the resetting of the coordinate does not influence the waiting time until the next jump we focus on the behavior of the mean squared displacement msd of the walker from its initial position on the conditions under which the probability density functions of the walker s displacement show universal behavior and on this universal behavior itself we show that the behavior of the msd is the same as in the scaled brownian motion sbm being the mean field model of the ctrw the intermediate asymptotics of the probability density functions pdf for ctrw under complete resetting provided they exist are also the same as in the corresponding case for sbm for incomplete resetting however the behavior of the pdf for ctrw and sbm is vastly different

properties of additive functionals of brownian motion with resetting we study the distribution of additive functionals of reset brownian motion a variation of normal brownian motion in which the path is interrupted at a given rate and placed back to a given reset position our goal is two fold 1 for general functionals we derive a large deviation principle in the presence of resetting and identify the large deviation rate function in terms of a variational formula involving large deviation rate functions without resetting 2 for three examples of functionals positive occupation time area and absolute area we investigate the effect of resetting by computing distributions and moments using a formula that links the generating function with resetting to the generating function without resetting

first passage times for random walks in bounded domains we present a novel computational method of first passage times between a starting site and a target site of regular bounded lattices we derive accurate expressions for all the moments of this first passage time validated by numerical simulations their range of validity is discussed we also consider the case of a starting site and two targets in addition we present the extension to continuous brownian motion these results are of great relevance to any system involving diffusion in confined media

effects of refractory period on stochastic resetting we consider a stochastic process undergoing resetting after which a random refractory period is imposed in this period the process is quiescent and remains at the resetting position using a first renewal approach we compute exactly the stationary position distribution and analyse the emergence of a delta peak at the resetting position in the case of a power law distribution for the refractory period we find slow relaxation we generalise our results to the case when the resetting period and the refractory period are correlated by computing the laplace transform of the survival probability of the process and the mean first passage time i e the mean time to completion of a task we also compute exactly the joint distribution of the active and absorption time to a fixed target

diffusion with resetting in bounded domains we consider the one dimensional diffusion in a bounded domain with stochastic resetting we start our analysis by presenting a method to derive the master equation for different resetting mechanisms in the next step we compute the non equilibrium steady state for a special case of this differential equation then we consider the existence of an absorbing point in the system and calculate the mean time to absorption of the diffusive particle by the target numerical and analytical calculations of the optimal resetting rate reveal a second order phase transition finally we discuss different special cases of the presented problem

optimization in first passage resetting we investigate classic diffusion with the added feature that a diffusing particle is reset to its starting point each time the particle reaches a specified threshold in an infinite domain this process is nonstationary and its probability distribution exhibits rich features in a finite domain we define a nontrivial optimization in which a cost is incurred whenever the particle is reset and a reward is obtained while the particle stays near the reset point we derive the condition to optimize the net gain in this system namely the reward minus the cost

slow levy flights among markovian processes the hallmark of levy flights is superdiffusion or faster than brownian dynamics here we show that levy laws as well as gaussian distributions can also be the limit distributions of processes with long range memory that exhibit very slow diffusion logarithmic in time these processes are path dependent and anomalous motion emerges from frequent relocations to already visited sites we show how the central limit theorem is modified in this context keeping the usual distinction between analytic and nonanalytic characteristic functions a fluctuation dissipation relation is also derived our results may have important applications in the study of animal and human displacements

optimal stochastic restart renders fluctuations in first passage times universal stochastic restart may drastically reduce the expected run time of a computer algorithm expedite the completion of a complex search process or increase the turnover rate of an enzymatic reaction these diverse first passage time fpt processes seem to have very little in common but it is actually quite the other way around here we show that the relative standard deviation associated with the fpt of an optimally restarted process i e one that is restarted at a constant nonzero rate which brings the mean fpt to a minimum is always unity we interpret further generalize and discuss this finding and the implications arising from it

first passage under stochastic resetting in an interval we consider a brownian particle diffusing in a one dimensional interval with absorbing end points we study the ramifications when such motion is interrupted and restarted from the same initial configuration we provide a comprehensive study of the first passage properties of this trapping phenomena we compute the mean first passage time and derive the criterion on which restart always expedites the underlying completion we show how this set up is a manifestation of a success failure problem we obtain the success and failure rates and relate them with the splitting probabilities namely the probability that the particle will eventually be trapped on either of the boundaries without hitting the other one numerical studies are presented to support our analytic results

proteins coexistence of stability and flexibility we introduce an equation for protein native topology based on recent analysis of data from the protein data bank and on a generalization of the landau peierls instability criterion for fractals the equation relates the protein fractal dimension df the spectral dimension ds and the number of amino acids n deviations from the equation may render a protein unfolded the fractal nature of proteins is shown to bridge their seemingly conflicting properties of stability and flexibility over 500 proteins have been analyzed df ds and n and found to obey this equation of state

anomalous diffusion under stochastic resettings a general approach we present a general formulation of the resetting problem which is valid for any distribution of resetting intervals and arbitrary underlying processes we show that in such a general case a stationary distribution may exist even if the reset free process is not stationary as well as a significant decreasing in the mean first passage time we apply the general formalism to anomalous diffusion processes which allow simple and explicit expressions for poissonian resetting events

path integral formalism for stochastic resetting exactly solved examples and shortcuts to confinement we study the dynamics of overdamped brownian particles diffusing in conservative force fields and undergoing stochastic resetting to a given location at a generic space dependent rate of resetting we present a systematic approach involving path integrals and elements of renewal theory that allows us to derive analytical expressions for a variety of statistics of the dynamics such as i the propagator prior to first reset ii the distribution of the first reset time and iii the spatial distribution of the particle at long times we apply our approach to several representative and hitherto unexplored examples of resetting dynamics a particularly interesting example for which we find analytical expressions for the statistics of resetting is that of a brownian particle trapped in a harmonic potential with a rate of resetting that depends on the instantaneous energy of the particle we find that using energy dependent resetting processes is more effective in achieving spatial confinement of brownian particles on a faster time scale than performing quenches of parameters of the harmonic potential

analysis of reflected diffusions via an exponential time based transformation let x t be a time homogeneous diffusion process with state space 0 infty where 0 is a reflecting or entrance endpoint and let z denote a random variable that describes the process x t evaluated at an exponentially distributed random time we propose a method to obtain closed form expressions for the conditional density and the mean of a new diffusion process y t with the same state space and with the same infinitesimal variance whose drift depends on the infinitesimal moments of x t and on the hazard rate function of z this method also allows us to obtain the laplace transform of the first passage time density of y t through a lower constant boundary we then discuss the relation between y t and the process x t subject to catastrophes as well as the interpretation of y t as a diffusion in a decreasing potential we study in detail some special cases concerning diffusion processes obtained when x t is the wiener ornstein uhlenbeck bessel and rayleigh process

a note on discontinuity waves in type iii thermoelasticity two recent nonlinear theories of thermoelasticity developed by green and naghdi are examined it is shown that in type ii theory second sound is permissible and both mechanical and temperature waves may propagate in type iii theory we show that the situation is more analogous to that in classical nonlinear thermoelasticity one wave propagates and a homothermal temperature wave is allowed

diffusion with optimal resetting we consider the mean time to absorption by an absorbing target of a diffusive particle with the addition of a process whereby the particle is reset to its initial position with rate r we consider several generalisations of the model of m r evans and s n majumdar 2011 diffusion with stochastic resetting phys rev lett 106 160601 i a space dependent resetting rate r x ii resetting to a random position z drawn from a resetting distribution p z iii a spatial distribution for the absorbing target pt x as an example of i we show that the introduction of a non resetting window around the initial position can reduce the mean time to absorption provided that the intial position is sufficiently far from the target we address the problem of optimal resetting that is minimising the mean time to absorption for a given target distribution for an exponentially decaying target distribution centred at the origin we show that a transition in the optimal resetting distribution occurs as the target distribution narrows

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

first passage statistics under stochastic resetting in bounded domains we investigate the first passage problem where a diffusive searcher stochastically resets to a fixed position at a constant rate in a bounded domain we put forward an analytical framework for this

michaelis menten reaction scheme as a unified approach towards the optimal restart problem we study the effect of restart and retry on the mean completion time of a generic process the need to do so arises in various branches of the sciences and we show that it can naturally be addressed by taking advantage of the classical reaction scheme of michaelis and menten stopping a process in its midst only to start it all over again may prolong leave unchanged or even shorten the time taken for its completion here we are interested in the optimal restart problem i e in finding a restart rate which brings the mean completion time of a process to a minimum we derive the governing equation for this problem and show that it is exactly solvable in cases of particular interest we then continue to discover regimes at which solutions to the problem take on universal details independent forms which further give rise to optimal scaling laws the formalism we develop and the results obtained can be utilized when optimizing stochastic search processes and randomized computer algorithms an immediate connection with kinetic proofreading is also noted and discussed

dynamical transition in the temporal relaxation of stochastic processes under resetting a stochastic process when subject to resetting to its initial condition at a constant rate generically reaches a nonequilibrium steady state we study analytically how the steady state is approached in time and find an unusual relaxation mechanism in these systems we show that as time progresses an inner core region around the resetting point reaches the steady state while the region outside the core is still transient the boundaries of the core region grow with time as power laws at late times with new exponents alternatively at a fixed spatial point the system undergoes a dynamical transition from the transient to the steady state at a characteristic space dependent timescale t x we calculate analytically in several examples the large deviation function associated with this spatiotemporal fluctuation and show that generically it has a second order discontinuity at a pair of critical points characterizing the edges of the inner core these singularities act as separatrices between typical and atypical trajectories our results are verified in the numerical simulations of several models such as simple diffusion and fluctuating one dimensional interfaces

resetting processes with noninstantaneous return we consider a random two phase process which we call a reset return one the particle starts its motion at the origin the first displacement phase corresponds to a stochastic motion of a particle and is finished at a resetting event the second return phase corresponds to the particle s motion toward the origin from the position it attained at the end of the displacement phase this motion toward the origin takes place according to a given equation of motion the whole process is a renewal one we provide general expressions for the stationary probability density function of the particle s position and for the mean hitting time in one dimension we perform explicit analysis for the brownian motion during the displacement phase and three different types of the return motion return at a constant speed return at a constant acceleration with zero initial speed and return under the action of a harmonic force we assume that the waiting times for resetting events follow an exponential distribution or that resetting takes place after a fixed waiting period for the first two types of return motion and the exponential resetting the stationary probability density function of the particle s position is invariant under return speed acceleration while no such invariance is found for deterministic resetting and for exponential resetting with return under the action of the harmonic force we discuss necessary conditions for such invariance of the stationary pdf of the positions with respect to the properties of the return process and we demonstrate some additional examples when this invariance does or does not take place

a continuous time ehrenfest model with catastrophes and its jump diffusion approximation we consider a continuous time ehrenfest model defined over the integers from n to n and subject to catastrophes occurring at constant rate the effect of each catastrophe istantaneously resets the process to state 0 we investigate both the transient and steady state probabilities of the above model further the first passage time through state 0 is discussed we perform a jump diffusion approximation of the above model which leads to the ornstein uhlenbeck process with catastrophes the underlying jump diffusion process is finally studied with special attention to the symmetric case arising when the ehrenfest model has equal upward and downward transition rates

on the growth and decay of acceleration waves in random media we study the effects of material spatial randomness on the growth to shock or decay of acceleration waves in the deterministic formulation such waves are governed by a bernoulli equation d d x x x 2 in which the material coefficients and represent the dissipation and elastic nonlinearity respectively in the case of a random microstructure the wavefront sees the local details it is a mesoscale window travelling through a random continuum upon a stochastic generalization of the bernoulli equation both coefficients become stationary random processes and the critical amplitude c as well as the distance to form a shock x become random variables we study the character of these variables especially as compared to the deterministic setting for various cases of the random process i one white noise ii two independent white noises iii two correlated gaussian noises and iv an ornstein uhlenbeck process situations of fully positively negatively or zero correlated noises in and are investigated in detail particular attention is given to the determination of the average critical amplitude c equations for the evolution of the moments of the probability of formation of a shock wave within a given distance x and the average distance to form a shock wave specific comparisons of these quantities are made with reference to a homogeneous medium defined by the mean values of the x process

large deviations for markov processes with resetting markov processes restarted or reset at random times to a fixed state or region in space have been actively studied recently in connection with random searches foraging and population dynamics here we study the large deviations of time additive functions or observables of markov processes with resetting by deriving a renewal formula linking generating functions with and without resetting we are able to obtain the rate function of such observables characterizing the likelihood of their fluctuations in the long time limit we consider as an illustration the large deviations of the area of the ornstein uhlenbeck process with resetting other applications involving diffusions random walks and jump processes with resetting or catastrophes are discussed

one dimensional shock waves in simple materials with memory asymptotic evolution laws for plane dilatational shock waves travelling in simple materials with memory are derived in this paper by using two approximation methods the first method is a combination of singular surface theory and perturbation methods a system of two coupled first order ordinary differential equations is derived for the shock amplitude and the amplitude of the accompanying second order discontinuity the shock amplitude is assumed to be small but the accompanying second order discontinuity may be taken either to be finite or to be small with the shock amplitude the first case corresponds to the situation in which the duration time of the applied load is small compared with the viscous relaxation time and we show that the evolutionary behaviour of the two discontinuities is strongly affected by material nonlinearity the second case however corresponds to the situation in which the duration time is comparable with the viscous relaxation time and we are able to show that the evolutionary behaviour is as predicted by the linear theory of viscoelasticity in both cases the corresponding elastic results are obtained on allowing the viscous relaxation time to tend to infinity the second approximation method is the shock fitting method applied to a modulated simple wave theory which is itself an approximation based on a small amplitude finite rate assumption equivalent to the first case discussed above the two approximation methods are shown to yield the same evolution laws within their common range of validity

localized partial traps in diffusion processes and random walks reaction diffusion equations in which the reaction is described by a sink term consisting of a sum of delta functions are studied it is shown that the laplace transform of the reactive green s function can be analytically expressed in terms of the green s function for diffusion in the absence of reaction moreover a simple relation between the green s functions satisfying the radiation boundary condition and the reflecting boundary condition is obtained several applications are presented and the formalism is used to establish the relationship between the time dependent geminate recombination yield and the bimolecular reaction rate for diffusion influenced reactions finally an analogous development for lattice random walks is presented