A spatial model for dormancy in random environment
Helia Shafigh


Helia Shafigh]
A Spatial Model for Dormancy in Random Environment
    [
    September 9, 2024
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Helia Shafigh[WIAS Berlin, Mohrenstraße 39, 10117 Berlin, Germany, shafigh@wias-berlin.de]



WIAS Berlin







§ ABSTRACT 
In this paper, we introduce a spatial model for dormancy in random environment via a two-type branching random walk in continuous-time, where individuals can switch between dormant and active states through spontaneous switching independent of the random environment. However, the branching mechanism is governed by a random environment which dictates the branching rates. We consider three specific choices for random environments composed of particles: (1) a Bernoulli field of immobile particles, (2) one moving particle, and (3) a Poisson field of moving particles. In each case, the particles of the random environment can either be interpreted as catalysts, accelerating the branching mechanism, or as traps, aiming to kill the individuals. The different between active and dormant individuals is defined in such a way that dormant individuals are protected from being trapped, but do not participate in migration or branching.

We quantify the influence of dormancy on the growth resp. survival of the population by identifying the large-time asymptotics of the expected population size. The starting point for our mathematical considerations and proofs is the parabolic Anderson model via the Feynman-Kac formula. Especially, the quantitative investigation of the role of dormancy is done by extending the Parabolic Anderson model to a two-type random walk.  
 






Keywords and phrases. Parabolic Anderson model, dormancy, populations with seed-bank, branching random walk, Lyapunov exponents, Rayleigh-Ritz formula, switching diffusions, Feynman-Kac formula, large deviations for two-state Markov chains




§ INTRODUCTION AND MAIN RESULTS



 §.§ Biological Motivation
 Dormancy is an evolutionary trait that has developed independently across various life forms and is particularly common in microbial communities. To give a definition, we follow <cit.> and refer to dormancy as the ability of individuals to enter a reversible state of minimal metabolic activity. The collection of all dormant individuals within a population is also often called a seed-bank. Maintaining a seed-bank leads to a decline in the reproduction rate, but it also reduces the need for resources, making dormancy a viable strategy during unfavourable periods. Initially studied in plants as a survival strategy (cf. <cit.>), dormancy is now recognized as a prevalent trait in microbial communities with significant evolutionary, ecological, and pathogenic implications, serving as an efficient strategy to survive challenging environmental conditions, competitive pressure, or antibiotic treatment. However, it is at the same time a costly trait whose maintenance requires energy and
a sophisticated mechanisms for switching between active and dormant states. Moreover, the increased survival rate of dormant individuals must be weighed against their low reproductive activity. Despite its costs, dormancy still seems to provide advantages in variable environments. For a recent overview on biological dormancy and seed-banks we refer to <cit.>.


The existing stochastic models for dormancy can be roughly categorized into two approaches: population genetic models and population dynamic models. While the first approach assumes a constant population size and focusses on the genealogical implications of seed-banks, the latter is typically concerned with individual-based modelling through the theory of branching processes. Following a brief example in the book <cit.>, a two-type branching process (without migration) in a fluctuating random environment has been introduced in <cit.>, which served as a motivation for this paper. In <cit.>, the authors consider three different switching strategies between the two types (dormant and active), namely the stochastic (or: spontaneous; simultaneous) switching, responsive switching and anticipatory switching. In the latter two strategies, individuals adapt to the fluctuating environment by selecting their state (dormant or active) based on environmental conditions via e.g. an increased reproduction activity during beneficial phases and a more extensive seed-bank during unfavourable ones (resp. vice versa). In contrast, the stochastic switching strategy, which remains unaffected by environmental changes, proves especially advantageous during catastrophic events, as it, with high probability, ensures the existence of dormant individuals, which may contribute to the survival of the whole population, when a severely adverse environment might eradicate all active ones. As an example, it is estimated that more than 80% of soil bacteria are
metabolically inactive at any given time, forming extensive seed-banks of dormant individuals independent of the current conditions (cf. <cit.> and <cit.>). This makes the understanding of the stochastic switching strategy an interesting and important task. 



 §.§ Modelling Approach and Goals
  
The aim of this paper is to investigate the stochastic switching strategy in order to quantitatively compare the long-term behaviour of populations with and without this dormancy mechanism. 

Inspired by the Galton-Watson process with dormancy introduced in <cit.>, our first goal was to extend this model to a continuous-time spatial model on ℤ^d.It is worth noting that spatial models for dormancy have already been considered in the setting of population genetics (cf. <cit.>), where the population consists of different genetics types being inherited from parents to children. In such models, the total population size is fixed, so that the questions that arises are not about the extinction and survival of the whole population but rather about the evolution of the fraction of the different types. Notably, one of the goals in <cit.> is to determine criteria for co-existence resp. clustering of types in the limit of large population sizes. Another similar population genetics model, but this time with a (static) random environment, has been introduced in <cit.>, in which the authors investigate the influence of dormancy again on co-existence and clustering. To the best of our knowledge, corresponding spatial models for dormancy in the setting of population size models are still missing, such that the extension of the branching process in <cit.> seems to be a natural step. At the same time, there is a large repertoire of branching random walk models in random environment in the literature (cf. <cit.> for a survey), even though none of them incorporates dormancy. Hence, by introducing a continuous-time spatial model with migration, branching resp. extinction driven by a random environment, as well as a Markovian switching between the two states active and dormant, we bridge the gap between  (non-spatial) two-type branching processes with dormancy on one side, and spatial branching random walks in random environments (without dormancy) on the other side of the existing literature. Especially, our main interest lies in a quantitative comparison of the population size of our model to those of existing branching random walk models without dormancy. 


 §.§ Description of the Model

In our model, the population lives on ℤ^d and consists of two different types i∈{0,1} of particles, where we refer to 0 as dormant and to 1 as active. Let η(x,i,t) be the number of particles in spatial point x∈ℤ^d and state i at time t≥ 0, which shall evolve in time according to the following rules:


  * at time t=0, there is only one active particle in 0∈ℤ^d and all other sites are vacant;

  * all particles act independently of each other;

  * active particles become dormant at rate s_1≥ 0 and dormant particles become active at rate s_0≥ 0;

  * active particles split into two at rate ξ^+(x,t)≥ 0 and die at rate ξ^-(x,t)≥ 0, depending on their spatial location x and on time t, where both ξ^+ and ξ^- are random fields;

  * active particles jump to one of the neighbour sites with equal rate κ≥ 0;

  * dormant particles do not participate in branching, dying or migration.


Write η_0(x,i):=η(x,i,0)=δ_(0,1)(x,i) and ℙ_η_0 for the corresponding probability measure with start in η_0. Then (η,ℙ_η_0) describes a Markov process on ℕ^ℤ^d. In the following, we abbreviate ξ(x,t):=ξ^+(x,t)-ξ^-(x,t) for the balance between branching and dying and refer to ξ as the underlying random environment. Let

    u(x,i,t):=𝔼_η_0[η(x,i,t)]

denote the expected number of particles in x∈ℤ^d and state i∈{0,1} at time t with initial condition 

    u(x,i,0)=δ_(0,1)(x,i).

Note, that the expectation is only taken over switching, branching and dying and not over the random environment ξ. If we average over ξ as well, what we will denote in the following by <·>, then we refer to

    <u(x,i,t)>=<𝔼_η_0[η(x,i,t)]>

as the annealed number of particles in x∈ℤ^d and in state i∈{0,1} at time t. 



 §.§ Choices of the Random Environment

In this paper we are going to consider three specific choices of the random random environment ξ which derives the evolution of the particles:


  * Bournoulli field of immobile particles. Place in each point x∈ℤ^d independently and with probability p∈(0,1) one single particle which does not exhibit any movement. This results in a stationary field of immobile particles with a Bernoulli distribution over ℤ^d. 

  * One moving particle. Here the random environment is dynamic and consists of one single particle starting in the origin and moving around according to a simple symmetric random walk with jump rate 2dρ. 

  * Poisson field of moving particles. At point x∈ℤ^d, independently and according to a Poisson distribution with intensity ν, place a random number of particles. The particles move independently of each other, each performing a simple symmetric random walk with jump rate 2dρ. This setup generates a field of moving particles starting from a Poisson cloud.   

Clearly, for each of the choices above, ξ is a non-negative number, which results always in an positive balance between branching and killing. To allow for negative rates as well, we multiply ξ with some factor γ∈[-∞,∞) and will consider γξ as the underlying random environment. Thus, each of our three choices can be either interpreted as a field of traps, which corresponds to γ<0, or catalysts, if γ>0. In the first case, active individuals will die with rate |γ| if they encounter one of the traps, whereas they branch into two with rate γ in the presence of catalysts in the latter case.   



 §.§ Results

Recall the number of particles u(x,i,t) in point x∈ℤ^d and state i∈{0,1} at time t, as defined in (<ref>). The quantity we are interested in at most in the current paper is the annealed total number of particles

    <U(t)>:=∑_x∈ℤ^d∑_i∈{0,1}<u(x,i,t)>,

which turns into the annealed survival probability up to time t for γ<0. Our results concern the large-time asymptotics of <U(t)> in case of both positive and negative γ and for the three specific choices of the random environment mentioned above. 
Our first theorem quantifies the asymptotic behaviour of <U(t)> if the environment consists of a Bernoulli field of particles:

Let ξ be chosen according to (1) and d≥ 1.


  (a) If γ=-∞, then the annealed survival probability converges to 0, as t→∞, and obeys the asymptotics

    log<U(t)>=-c_d(log(1-p))^2/d+2(κ 
    s_0/s_0+s_1)^d/d+2t^d/d+2(1+o(1)),    t→∞,

for some constant c_d depending only on the dimension d. 

  (b) If γ∈(0,∞), then the annealed number of particles satisfies

    lim_t→∞1/tlog<U(t)>=γ-s_1-(γ+s_0-s_1)^2-s_0s_1/√(γ^2+2γ(s_0-s_1)+(s_0+s_1)^2).



Our second theorem deals with the case of one moving particle:

Let ξ be chosen according to (2).


  (a) If γ∈(-∞,0), then the annealed survival probability converges to 0 in the dimensions d∈{1,2}, as t→∞, and satisfies the asymptotics

    <U(t)>={[ 2√((s_0+s_1)(s_0(ρ+κ)+s_1ρ))/√(π)s_0|γ|1/√(t)(1+o(1)),                                                   d=1,;              4π(s_1ρ+s_0(ρ+κ))/s_0|γ|1/log(t)(1+o(1)),                                                    d=2 ].

as t→∞. 

In dimensions d≥ 3 the annealed survival probability admits the limit

    lim_t→∞<U(t)>=1/1+|γ| G_d∈(0,1)

for a constant G_d∈(0,∞).


  (b) If γ∈(0,∞), then for all d≥ 1, the annealed number of particles satisfies

    lim_t→∞1/tlog<U(t)>=sup_f∈ℓ^2(ℤ^d×{0,1}),f_2=1(A_1(f)-A_2(f)-A_3(f))+√(s_0s_1)

where

    A_1(f):=   γ f(0,1)^2,
    
    A_2(f):=   1/2∑_i∈{0,1}∑_x,y∈ℤ^d, x∼ y(iκ+ρ)(f(x,i)-f(y,i))^2,
    
    A_3(f):=   ∑_x∈ℤ^d√(s_0s_1)(f(x,1)-f(x,0))^2+∑_i∈{0,1}∑_x∈ℤ^ds_i f(x,i)^2.



Finally, we establish the asymptotics of <U(t)> for the third choice of the environment, namely a Poisson field of moving particles:

Let ξ be chosen according to (3).


  (a) If γ∈[-∞,0), then the annealed survival probability converges exponentially fast to 0 as t→∞ in all dimensions d≥ 1 and obeys the asymptotics

    log<U(t)>={[ -4ν√(ρ s_0/(s_0+s_1)π)√(t)(1+o(1)),                                d=1,;  -4νρπ s_0/s_0+s_1t/log(t)(1+o(1)),                                d=2,;      -λ_d,γ, ρ,ν,s_0,s_1 t(1+o(1)),                               d≥ 3, ].

as t→∞, for some constant λ_d,γ, ρ,ν,s_0,s_1>0 depending on all the parameters.










  (b) If γ∈(0,∞), then for all dimensions d≥ 1 the annealed number of particles satisfies the double-exponential asymptotics

    lim_t→∞1/tloglog<U(t)>=sup_f∈ℓ^2(ℤ^d),f_2=1(γ f(0)^2-1/2∑_x,y∈ℤ^d, x∼ yρ(f(x)-f(y))^2).





 §.§ Relation to the Parabolic Anderson Model

Recall the number of particles u(x,i,t) in point x∈ℤ^d and state i∈{0,1} at time t as defined in (<ref>). It is already known (cf. <cit.>) that u:ℤ^d×{0,1}×[0,∞)→ℝ solves the partial differential equation

    {[                                  /ṭu(x,i,t)                                           = iκΔ u(x,i,t) + Q u(x,i,t)+iγξ(x,t)u(x,i,t),                                        t>0,;                                    u(x,i,0)                                           =                               δ_(0,1)(x,i), ].

where

    Q u(x,i,t):= s_i(u(x,1-i,t)-u(x,i,t))

and Δ is the discrete Laplacian

    Δ f(x):=∑_y∈ℤ^d, x∼ y[f(y)-f(x)]

acting on functions f:ℤ^d→ℝ, such that

    Δ u(x,i,t):=∑_y∈ℤ^d, x∼ y[u(y,i,t)-u(x,i,t)].

We call (<ref>) the parabolic Anderson model with switching. If we consider a one-type branching random walk with only active particle evolving under the same evolution rules except of the switching mechanism, starting from one single particle in the origin, then in is well-known and has been shown first in <cit.> that the corresponding expected number of particles solves the  Parabolic Anderson model (without switching)

    {[                   /ṭu(x,t)                          = κΔ u(x,t) + γξ(x,t)u(x,t),                 t>0, x∈ℤ^d;                     u(x,0)                          =                    δ_0(x),                     x∈ℤ^d. ].

The parabolic Anderson model has been studied intensely during the past years and a comprehensive overview of results can be found in <cit.>. One of the most powerful tools and often the starting point of the analysis of the PAM is the Feynman-Kac formula

    u(x,t)=𝔼_x^X[exp(∫_0^tγξ(X(s),t-s) ṣ)δ_0(X(t))],

where 𝔼_x^X denotes the expectation over a simple symmetric random walk X with start in x and generator κΔ. In other words, the Feynman-Kac formula asserts that the time evolution of all particles can be expressed as an expectation over one single particle moving around according to the same migration kernel and with a varying mass, representing the population size. As we can see on the right hand-side of (<ref>), the mass of X changes exponentially depending on the random environment ξ. Note, that if γ<0, then the right hand-side of (<ref>) lies in [0,1] and represents the survival probability of a single particle up to time t. Now, since the Feynman-Kac formula is a mighty tool for the study of the parabolic Anderson model, it is only natural to pursue an analogous formulation in case of our two-type process with switching. To this end, let α=(α(t))_t≥ 0 be a continuous-time Markov process with state space {0,1} and generator

    Qf(i):=s_i(f(1-i)-f(i))

for f:{0,1}→ℝ. Conditioned on the evolution of α, we define a continuous-time random walk X=(X(t))_t≥ 0 on ℤ^d which is supposed to stay still at a time t, if α(t)=0, or perform a simple symmetric walk with jump rate 2dκ, if α(t)=1. In other words, the joint process (X,α) is the Markov process with the generator

    ℒf(x,i):=iκ∑_y∼ x(f(y,i)-f(x,i))+s_i(f(x,1-i)-f(x,i))
 
for x∈ℤ^d, i,j∈{0,1} and a test function f:ℤ^d×{0,1}→ℝ. Note, that the random walk X itself is not Markovian due to the dependence on α. Then, we call (X,α) a regime-switching random walk (cf. <cit.> for the continuous-space version) and interpret X as a particle which is active at time t, if α(t)=1, and dormant otherwise. Then, given a fixed realization of ξ, the formal solution of (<ref>) is given by the Feynman-Kac formula

    u(x,i,t)=𝔼_(x,i)^(X,α)[exp(∫_0^tγα(s)ξ(X(s),t-s) ṣ)δ_(0,1)(X(t),α(t))],

where 𝔼_(x,i)^(X,α) denotes the expectation over the joint process (X,α) starting in (x,i) (cf. <cit.>). Thus, the study of our two-type branching process can be reduced to the analysis of only one particle with the same migration, branching and switching rates.



 §.§ Related Results

The parabolic Anderson model without switching has been a topic of great interest during the past years. For a recent overview of results related to the classical model as well as many extensions we refer to <cit.>. In this section we recall few results according to the parabolic Anderson model on ℤ^d which are most relevant for and most related to our models. Let us start with the case γ<0, i.e. with the case of a trapping random environment. The first model is best known as random walk among Bernoulli obstacles and is the analogous version of our model (1) without switching, i.e. the time-independent potential ξ, which represents the traps or obstacles, is Bernoulli distributed with some parameter p>0. After getting trapped, the random walk with generator κΔ dies immediately, which corresponds to the hard trapping case γ=-∞. It has been shown in <cit.> using a coarse graining technique known an method of
enlargement of obstacles that the annealed survival probability up to time t decays asymptotically as

    exp(-c_d(log(1-p))^2/d+2(κ t)^d/d+2(1+o(1))),    t→∞,

with the same constant c_d as in (<ref>). In the setting of time-dependent potentials, the case of one moving trap with generator ρΔ with soft killing (γ∈(-∞,0)) has been studied in <cit.> for which it has been proven that the survival probability of a random walk with generator κΔ among this mobile traps has the asymptotics

    {[ 2√(ρ+κ)/√(π)γ1/√(t)(1+o(1)),                         d=1,;   4π(ρ+κ)/|γ|log(t)(1+o(1)),                         d=2,;     1-G_d(0)/ρ+κ+|γ| G_d(0),                        d≥ 3, ].

for t→∞ where G_d denotes the Green's function of a random walk with generator Δ. Hence, in dimension d∈{1,2}, the survival probability converges polynomially to zero, where the rate of convergence depends on all the parameters ρ,κ and γ, whereas in dimensions d≥ 3 the survival probability converges to some number in (0,1) and depends on the averaged total time spent in the origin, represented by the Green's function, as well. Finally, the case of a Poisson field of moving traps with the random potential according to (3) has been investigated in <cit.>. Here, the survival probability of the random walk with jump rate 2dκ is known to decay asymptotically as

    {[ exp(-4ν√(ρ/π)√(t)(1+o(1))),                        d=1,; exp(-4νρπt/log(t)(1+o(1))),                        d=2,;  exp(-λ_d,γ,ρ,ν t(1+o(1))),                       d≥ 3, ].

for some λ_d,γ,ρ,ν>0, in both case of hard and soft trapping rates γ∈[-∞,0).[In <cit.> the authors have considered slightly different migration rates resulting in slightly different prefactors in the asymptotics, namely the normalized Laplacian 1/2dΔ instead of Δ. For better comparison, we stated (<ref>) in case of the non-normalized Laplacian Δ.] That the survival probability, at least in dimensions d∈{1,2}, seems to be independent of the jump rate κ of the individuals, is due to the fact that the asymptotics (<ref>) come from the behaviour that the individuals stay in the origin throughout the time, which corresponds to κ=0.

In the case γ>0 of catalysts, the analogous version without switching of our first model regarding Bernoulli distributed immobile catalysts is covered in <cit.> as an example of a static bounded potential ξ. Although the results in <cit.> are held very general, applying them to the Bernoulli potential yields that

    lim_t→∞1/tlog<U(t)>=γ,

which come from the behaviour that the individuals find a region full of catalysts and stay there the whole time, such that they can continue branching throughout the time. 

For the case of one moving catalyst with jump rate 2dρ, is has been shown in <cit.> that the annealed number of particles increases exponentially as well and the exponential growth rate it given by the variational formula

    lim_t→∞1/tlog<U(t)>=sup_f∈ℓ^2(ℤ^d),f_2=1(γ f(0)^2-1/2∑_x,y∈ℤ^d, x∼ y(κ+ρ)(f(x)-f(y))^2).
 
A similar result but with double-exponential growth has been proven in <cit.> for the case of a Poisson field of moving catalysts, where the limit

    lim_t→∞1/tloglog<U(t)>=sup_f∈ℓ^2(ℤ^d),f_2=1(γ f(0)^2-1/2∑_x,y∈ℤ^d, x∼ yρ(f(x)-f(y))^2).

has been shown to be finite in all dimensions d≥ 1. The absence of κ is due to the fact that it is most favourable for the population growth if the individuals stay immobile, which corresponds to κ=0. Moreover, is has been shown in <cit.> that there are cases, where already the quantity 1/tlog<U(t)> converges to a finite limit, namely in the transient dimensions d≥ 3 under the assumption that 0≤γ/ρ≤ G_d(0)^-1, where G_d(0) denotes the Green's function again.


 §.§ Discussion

In this section, we discuss the extent to which the stochastic dormancy strategy, as defined in our model, affects the long-time dynamics of the population. Let us start with the case of catalytic random environments. 
Recall from (<ref>) that if the environment is chosen according to a static Bernoulli field, then in the analogous model without dormancy the population grows exponentially fast in time t with rate γ. Theorem 1.1.(b) asserts that the population growth still occurs exponentially in t when dormancy is incorporated; however, the growth rate is no longer γ any more but rather a smaller constant, as seen from (<ref>). Indeed, an easy calculation shows that 

    s_1+(γ+s_0-s_1)^2-s_0s_1/√(γ^2+2γ(s_0-s_1)+(s_0+s_1)^2)>0

for all γ,s_0,s_1>0, such that the growth rate is strictly decreased after incorporating the stochastic dormancy strategy. It is worth mentioning that this effect comes from the probability for large deviations of the time spent in the active state. Indeed, as we will see later in the proofs, (<ref>) can also be expressed as

    lim_t→∞1/tlog<U(t)>=γ-inf_[a∈[0,1]{I(a)+γ(1-a)},

where I is a non-negative and strictly convex function defined as

    I(a)=-2√(s_0s_1a(1-a))+(s_1-s_0)a+s_0.

Next, note that the population only grows in those time intervals in which the individuals are active. Now, on one hand, we have a law of large numbers for the fraction of time spent in the active state, which asserts that the average time proportion each individual spends in the active state equal s_0/(s_0+s_1) with probability one. On the other hand, as we will prove in Section 2, there is a large deviation principle asserting that for any other fraction of time a∈[0,1], the probability of spending a· t time units in the active state up to time t decays exponentially in t for t→∞, where the decay rate depends on the proportion a. This decays rate is nothing but the function I defined in (<ref>). Hence, (<ref>) tells us that the original growth rate γ is decreased by γ multiplied with the proportion 1-a of time spent in the dormant state on one hand, as there is no reproduction in this case, and by I(a) on the other hand, as it represents the probabilistic cost to spend exactly the proportion a of time in the active state. At the end, this probabilistic cost has to be weighed against the positive contribution to reproduction, such that I(a)+γ(1-a) has to be optimized over α. 

Continuing with one moving catalyst and comparing (<ref>) to (<ref>), we see that the population again grows exponentially in t and the rate is affected by all the involved mechanisms. While the first term in the variational formula (<ref>) shows that branching with rate γ occurs whenever the distance between individuals and catalytic particles is equal to 0, the term γ f(0,1)^2 appearing in (<ref>) has the interpretation that in case of dormancy, individuals can only branch with rate γ if, first, the distance between them and the catalyst is equal to 0, and second, if they are in the active state 1. To highlight another difference, we see that the only probabilistic cost that appears in the variational formula (<ref>) is the one coming from the movement of the individuals (with rate 2dκ) as well as of the catalyst (with rate 2dρ), whereas in (<ref>) besides the movements appearing  in the term A_2, also the exchange between states, represented by A_3, has to be taken into account. The additional term √(s_0s_1) comes from a change of measure , which will be clarified in Section 2. 

In case of a Poisson field of moving catalysts, we see that our asymptotics (<ref>) is on a double-exponential scale and equals the variational formula (<ref>) for the corresponding model without dormancy. In other words, although the stochastic dormancy strategy slows down the population growth due to the lack of reproduction in dormant phases in the first two choices of the environment, this inactivity does not seem to influence the population growth at all, if the moving catalysts start from a Poisson cloud. As will be revealed in Section 5, this is due to the fact that if the individuals manage to find favourable regions with a high density of catalysts, then the reproduction rate in the active state is on such a high scale, namely double-exponentially in time t, that the exponential probabilistic cost to stay active is negligible in comparison to the high positive outcome. Thus, the variational formula (<ref>) does not take dormancy into account and depends only on the branching rate and movement of the catalysts. 
As we will see later in the proofs, the independence of κ arises from the fact that it is most favourable for the population growth if the individuals stay immobile, which corresponds to κ=0, and matches the behaviour in the model without dormancy, as mentioned in the last section. 

Next, we discuss the case γ<0 of trapping environments. Our results concerning this case can be summarized more briefly, since they share a similarity regarding the dependence on dormancy: At least in dimensions, in which we have an explicit expression for the asymptotic survival probability, we see that the latter is increased after incorporating the stochastic dormancy strategy in comparison to the corresponding models without dormancy, and is monotone in the average time s_1/(s_0+s_1) spent in the dormant state. Moreover, setting s_1=0 yields exactly the same asymptotics as for the models without dormancy, making our results a generalization for arbitrary s_1≥ 0. This is immediate clear for the Bernoulli field of immobile traps in all dimensions, as comparing (<ref>) and (<ref>) shows. As will be addressed in the proof, formula (<ref>) also indicates that the higher survival probability results from a time-change, since individuals can only move towards the immobile traps during active phases. Therefore, if we only take into account those time intervals in which the individuals are mobile, which in average accounts for a proportion of s_0/(s_0+s_1) of the whole time due to the law of large numbers, then (<ref>) translates into (<ref>). Note that here the law of large numbers dictates the behaviour of the time spent in the active state and not the large deviation principle, since the scale t^d/(d+2) is much smaller than the large deviation scale t. 

The law of large number seems to play a role also in the case of one moving catalyst. However, comparing (<ref>) and (<ref>) demonstrates that in this case the positive effect of dormancy on the survival probability does not only come from a time-change. Rewriting the pre-factor

    2√((s_0+s_1)(s_0(ρ+κ)+s_1ρ))/√(π)s_0|γ|=2√(s_0/s_0+s_1κ+ρ)/√(π)s_0/s_0+s_1|γ|

of the polynomial asymptotics (<ref>) in dimension d=1 and comparing it to (<ref>) suggests that, although the time-change is still present as a pre-factor of the jump rate κ, the killing rate γ is reduced as well, since the individuals are again only a proportion of s_0/(s_0+s_1) of the time active and therefore vulnerable to the traps. For d=2, we can see both effects as well. Also the monotonicity of the survival probability in the average dormancy proportion s_1/(s_0+s_1) becomes evident  through a simple calculation. 


We see similar  effects also in case of a Poisson field of moving traps by comparing (<ref>) to (<ref>), where the reduction of the exponential decay rate by factor √(s_0/(s_0+s_1)) in dimension 1 and by factor s_0/(s_0+s_1) in dimension 2 comes from the law of large numbers as well, whereas in the dimensions d≥ 3 the large deviation principle dictates the asymptotics due to the joint time scale t. The (surprising) independence of the survival probability of the killing rate γ as well as the jump rate κ has been discussed in <cit.> for the corresponding model without dormancy and underlies the same reasons in our case. 

To summarize, for γ>0 the number of particles up to time t is either unchanged or reduced due to dormancy in our models, while for γ<0, at least in those cases in which an explicit expression is given, the survival probability is increased with dormancy and is monotone in the average time spent in the dormant state. 


 §.§ Outline

The rest of this paper is organized as follows. In section 2 we will convert our branching model into a switching random walk via the Feynman-Kac formula in order to obtain a more convenient representation of <U(t)>. Further, we will collect some results related to the distribution and large deviations of the local times of α. Section 3, 4 and 5 respectively are dedicated to the proofs of our main Theorems 1.1, 1.2 and 1.3 respectively.


§ PREPARATORY FACTS



 §.§ Feynman-Kac formula

As discussed in the introduction, our dormancy model is motivated by population dynamics and initially defined as a two-type branching random walk with Markovian switching between the types. However, all our proofs and considerations are founded on the Feynman-Kac formula (<ref>), which serves as the cornerstone for the subsequent steps throughout the remainder of this paper. Note that our choices of our dynamic random environments (2) and (3) are reversible in time, in the sense that (ξ(·,t))_0≤ t≤ T is equally distributed to (ξ(·,T-t))_0≤ t≤ T, for all T>0. Hence, taking the expectation with respect to ξ and changing the order of integrals, we can write

    <U(t)>   =∑_x∈ℤ^d∑_i∈{0,1}𝔼_(x,i)^(X,α)[<exp(∫_0^tγα(s)ξ(X(s),t-s) ṣ)δ_(0,1)(X(t),α(t))>]
    
       =∑_x∈ℤ^d∑_i∈{0,1}𝔼_(x,i)^(X,α)[<exp(∫_0^tγα(s)ξ(X(s),s) ṣ)δ_(0,1)(X(t),α(t))>].

Moreover, we can change the starting and end point of the path (X,α) to obtain

    <U(t)>   =∑_x∈ℤ^d∑_i∈{0,1}𝔼_(0,1)^(X,α)[<exp(∫_0^tγα(s)ξ(X(s),s) ṣ)δ_(x,i)(X(t),α(t))>]
       =𝔼_(0,1)^(X,α)[<exp(∫_0^tγα(s)ξ(X(s),s) ṣ)>]
       =<𝔼_(0,1)^(X,α)[exp(∫_0^tγα(s)ξ(X(s),s) ṣ)]>.

Especially, U(t) can also be interpreted as the solution ũ of the partial differential equation

    {[                                                /ṭũ(x,i,t)                                                         = iκΔũ(x,i,t) + s_i (ũ(x,1-i,t)-ũ(x,i,t))+iγξ(x,t)ũ(x,i,t),                                                     t≥ 0,;                                                  ũ(x,i,0)                                                         =                                                         1 ].

in point (0,1), which differs from (<ref>) only in the initial distribution. From now on, we will work with the representation (<ref>). Note, that (<ref>) also holds for the static choice (1), as ξ does not depend on time. 


 §.§ Large deviation principle for α

In this section we establish a large deviation principle for the normalized local times

    1/tL_t(i)=1/t∫_0^tδ_i(α(s))ṣ

of α in state i∈{0,1}. Where such a principle is already well-known in the literature for discrete-space Markov processes with symmetric transition rates (cf. <cit.>), the corresponding large deviation principle in case of asymmetric rates is still missing. In our case, we can obtain the probability for large deviations directly by computing the exact distribution of the local times in:

For all s_0,s_1>0 the probability density function of the local times (L_t(1))_t>0 of α in state 1 is given by

    ℙ(L_t(1)∈ỵ)=   s_1^-s_0t-(s_1-s_0)y(∑_k=0^∞(s_0s_1y(t-y))^k/k!k!(s_0y/k+1+1)).



Let N(t) be the number of jumps of the Markov chain α up to time t>0 when α starts in state 1. We denote by a_i, i∈ℕ, the waiting times of transitions from 0 to 1 and by b_i those from 1 to 0 such that a_i resp.  b_i are independent and exponentially distributed with parameter s_0 resp.  s_1. If N(t) is even, α will be in state 0 after the last jump before t. Then

    ℙ(L_t(0)∈ỵ, N(t)=2k)=   ℙ(∑_i=1^ka_i∈ỵ, ∑_i=1^ka_i+∑_i=1^kb_i<t, b_k+1>t-(∑_i=1^ka_i+b_i))
    
    =   ∫_x=0^t-yℙ(∑_i=1^kb_i∈x̣, ∑_i=1^ka_i∈ỵ, b_k+1>t-x-y)
    
    =   ∫_0^t-ys_1^kx^k-1e^-s_1x/(k-1)!s_0^ky^k-1^-s_0y/(k-1)!^-s_1(t-x-y) x̣ ỵ
    =   (s_0s_1)^k/(k-1)!k!^-s_1t-(s_0-s_1)y(t-y)^ky^k-1

for y∈[0,t] and therefore

    ℙ(L_t(1)∈ỵ, N(t)=2k)=(s_0s_1)^k/(k-1)!k!^-s_0t-(s_1-s_0)y(t-y)^k-1y^k.

In case of an odd number of jumps, where α is in state 1 after the last jump before t, the joint distribution of L_t(1) and N(t) reads

    ℙ(L_t(1)∈ỵ, N(t)=2k+1)=   ∫_x=0^t-yℙ(∑_i=1^k+1b_i∈ỵ, ∑_i=1^ka_i∈x̣, a_k+1>t-x-y)
    
    =   s_0^ks_1^k+1/k!k!^-s_0t-(s_1-s_0)y(t-y)^ky^k.

The claim follows after summing over all k∈ℕ. 

Making use of the exact distribution of the local times of α, we are able to establish the following large deviation principle:

For all choices of the transition rates s_0,s_1≥ 0, the normalized local times (1/tL_t(1))_t≥ 0 of α in state 1 satisfy a large deviation principle on [0,1] with rate function I:[0,1]→ℝ given by

    I(a)=-2√(s_0s_1a(1-a))+(s_1-s_0)a+s_0.



This follow immediately from lemma <ref>. Indeed, for all t>0 and a∈[0,1] we can write

    ℙ^α_1(L_t(1)∈(̣at))=^-s_0t-(s_1-s_0)at(s_0s_1at/k+1+s_1)I_0(2t√(s_0s_1a(1-a)))

where 
I_0(x)=∑_k=0^∞(1/4x^2)^k/k!k! is the modified Bessel function with parameter 0, which can also be written as

    I_0(x)=1/π∫_0^π^xcos(θ) θ̣.

Hence,

    I(a)=   -lim_t→∞1/tlogℙ^α_1(L_t(1)∈(̣at))
    
    =   s_0+(s_1-s_0)a-lim_t→∞1/tlog(∫_0^π^t2√(s_0s_1a(1-a))cos(θ) θ̣)

Now, ppplying the method of Laplace for integrals (cf.  <cit.>) to the representation (<ref>) yields

    lim_t→∞1/tlog(∫_0^π^t2√(s_0s_1a(1-a))cos(θ) θ̣)=max_θ∈[0,π]2√(s_0s_1a(1-a))cos(θ)=2√(s_0s_1a(1-a))

and thus

    I(a)=s_0+(s_1-s_0)a-2√(s_0s_1a(1-a)).



Note, that the rate function I is strictly convex and, as an easy calculations shows, attends its unique minimizer at a_min=s_0
/s_0+s_1. This implies a law of large numbers for the proportion of time spent in the active state, i. e. 

    lim_t→∞1/tL_t(1)=s_0/s_0+s_1    ℙ^α- almost surely.

Further, if we choose s:=s_0=s_1 equally, the rate function becomes

    I(a)=s-2s√(a(1-a))=s(1-2√(a(1-a)))=s(√(a)-√(1-a))^2

which is the well-known large deviation rate function in the case of symmetric transition rates (cf. <cit.>).

The next lemma shows that if we look at the exponential moments of the normalized local times (1/tL_t(1))_t≥ 0 of α with a smaller exponential rate than t, then the best (1/tL_t(1))_t≥ 0 can do in order to maximize the exponent, is to take its long-term average:

Let (f(t))_t≥ 0 be a sequence of positive real numbers with lim_t→∞f(t)=∞ and lim_t→∞f(t)/t=0. Then, for any continuous and bounded function F[0,1]→ℝ,

    lim_t→∞1/f(t)log𝔼_1^α[^f(t)F(1/tL_t(1))]=F(s_0/s_0+s_1).



For the lower bound, fix δ>0 and let G be some open ball around x_0:=s_0/s_0+s_1 such that F(x_0)-δ≤inf_G F(x_0)≤  F(x_0)+δ. Then we use the law of large numbers for the sequence (1/tL_t(1))_t≥ 0 to obtain

    lim inf_t→∞1/f(t)log𝔼_1^α[^f(t)F(1/tL_t(1))]≥   lim inf_t→∞1/f(t)log(^f(t)inf_a∈ GF(a)ℙ_1^α(1/tL_t(1)∈ G))
    
    ≥   inf_a∈ GF(a)+lim inf_t→∞1/f(t)logℙ_1^α(1/tL_t(1)∈ G)
    
    ≥    F(x_0)-δ.

For the upper bound, the boundedness of F by some M>0 gives

    𝔼_1^α[^f(t)F(1/tL_t(1))]≤   ^f(t)(F(x_0)+δ)ℙ_1^α(1/tL_t(1)∈G̅)+^f(t)Mℙ_1^α(1/tL_t(1)∈ G^c).

Now, as

    lim sup_t→∞1/f(t)logℙ_1^α(1/tL_t(1)∈ G^c)=-lim sup_t→∞t/f(t)inf_a∈ G^cI(a)=-∞,

where I is the rate function defined in (<ref>), and using the law of large numbers again as well as the method of Laplace for sums, we deduce that

    lim sup_t→∞1/f(t)log𝔼_1^α[^f(t)F(1/tL_t(1))]≤max{F(x_0)+δ, -∞}=F(x_0)+δ.

The claim follows after letting δ→ 0. 



 §.§ Change of measure for α

One of the proof methods we will use in section 4 to obtain the representation (<ref>) is the Perron-Frobenius spectral theory for bounded self-adjoint operators, which we would like to apply to the generator

    ℒ̅f(z,i)=∑_y∼ z(iκ+ρ)(f(y,i)-f(z,i))+s_i(f(z,1-i)-f(z,i))
 
of a Markov process (Z,α), where fℤ^d×{0,1}→ℝ is a suitable test function. However, this may be a problem, as the matrix Q defined in (<ref>) is not symmetric and therefore the generator (<ref>) of (Z,α) is not self-adjoint. In order to fix this, we will use a result from <cit.> which we formulate here for the convenience of the reader:

Let α=(α(t))_t≥ 0 be any Markov process on a finite state space ℳ with transition rates q_ij from state i∈ℳ to j∈ℳ. For a positive function hℳ→(0,∞), let α̃=(α̃(t))_t≥ 0 be another Markov process on ℳ defined on the same filtered space (Ω, (ℱ_t)_t≥ 0)) with transition rates q̃_ij given by

    q̃_ij=q_ijh(j)/h(i)

for i≠ j and q̃_ii=-∑_k≠ iq_ikh(k)/h(i). Denote by ℙ^α resp. ℙ^α̃ the distribution of α resp. α̃. Then ℙ^α is absolutely continuous with respect to ℙ^α̃ with the Radon-Nikodym derivative 

    P^α/P^α̃|_ℱ_t=h(α̃(t))/h(α̃(0))exp(-∫_0^tQ̃h(α̃(s))/h(α̃(s)) ṣ).



Cf. proof of <cit.>.

We can now apply <ref> to build our favourite Markov process with symmetric rates out of α:

Let α̃ be a Markov process on {0,1} with generator

    Q̃f(i):=√(s_0s_1)(f(1-i)-f(i))

for f{0,1}→ℝ, and write ℙ^α̃ for its distribution with start in state 1. Further, denote by ℙ^α the distribution of the Markov chain α with generator defined in (<ref>) and start in 1. Then,

    P^α/P^α̃|_ℱ_t=exp(√(s_0s_1)t-s_0L̃_t(0)-s_1L̃_t(1))

where we wrote L̃_t(i)=∫_0^tδ_i(α̃(s)) ṣ for the local times of α̃ in state i∈{0,1} up to time t.


Define h:{0,1}→ℝ as h(0)=√(s_1) and h(1)=√(s_0). Conditioned on α̃(t)=1, this yields

    P^α/P^α̃|_ℱ_t=   h(α̃(t))/h(α̃(0))exp(-∫_0^tQ̃h(α̃(s))/h(α̃(s)) ṣ)
    
    =   exp(-L̃_t(0)√(s_0s_1)(√(s_0)-√(s_1))/√(s_1)-L̃_t(1)√(s_0s_1)(√(s_1)-√(s_0))/√(s_0))
    
    =   exp(-L̃_t(0)(s_0-√(s_0s_1))-L̃_t(1)(s_1-√(s_0s_1)))
    
    =   exp(√(s_0s_1)t-s_0L̃_t(0)-s_1L̃_t(1)).



Especially, if (Z̃,α̃) is the Markov process with symmetric generator

    L̃f(x,i):=(iκ+ρ)κ∑_y∼ x(f(y,i)-f(x,i))+√(s_0s_1)(f(x,1-i)-f(x,i))

for test functions fℤ^d×{0,1}→ℝ and if we write ℙ^(Z,α)_(0,1) resp. ℙ̃^(Z,α̃)_(0,1) for the distribution of (Z,α) resp. (Z̃,α̃) with start in (0,1), then

    P^(Z,α)_(0,1)/̣̃ℙ^(Z,α̃)_(0,1)|_ℱ_t=exp(√(s_0s_1)t-s_0L̃_t(0)-s_1L̃_t(1))

as well, since the generator of Z, conditioned on α, equals that of Z̃, conditioned on α̃, and since α resp. α̃ is independent of Z resp. Z̃. 



 §.§ Hitting probabilities

The following lemma asserts that the hitting probabilities of the two-type switching random walk (Z,α) defined in (<ref>) can be expressed as time-changed hitting probabilities of a simple symmetric random walk without switching:

Let (Z,α) be the switching random walk with generator (<ref>) and Z̃ a simple symmetric rate random walk with generator Δ. Then, for any fixed realization of (α(t))_t≥ 0,

    ℙ_y^Z(∃ s≤ t:(Z(s),α(s))=(0,1))=ℙ_y^Z̃(∃ s≤ L_t(1): Z̃(ρ s+bκ L_s(1))=0),

where ℙ_y^Z resp. ℙ_y^Z̃ denotes the probability with respect to Z resp. Z̃ with start in y.


Write

    ψ_α(y,t):=ℙ_y^Z(∃ s∈[0,t]: Z(s)=0, α(s)=1)=ℙ_y^Z(∃ s∈ A_t: Z(s)=0)

for a fixed realization of α with A_t:={s∈[0,t]:α(s)=1}. Note, that

    ψ_α(y,t)=∫_0^tℙ_y^Z(Z(s)=0)1_{α(s)=1} ṣ

and let N_t denote the number of jumps of α till t and write s_1,s_2,⋯,N_t for the jump times of α and τ_k:=s_k-s_k-1 for the corresponding waiting times, such that A_t=[0,s_1]∪[s_2,s_3]∪⋯∪[s_N_t,t] and

    ψ_α(y,t)=   ∫_0^s_1ℙ_y^Z(Z(s)=0) ṣ+∫_s_2^s_3ℙ_y^Z(Z(s)=0) ṣ+⋯+∫_s_N_t^tℙ_y^Z(Z(s)=0) ṣ

where we w.l.o.g. assume that N_t is even. For each k-th intervall, starting with the intervall [s_2,s_3], we use a linear substitution of the integration variable to obtain that each integration interval starts with the endpoint of the previous one, i.e.

    ψ_α(y,t)=   ∫_0^s_1ℙ_y^Z(Z(s)=0) ṣ+∫_s_1^s_3-s_2+s_1ℙ_y^Z(Z(s)=0) ṣ+∫_s_3-s_2+s_1^s_5-s_4+s_3-s_2+s_1ℙ_y^Z(Z(s)=0) ṣ
       +⋯+∫_∑_j=0^N_t-1(-1)^j+1s_j^∑_j=0^N_t+1(-1)^j+1s_jℙ_y^Z(Z(s)=0) ṣ
    =   ∫_0^t-∑_j=0^1/2N_tτ_2jℙ_y^Z(Z(s)=0) ṣ=∫_0^t-L_t(0)ℙ_y^Z(Z(s)=0) ṣ=∫_0^L_t(1)ℙ_y^Z(Z(s)=0) ṣ

with N_t+1:=t, where we used that the sum ∑_j=0^1/2N_tτ_2j corresponds to the waiting times of α in state 0.
Now, we use the fact that the end point Z(s) has the same distribution as Z̃(ρ s+κ L_s(1)), given α, which yields

    ψ_α(y,t)=∫_0^L_t(1)ℙ_y^Z̃(Z̃(ρ s+κ L_s(1))=0)ṣ=ℙ_y^Z̃(∃ s≤ L_t(1): (Z̃(ρ s+κ L_s(1))=0).




§ PROOF OF THEOREM <REF>

This section is dedicated to the proof of Theorem <ref>. For this, let ξ=(ξ(x))_x∈ℤ^d be a static field built out of Bernoulli distributed particles, i. e. , for each x∈ℤ^d the random variable ξ(x) is independent and Bernoulli distributed with ℙ(ξ(x)=1)=p=1-ℙ(ξ(x)=0). As the random environment is static, we only have to average over the movement of the switching random walk X and the initial distribution of the Bernoulli field in order to determine the long-term behaviour of <U(t)>. Thus, the proof of theorem <ref> is based on a time-change combined with existing results regarding the behaviour of random walks (without switching) in a Bernoulli field of particles. More precisely, let X̃ be a simple symmetric random walk without switching and with generator κΔ. Then it is well-known that, conditioned on (α(s))_s≤ t, the endpoint X(t) is equal to X̃(L_t(1)) in distribution. This will be the starting point of the following proof:


 §.§ Proof of Theorem <ref>
 For (x,i)∈ℤ^d×{0,1} let

    ℓ_t(x,i):=∫_0^tδ_(x,i)(X(s),α(s))ṣ

denote the local time of the process (X,α) in (x,i). Then, for an arbitrary γ∈[-∞, ∞), we can rewrite <U(t)>, using the independence of the Bernoulli distribution in each spatial point x∈ℤ^d, as

    <U(t)>=   <𝔼_(0,1)^(X,α)[exp(∑_x∈ℤ^d∑_i∈{0,1}ξ(x,i)ℓ_t(x,i))]>
    
    =   𝔼_(0,1)^(X,α)[∏_x∈ℤ^d<exp(∑_i∈{0,1}ξ(x,i)ℓ_t(x,i))>]
    
    =   𝔼_(0,1)^(X,α)[∏_x∈ℤ^d(p^γℓ_t(x,1)+1-p)]
    
    =   𝔼_(0,1)^(X,α)[exp(∑_x∈ℤ^dlog(p^γℓ_t(x,1)+1-p)1_{ℓ_t(x,1)>0})].

Now, let γ=-∞. Then the annealed survival probability up to time t reads

    <U(t)>=   𝔼_(0,1)^(X,α)[exp(∑_x∈ℤ^dlog(1-p)1_{ℓ_t(x,1)>0})]=𝔼_(0,1)^(X,α)[(1-p)^∑_x∈ℤ^d1_{ℓ_t(x,1)>0}].

Recall, that the walk X can only move in the active state 1 such that each newly visited point x∈ℤ^d is crossed by X for the first time in state 1. Therefore, 

    ∑_x∈ℤ^d1_{ℓ_t(x,1)>0}=∑_x∈ℤ^d1_{ℓ̅_t(x)>0}= R(t),

where we write ℓ̅_t(x) for the projection of ℓ_t(x,i), i∈{0,1}, on the first component and denote by R(t) the range of the random walk X up to time t, i.e. , the number of all distinct visited points up to time t by X.  Changing the order of the expectations yields

    <U(t)>=𝔼_1^α𝔼_0^X[(1-p)^R(t)]=𝔼_1^α𝔼_0^X̃[(1-p)^R̃(L_t(1))],

where 𝔼_0^X̃ denotes the expectation with respect to X̃ and R̃(t) the range of X̃ up to time t. For a fixed realization of α, the inner expectation is nothing but the survival probability of the simple random walk X̃ among Bernoulli traps up to the (deterministic) time L_t(1), to which we can apply the well-known result from <cit.> asserting that

    𝔼_0^X̃[(1-p)^R̃(L_t(1))]=exp(-c_dκ^d/d+2(log(1-p))^2/d+2L_t(1)^d/d+2(1+o(1)))

as t→∞, where the constant c_d is given by

    c_d:=(d+2)d^2/d+2-1λ_d^d/d+2

and λ_d denotes the principal Dirichlet eigenvalue of -Δ on [-1,1]^d⊆ℝ^d. Finally, application of lemma <ref> finishes the proof of part (a).

Now, let γ>0. Then

    <U(t)>   =𝔼_(0,1)^(X,α)[exp(∑_x∈ℤ^dlog(p^γℓ_t(x,1)+1-p)1_{ℓ_t(x,1)>0})]
    
       =𝔼_(0,1)^(X,α)[exp(∑_x∈ℤ^d(log(p)+log(^γℓ_t(x,1)+1-p/p))1_{ℓ_t(x,1)>0})]
    
       =𝔼_(0,1)^(X,α)[p^R(t)·exp(∑_x∈ℤ^dlog(^γℓ_t(x,1)+1-p/p)1_{ℓ_t(x,1)>0})].

Using the asymptotics (<ref>) of the first term in the expectation (with 1-p replaced by p) and time-change again, we can lower-bound

    <U(t)>≥   𝔼_(0,1)^(X,α)[p^R(t)·exp(∑_x∈ℤ^dlog(^γℓ_t(x,1))1_{ℓ_t(x,1)>0})]
    
    =   𝔼_1^α𝔼_0^X̃[p^R̃(L_t(1))exp(γ L_t(1))]
    
    =   𝔼_1^α[exp(γ L_t(1)-c_dκ^d/d+2log(p)^2/d+2L_t(1)^d/d+2(1+o(1)))],    t→∞.

Now, recall the large deviation principle <ref> for the normalized local times 1/tL_t(1) of α in 1 with rate function I defined in (<ref>). Using Varadhan's lemma (cf. <cit.>), we can deduce

    lim_t→∞1/tlog<U(t)>≥sup_a∈[0,1]{γ a-I(a)}=γ-inf_a∈[0,1]{(1-a)γ+I(a)},

as t→∞, since the term

    L_t(1)^d/d+2=t^-2/d+2(1/tL_t(1))^d/d+2

vanishes on scale t for t→∞. On the other hand, for large t, we can upper-bound

    <U(t)>≤   𝔼_(0,1)^(X,α)[p^R(t)·exp(∑_x∈ℤ^dlog(2max{^γℓ_t(x,1),1-p/p})1_{ℓ_t(x,1)>0})]
    
    =   𝔼_(0,1)^(X,α)[(2p)^R(t)·exp(γ L_t(1))]
    
    =   𝔼_1^α[exp(t(γ1/tL_t(1)-t^-2/d+2c_dκ^d/d+2log(2p)^2/d+2(1/tL_t(1))^d/d+2(1+o(1))))],    t→∞,

using the same considerations as before. The second term in the expectation is again negligible on the time scale t for t→∞ and thus

    lim_t→∞1/tlog<U(t)>≤γ-inf_a∈[0,1]{(1-a)γ+I(a)}.

For an explicit expression, we calculate the minimizer of the function f(a)=(1-a)γ+I(a) and find that

    γ-inf_a∈[0,1]{(1-a)γ+I(a)}=γ-s_1-(γ+s_0-s_1)^2-s_0s_1/√(γ^2+2γ(s_0-s_1)+(s_0+s_1)^2).



§ PROOF OF THEOREM <REF>

In this section, we prove Theorem <ref>, in which the underlying environment ξ is dynamic and consists of one single particle moving independently of X. More precisely, ξ is the Markov process on {0,1}^ℤ^d given by

    ξ(x,t)=δ_x(Y(t)),

where Y=(Y(t))_t≥ 0 is a continuous-time simple symmetric random walk on ℤ^d with jump rate 2dρ for a constant ρ>0 and starting in the origin. Hence,

    <U(t)>=𝔼_0^Y𝔼_(0,1)^(X,α)[exp(γ∫_0^tδ_(0,1)(X(s)-Y(s),α(s))  d s)],

where 𝔼_0^Y denotes the expectation with respect to Y. Set Z:=X-Y. Then (Z,α) has the generator

    ℒ̅f(z,i)=∑_y∼ z(iκ+ρ)(f(y,i)-f(z,i))+s_i(f(z,1-i)-f(z,i))
 
for z∈ℤ^d, i,j∈{0,1} and a test function f:ℤ^d×{0,1}→ℝ, and thus

    <U(t)>=𝔼_(0,1)^(Z,α)[exp(γ∫_0^tδ_(0,1)(Z(s),α(s))  d s)]=:v(0,1,t).

Especially, the new potential ξ̃(z,i):=δ_(0,1)(x,i) does not depend on the time any more. Using the Feynman-Kac formula, we further see that (<ref>) is the solution to

    {[                                       d/ dt v(y,i,t)                                                    = (iκ+ρ)Δ v(y,i, t)+Q v(y,i,t)+γ·δ_(0,1)(y,i)v(y,i,t),                                                  t>0;                                             v(y,i,0)                                                    =                                                   i, ].

with (y,i)=(0,1). In the following, we shall use either of the representations (<ref>), (<ref>) or (<ref>), depending on what is to be proven. We start with the proof of theorem <ref>(a) and show theorem <ref>(b) separately, as different methods are used in the case γ<0 and γ>0 respectively.


  §.§.§ Proof of Theorem <ref>(a)
 Let γ∈(-∞,0) and denote by p(y,i,t) the probability density function of the switching diffusion (Z,α) with start in (0,1). Then we get the representation

    v(0,1,t)=1+γ∫_0^t p(0,1,s)v(0,1,t-s) ṣ,

to which we want to apply the Laplace transform. Denoting by v̂_̂1̂(λ) resp. p̂_̂1̂(λ) the Laplace transform of v(0,1,t) resp. p(0,1,t), and solving (<ref>) for v̂_̂1̂(λ), we arrive at

    v̂_̂1̂(λ)=1/λ·1/1-γp̂_̂1̂(λ).

Our next aim is therefore to compute p̂_̂1̂(λ). For this, note that the probability density function p satisfies the system of equations 

    {[                                /ṭp(y,1,t)                                         = (κ+ρ)Δ p(y,1,t)+s_0p(y,0,t)-s_1p(y,1,t)),                                      t>0,;                                /ṭp(y,0,t)                                         =      ρΔ p(y,0,t)+s_1p(y,1,t)-s_0p(y,0,t),                                      t>0,;                                  p(y,i,0)                                         =                             δ_(0,1)(y,i). ].

Forth order systems of the form (<ref>) with two different diffusion constants have been studied in <cit.>. For the convenience of the reader, we will include the first steps and ideas to calculate the solution of (<ref>). We denote by p̂_i(y,λ) the Laplace transform of p(y,i,·) and apply this to (<ref>), using the initial condition, to obtain the new system

    0=   (κ+ρ)Δp̂_1(y,λ)-(λ+s_1)p̂_1(y,λ)+s_0p̂_0(y,λ)+δ_0(y),
    0=   ρΔp̂_0(y,λ)-(λ+s_0)p̂_0(y,λ)+s_1p̂_1(y,λ),

which, after solving (<ref>) for p̂_1(y,λ) and applying this to (<ref>), translates in to the forth-order equation

    (Δ^2-(s_1+λ/κ+ρ+s_0+λ/ρ)Δ+(s_1+λ)(s_0+λ)-s_0s_1/(κ+ρ)ρ)p̂_0(y,λ)=s_1/(κ+ρ)ρδ_0(y).

Forth order systems of the form (<ref>) are known to have the solution

    p̂_0(y,λ)=s_1/2(κ+ρ)ρ (a^2-b^2)(1/a^a|y|-1/b^b|y|),

in dimension d=1, where

    a,b=-√(λ(κ+ρ)+s_1ρ+s_0(κ+ρ)±√(κ^2+2λκ(s_1ρ-s_0(κ+ρ))+(s_1ρ+s_0(κ+ρ))^2))/√(2ρ(κ+ρ)).

Using the relation between p̂_1(y,λ) and p̂_0(y,λ) and inserting y=0 we obtain

    p̂_1(0,λ)= -(λ+s_0+ρ ab)/2ρ(κ+ρ)ab(a+b)∼1/√(λ)·s_0/2√((s_0+s_1)(s_0(ρ+κ)+s_1ρ)),   λ→ 0,

as long and tedious calculations show. In dimension d=2 we proceed in a similar way to find

    p̂_1(0,λ)∼s_0/4π(s_1ρ+s_0(κ+ρ))log(1/λ),  λ→ 0.

Thus, we deduce from (<ref>) that

    v̂_1(λ)∼{[ 1/√(λ)2√((s_0+s_1)(s_0(ρ+κ)+s_1ρ))/s_0|γ|,                                       d=1,;         4π(s_1ρ+s_0(κ+ρ))/s_0|γ|λlog(1/λ),                                       d=2, ].
 
as λ→ 0. 
Using the Littlewood-Hardy Tauberian theorem we finally arrive at

    v(1,0,t)∼{[ 2√((s_0+s_1)(s_0(ρ+κ)+s_1ρ))/√(π)s_0|γ|1/√(t),                                           d=1,;                4π(s_1ρ+s_0(κ+ρ))/s_0|γ|log(t),                                           d=2, ].
 
as λ→∞. Next, let d≥ 3 and denote by

    G_d(x,i):=∫_0^∞p_d(x,i,t) ṭ

the Green's function of (Z,α) in (x,i), which has the probabilistic representation

    G_d(x,i)=𝔼_(x,i)^(Z,α)[∫_0^∞δ_(0,1)(Z(s),α(s)) ṣ].

Hence, for all (x,i)∈ℤ^d×{0,1} the quantity

    v(x,i):=lim_t→∞v(x,i,t)=𝔼_(x,i)^(Z,α)[exp(γ∫_0^∞δ_(0,1)(Z(s),α(s)) ṣ)]

lies in (0,1). Moreover, v solves the boundary value problem

    {[                                   0                                   = (iκ+ρ)Δ v(x,i)+γδ_(0,1)(x,i)v(x,i),                    (x,i)∈ℤ^d×{0,1},;                                   1                                   =                      lim_x→∞v(x,i),                            i∈{0,1}, ].

and can therefore be written as

    v(0,1)=1+γ∫_0^∞p_d(0,1,t)v(0,1) ṭ=1+γ v(0,1)G_d(0,1)

in point (0,1). Solving for v(0,1) gives

    v(0,1)=1/1-γ G_d(0,1).

The survival probability converges therefore to a non-trivial limit in (0,1) in all dimensions d≥ 3.

We now continue with the case γ>0 of catalysts. Recall the two-state Markov chain α̃ with symmetric generator (<ref>). Before proving theorem <ref>(b), we need two statements that are highly inspired by <cit.>:

Let r(t)=tlog^2(t), Q_r(t)=[-r(t),r(t)]^d∩ℤ^d and V:ℤ^d×{0,1}→ℝ a bounded function. Further, abbreviate

    A_t:=∫_0^tV(Z(s),α̃(s)) ṣ.

Then, the following holds true:


  (a) As t→∞,

    𝔼^(Z,α̃)_(0,1)   [^A_t]=(1+o(1))∑_z∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_z^Y[^A_t·δ_0(Y(t))1_{X(t)∈ Q_r(t)}].


  (b) As t→∞,

    ∑_y∈ Q_r(t)𝔼_(0,1)^(X,α̃)   𝔼_0^Y[^A_t·δ_y(X(t))δ_y(Y(t))]=(1+o(1))∑_y∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_0^Y[^A_t·δ_y(X(t))δ_y(Y(t))].




Note that A_t∈[0, Mt] with M:=sup_(x,i)∈ℤ^d×{0,1}V(x,i). Then,

    𝔼^(Z,α̃)_(0,1)[^I_t]=∑_z∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_0^Y[^A_tδ_z(Y(t))]=∑_z∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_z^Y[^A_tδ_0(Y(t))]

using Fubini and a time reversal for Y. 
In order to prove part (a) we have to check that

    a(t):=∑_z∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_z^Y[^A_tδ_0(Y(t))]-∑_z∈ Q_r(t)𝔼_(0,1)^(X,α̃)𝔼_z^Y[^A_tδ_0(Y(t))1_{X(t)∈ Q_r(t)}]/∑_z∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_z^Y[^A_tδ_0(Y(t))],

converges to zero as t→∞. which is done in a similar way as in the proofs of <cit.>, such that we only highlight the differences. Splitting ℤ^d in Q_r(t) and it's complement, upper-bounding ^A_t by ^M t and using a time reversal for Y again, we obtain the bound

    a(t)≤   ^M t(ℙ_0^Y(Y(t)∉ Q_r(t))+ℙ_(0,1)^(X,α̃)(X(t)∉ Q_r(t))/ℙ_0^Y(Y(t)=0)
    =   ^M t(ℙ_0^Y(Y(t)∉ Q_r(t))+ℙ_1^α̃ℙ_0^X̃(X̃(L̃_t(1))∉ Q_r(t))/ℙ_0^Y(Y(t)=0),

where X̃ is a simple symmetric random walk without switching and with generator κΔ and L̃_t(1) denotes the local time of α̃ in state 1 up to time t. Now, <cit.> asserts that

    ℙ_0^Y(Y(t)∉ Q_r(t))≤ 2^d+1^-r(t)log(r(t)/dρ t)+r(t)
 
such that for our choice of r(t) and for sufficiently large t,

    ℙ_0^Y(Y(t)∉ Q_r(t))≤^-r(t),

as a quick estimation shows. Analogously,

    ℙ_1^α̃ℙ_0^X̃(X̃(L_t(1))∉ Q_r(t))≤   ℙ_1^α̃[2^d+1exp(-r(t)log(r(t)/dκL̃_t(1))+r(t))]
    
    =   2^d+1exp(-r(t)(log(r(t)/dκ)-1))𝔼_1^α̃[ exp(r(t)log(L̃_t(1)))]
    
    ≤   2^d+1exp(-r(t)log(r(t)/dκ t)+r(t)).

Thus, we have again

    ℙ_1^α̃ℙ_0^X̃(X̃(L̃_t(1))∉ Q_r(t))≤^-r(t)

for sufficiently large t. This shows that the nominator of (<ref>) converges exponentially in t to zero, whereas its dominator converges only polynomially. Hence, a(t)→ 0 for t→∞. 

In order to prove part (b), we define

    b(t):=∑_y∉ Q_r(t)𝔼_(0,1)^(X,α̃)𝔼_0^Y[^A_tδ_y(X(t))δ_y(Y(t))]/∑_y∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_0^Y[^A_tδ_y(X(t))δ_y(Y(t))]

and proceed in a similar way to obtain the upper bound

    b(t)≤^M t(ℙ_(0,1)^(X,α̃)(X(t)∉ Q_r(t))ℙ_0^Y(Y(t)∉ Q_r(t)))/ℙ_(0,1)^(X,α̃)(X(t)=0)ℙ_0^Y(Y(t)=0).

From the proof of part (a) we already know that the nominator decays exponentially in t as t→∞. Moreover,

    ℙ_(0,1)^(X,α̃)(X(t)=0)=ℙ_1^α̃ℙ_0^X̃(X̃(L̃_t(1))=0),

where ℙ_0^X̃(X̃(L̃_t(1))=0) decays polynomially in L̃_t(1). Hence, lemma <ref> asserts that

    ℙ_1^α̃ℙ_0^X̃(X̃(L̃_t(1))=0)=ℙ_0^X̃(X̃(s_0/s_0+s_1t)=0)

for sufficiently large t. Thus, the dominator of the right hand-side of (<ref>) decays polynomially in t and therefore b(t)→ 0 as t→∞. 

We are now ready to prove theorem <ref>(b).


 §.§ Proof of Theorem <ref>(b)

Let α̃ be the two-state Markov chain with symmetric generator Q̃ defined in (<ref>) and denote by ℓ_t(x,i) resp. ℓ̃_t(x,i) the local times of (Z,α) resp. (Z,α̃) in (x,i) up to time t. Then, combining the representation (<ref>) with corollary <ref> and remark <ref>, the annealed number of particles up to time t reads

    <U(t)>=   𝔼^(Z,α)_(0,1)[exp(γ∫_0^tδ_(0,1)(Z(s),α(s)) ṣ)]
    =   𝔼^(Z,α)_(0,1)[exp(γℓ_t(0,1))]
    =   𝔼^(Z,α̃)_(0,1)[exp(√(s_0s_1)t-s_0L̃_t(0)-s_1L̃_t(1)+γℓ̃_t(0,1))]
    =   𝔼^(Z,α̃)_(0,1)[exp(√(s_0s_1)t-s_0∑_y∈ℤ^dℓ̃_t(y,0)-s_1∑_y∈ℤ^dℓ̃_t(y,1)+γℓ̃_t(0,1))]
    =   ^√(s_0s_1)t𝔼^(Z,α̃)_(0,1)[exp(∫_0^tV(Z(s),α̃(s)) ṣ)]

where we define

    V(z,i):=-s_0δ_0(i)-s_1δ_1(i)+γδ_(0,1)(z,i)

for (z,i)∈ℤ^d×{0,1}. 
Let us start with the  upper bound, which is done in a similar way as in the proof of <cit.>. Applying lemma <ref>(a) yields

    𝔼^(Z,α̃)_(0,1)   [exp(∫_0^tV(Z(s),α̃(s)) ṣ)]
    =   (1+o(1))∑_z∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_z^Y[exp(∫_0^tV(X(s)-Y(s),α̃(s)) ṣ)δ_0(Y(t))1_{X(t)∈ Q_r(t)}]
    ≤    (1+o(1))∑_z∈ℤ^d𝔼_(0,1)^(X,α̃)𝔼_z^Y[exp(∫_0^tV(X(s)-Y(s),α̃(s)) ṣ)1_{X(t)-Y(t)∈ Q_r(t)}]
    =   (1+o(1))∑_z∈ℤ^d𝔼_(z,1)^(Z,α̃)[exp(∫_0^tV(Z(s),α̃(s)) ṣ)1_{Z(t)∈ Q_r(t)}].

Denote with (·,·) the inner product in ℓ^2(ℤ^d×{0,1}) with corresponding norm · and let 

    λ:=supSp(L̃+V)

be the largest eigenvalue of the bounded and self-adjoint operator L̃+V. Then, applying the spectral representation to the right hand-side of (<ref>) and proceeding in the standard way we obtain the upper bound

    𝔼^(Z,α̃)_(0,1)[exp(∫_0^tV(Z(s),α̃(s))ṣ)]≤    (1+o(1))(^(L̃+V)t1_Q_r(t),1_Q_r(t))
    
    ≤   (1+o(1))^tλ1_Q_r(t)^2
    
    ≤   (1+o(1))^tλ|Q_r(t)|
    
    =   (1+o(1))^tλ(2tlog^2(t))^d.

As (2tlog^2(t))^d grows only polynomially, we have

    lim_t→∞1/tlog𝔼^(Z,α̃)_(0,1)[exp(∫_0^tV(Z(s),α̃(s)) ṣ)]≤λ.

For the lower bound, we proceed as proof of <cit.> to obtain

    𝔼^(Z,α̃)_(0,1)[exp(∫_0^tV(Z(s),α̃(s))ṣ)]≥1/|Q_r(t)|(∑_y∈ Q_r(t)𝔼^(X,α̃)_(0,1)𝔼_0^Y[^A_t/2δ_y(X(t/2)δ_y(Y(t/2))])^2

for A_t:=exp(∫_0^t V(X(s)-Y(s),α̃(s) ṣ). Then, applying lemma <ref>(b) yields

    𝔼^(Z,α̃)_(0,1)[exp(∫_0^tV(Z(s),α̃(s)) ṣ)]≥   (1+o(1))/|Q_r(t)|(∑_y∈ℤ^d𝔼^(X,α̃)_(0,1)𝔼_0^Y[^A_t/2δ_y(X(t/2)δ_y(Y(t/2))])^2
    
    =   (1+o(1))/|Q_r(t)|(𝔼^(X,α̃)_(0,1)𝔼_0^Y[^A_t/2δ_0(X(t/2)-Y(t/2))])^2
    
    ≥   (1+o(1))/|Q_r(t)|(^(L̃+V)t/2δ_0,δ_0)^2.

Now, we restrict the operator L̃+V to finite boxes Q_n:=([-n,n]^d∩ℤ^d)×{0,1} and apply the Perron-Frobenius theorem for non-negative irreducible matrices to derive the existence of a largest eigenvalue λ_n of L̃+V on Q_n:=([-n,n]^d∩ℤ^d)×{0,1}, for which

    lim_t→∞1/tlog𝔼^(Z,α̃)_(0,1)[exp(∫_0^tV(Z(s),α̃(s)) ṣ)]≥λ_n

holds for every n∈ℕ, and show that lim_n→∞λ_n=λ. We omit the details as refer to the proof of <cit.>. Altogether, we have shown that

    lim_t→∞1/tlog𝔼^(Z,α̃)_(0,1)[exp(∫_0^tV(Z(s),α̃(s)) ṣ)]=λ,

where, according to the Rayleigh-Ritz formula, λ is given by

    λ=   sup_f∈ℓ^2(ℤ^d×{0,1}),f_2=1<(L̃+V)f,f>.

Let us calculate the inner product. We have

    <V f, f>=-s_0∑_x∈ℤ^df(x,0)^2-s_1∑_x∈ℤ^df(x,1)^2+γ f(0,1)^2

and

    <L̃f,f>=   ∑_i∈{0,1}∑_x∈ℤ^d((iκ+ρ)∑_y∼ x(f(y,i)-f(x,i))f(x,i)+√(s_0s_1)(f(x,1-i)-f(x,i))f(x,i))
    
    =   ∑_i∈{0,1}∑_j=1^d(iκ+ρ)∑_x∈ℤ^d(f(x+e_j,i)-f(x,i))f(x,i)+(f(x,i)-f(x+e_j,i))f(x+e_j,i)
    
    -   ∑_x∈ℤ^d√(s_0s_1)(f(x,1)-f(x,0))^2
    
    =   -1/2∑_i∈{0,1}(iκ+ρ)∑_x,y∈ℤ^d, x∼ y(f(x,i)-f(y,i))^2-∑_x∈ℤ^d√(s_0s_1)(f(x,1)-f(x,0))^2,

where the factor 1/2 comes from summing over ordered pairs (x,y). Now, recall from (<ref>) that

    lim_t→∞1/tlog<U(t)>=√(s_0s_1)+λ

to conclude. 

















§ PROOF OF THEOREM <REF>

In this chapter, we give a proof for Theorem <ref> and consider a dynamic random environment given by a field of independent random walks with equal jump rate 2dρ starting from a Poisson cloud on ℤ^d with intensity ν. More precisely, we define the potential ξ to be

    ξ(x,t)=∑_y∈ℤ^d∑_j=1^N_yδ_x(Y_j^y(t)),

where N_y is a Poisson random variable with intensity ν>0 for each y∈ℤ^d and {Y_j^y: y∈ℤ^d, j=1,⋯,N_y, Y_j^y(0)=y} is the collection of random walks with jump rate 2dρ>0. 

Our first lemma, which provides a more convenient representation of <U(t)>, is an adaptation of <cit.> to our setting for switching random walks:

For all t≥ 0 and all γ∈[-∞,∞),

    <U(t)>=𝔼_(0,1)^(X,α)[exp(νγ∫_0^tα(s)v_(X,α)(X(s),s) ds)],

where v_(X,α)(y,t):ℤ^d×[0,∞)→ℝ is the solution of

    {[                                d/d tv_(X,α)(y,t)                                                = ρΔ v_(X,α)(y,t)+γδ_(X(t),α(t))(y,1)v_(X,α)(y,t),                                             t>0,;                                     v_(X,α)(y,0)                                                =                                                1 ].

conditioned on a fixed realization of (X,α). 


The proof is similar to the proof of <cit.>, but with the additional component α. Write 𝔼_ν for the expectation of a Poisson random variable with intensity ν. As in <cit.>, we integrate out the Poisson system ξ to obtain

    <U(t)>   =<𝔼_(0,1)^(X,α)[exp(γ∫_0^t∑_kδ_(Y^k(s),1)(X(s),α(s)) ṣ) ]>
       =<𝔼_(0,1)^(X,α)∏_k[exp(γ∫_0^tδ_(Y^k(s),1)(X(s),α(s)) ṣ) ]>
       =𝔼_(0,1)^(X,α)∏_y∈ℤ^d𝔼^ν𝔼_y^Y[exp(γ∫_0^tδ_(Y(s
    ),1)(X(s),α(s))  d s)]
    
       =𝔼_(0,1)^(X,α)∏_y∈ℤ^d𝔼^ν[v_(X,α)(y,t)].

Then, taking the expectation with respect to a Poisson random variable yields

    <U(t)>   =𝔼_(0,1)^(X,α)[∏_y∈ℤ^d∑_n(ν v_(X,α)(y,t))^n/n!e^-ν]
       =𝔼_(0,1)^(X,α)[∏_y∈ℤ^dexp(-ν(1-v_(X,α)(y,t)))]
       =𝔼_(0,1)^(X,α)[exp(-ν∑_y∈ℤ^dw_(X,α)(y,t))]

for w_(X,α):=1-v_(X,α). Note, that for γ<0, the quantity v_(X,α)(y,t) represents the survival probability of Y with start in y up to time t, where the (fixed) trajectory of X is seen as a trap, which tries to capture Y with rate γ whenever it crosses the latter and if α takes the value 1 in that moment. For γ>0, v_(X,α)(y,t) represents the number of particles build out of one single particle starting in 0 which moves around with jump rate 2dκ and branches into two, whenever it meets the random Walk X and if α equals to 1 at this time. Next, we see that

    ∑_y∈ℤ^d d/ d tw_(X,α)(y,t)   =-∑_y∈ℤ^d d/ d tv_X(y,t)
    
       =-∑_y∈ℤ^d(ρΔ v_(X,α)(y,t)+γδ_(X(t),α(t))(y,1)v_(X,α)(y,t))
    
       =-γα(t)v_(X,α)(X(t),t),

which together with the initial condition ∑_y∈ℤ^dw_(X,α)(0)=0 yields

    ∑_y∈ℤ^dw_(X,α)(y,t)=-γ∫_0^tα(s)v_(X,α)(X(s),s)  ds.

Combined with (<ref>), this proofs the claim.

Before continuing our investigations regarding to the asymptotics of (<ref>), we will first consider the case κ=0, i. e. , the case of an immobile particle X staying in 0 the whole time. This idea is highly inspired by <cit.> and <cit.> and will be extended here to the case of switching random walks. For κ=0 the equation (<ref>) with X≡ 0 reduces to

    {[                           /ṭv_(0,α)(y,t)                                        = ρΔ v_(0,α)(y,t)+γα(t)δ_0(y)v_(0,α)(y,t),                              y∈ℤ^d, t>0,;                             v_(0,α)(y,0)                                        =                                       1,                                   y∈ℤ^d, ].

such that the annealed survival probability becomes

    <U(t)>=𝔼_1^α[exp(νγ∫_0^tα(s)v_(0,α)(0,s) ṣ)].

As we will see later, the following two propositions will help us with the general case κ≥ 0:

Let γ∈[-∞,0) and κ=0. Then, as t→∞,

    log<U(t)>={[ -4ν√(ρ s_0/(s_0+s_1)π)√(t)(1+o(1)),                                d=1,;  -4νρπ s_0/s_0+s_1t/log(t)(1+o(1)),                                d=2,;                  -λ̃_d,γ t(1+o(1)),                               d≥ 3, ].

with

    λ̃_d,-∞=inf_a∈[0,1]{s_0-√(s_0s_1)+(s_1-s_0+2dνρ G_d(0)^-1)a-2√(s_0s_1a(1-a))},

where G_d(0) is the Green's function of a simple symmetric random walk in 0, and

    λ̃_d,γ=inf_a∈[0,1]{s_0-√(s_0s_1)+(s_1-s_0+2dνρ/2dρ/|γ|+G_d(0))a-2√(s_0s_1a(1-a))}

for γ∈(-∞,0). 


We start with the case of hard traps, i. e. , γ=-∞, where the random walk X is immediately killed after crossing one of the traps, if α takes the value 1 at this times. Then, in an analogous manner as in (<ref>), the annealed survival probability up to time t can be written as

    <U(t)>=𝔼_1^α[exp(-ν∑_y∈ℤ^dψ_α(y,t))],
 
where 

    ψ_α(y,t)=ℙ_y^Y(∃ s∈[0,t]: Y(s)=0, α(s)=1)

which for γ=∞ corresponds to the probability that Y has been killed up to time t, if we consider 0 as a trap which is only open if α is equal to 1. Recall from Lemma <ref> that ψ_α(y,t)=ψ(y,L_t(1)) with

    ψ(y,t):=ℙ_y^Y(∃ s∈[0,t]:Y(s)=0)

the probability, that Y with start in y does not hit 0 up to time t, regardless of values of α, which solves the differential equation

    {[   /ṭψ(y,t)          =  ρΔψ(y,t), t>0, y≠ 0,;     ψ(0,t)          =         1,       t>0,;     ψ(y,0)          =         0,      y≠ 0. ].

Observe that due to the chain rule,

    /ṭψ(y,L_t(1))=/ṭL_t(1)·/ṭψ(y,·)(L_t(1))=α(t)·Δψ(y,L_t(1)).

Hence,

    /ṭ∑_y∈ℤ^dψ_α(y,t)   =∑_y∈ℤ^dα(t)Δψ(y,L_t(1))-α(t)Δψ(0,t)
    
       =-α(t)ρ∑_y∼ 0(ψ(y,L_t(1))-ψ(0,L_t(1))
    
       =-2dρα(t)(ψ(e_1,L_t(1))-1)

with e_1=(1,0,⋯,0)^T the first unit vector, where we used the symmetry of the random walk as well as the fact that ∑_yΔψ_α(y,t)=0. Thus, setting ϕ(e_1,t):=1-ψ(e_1,t), 

    ∑_y∈ℤ^dψ_α(y,t)=∫_0^t2dρϕ(e_1,L_s(1))α(s)ṣ=∫_0^L_t(1)2dρϕ(e_1,s)ṣ

where we substituted L_s(1) in the last step. Now, the quantity ϕ(e_1,t), which represents the probability that a random walk with jump rate 2dρ and start in e_1 does not hit the origin up to time t, is known (see e.g. <cit.>) to have the asymptotics

    ϕ(e_1,t)={[ √(1/πρ t)(1+o(1)),               d=1,;  π/log(t)(1+o(1)),               d=2,; G_d(0)^-1(1+o(1)),              d≥ 3, ].

as t→∞, where G_d is the Green's function of a d-dimensional symmetric random walk with generator Δ. Thus,

    ∑_y∈ℤ^dψ^α(y,t)={[      4√(ρ/π)√(L_t(1))(1+o(1)),                           d=1,; ; 4πρL_t(1)/log(L_t(1))(1+o(1)),                           d=2,; ;   2dρ G_d(0)^-1L_t(1)(1+o(1)),                          d≥ 3, ].

as t→∞. In the following, we distinguish between the recurrent dimensions d∈{1,2} on one side and the transient dimensions d≥ 3 on the other side. Let us start with the case d≥ 3 and recall the large deviation principle for the normalized local times (1/tL_t(1))_t≥ 0 of α in state 1 from theorem <ref> with rate function  I given by (<ref>), which has a unique zero at s_0/s_0+s_1. As the function g [0,1]→ℝ, x↦ -νρ G_d(0)^-1x is continuous and bounded, we can apply Varadhan's lemma to deduce the limit

    -lim_t→∞1/tlog𝔼_1^α[exp(-ν2dρ/G_d(0)L_t(1))]=inf_a∈[0,1]{I(a)+2dνρ/G_d(0)a}=:λ̃_d,∞.

This establishes the asymptotics (<ref>) for d≥ 3. For d=1, we apply lemma <ref> with f(t)=√(t) to obtain

    lim_t→∞1/√(t)log𝔼_1^α[exp(-ν√(8ρ/π)√(L_t(1)))]=-4ν√(ρ/π)√(s_0/s_0+s_1).












The case d=2 is similar with f(t)=t/log(t) and F(a)=-4νπρ a. Next, we continue with γ∈(-∞,0). Recall the representation (<ref>) of the annealed survival probability as well as the solution v_(0,α)(y,t) of (<ref>), which is the survival probability of Y up to time t, if we interpret 0 as a trap which tries to kill Y with rate |γ| at time s if (Y(s),α(s))=(0,1). Here, v_(0,α) plays the analogous role of ϕ_α, and w_(0,α)=1-v_(0,α) the role of ψ_α. Denote by w(y,t) the killing probability of a random walk Y with start in y up to time t, if Y is killed in 0 at rate γ independent of α, and v:=1-w, which solves the differential equation

    {[               /ṭv(y,t)                      = ρΔ v(y,t)+γδ_0 v(y,t),            y∈ℤ^d, t>0,;                 v(y,0)                      =                     1,                  y∈ℤ^d ]..

Then, applying lemma <ref> again yields w_(0,α)(y,t)=w(y,L_t(1)) and therefore

    v_(0,α)(y,t)=v(y,L_t(1)).

In <cit.> it has been shown that v(0,t) has the asymptotics

    v(0,t)={[ 1/|γ|√(ρ/π)1/√(t)(1+o(1)),                       d=1,;   4πρ/|γ|1/log(t)(1+o(1)),                       d=2,;  2dρ/2dρ-γ G_d(0)(1+o(1)),                      d≥ 3, ].

as t→∞. Moreover,

    ∫_0^tα(s)v_(0,α)(0,s) ṣ=∫_0^tα(s)v(0,L_s(1)) ṣ=∫_0^L_t(1))v(0,s) ṣ

as in (<ref>). This yields

    νγ∫_0^tα(s)v_(0,α)(0,s)ṣ={[       -4ν√(ρ/π)√(L_t(1))(1+o(1)),                              d=1,;  -4νπρL_t(1)/log(L_t(1))(1+o(1)),                              d=2,; νγ2dρ/2dρ-γ G_d(0)L_t(1)(1+o(1)),                             d≥ 3, ].

for t→∞. In the dimensions d∈{1,2}, we can apply Lemma <ref> as in the case γ=∞, whereas in dimension d≥ 3 Varadhan's Lemma again tells us that

    -lim_t→∞1/tlog𝔼_1^α[exp(2dνρ/2dρ/γ-G_d(0)L_t(1))]=inf_a∈[0,1]{I(a)-2dνρ/2dρ/γ-G_d(0)a}=:λ̃_d,γ.

This proofs the proposition for the case γ∈(-∞,0). 


Note that in the first two dimensions the survival probability decays sub-exponentially and does not depend on γ and in higher dimensions d≥ 3 the asymptotics for γ=∞ are consistent with those of the case γ<∞, as lim_γ→∞λ̃_d,γ=λ̃_d,∞.

The next proposition deals with the case γ>0 of catalysts, still under the assumption κ=0.

Let γ∈(0,∞) and κ=0. Then for all dimensions d≥ 1 the annealed number of particles satisfies the double-exponential asymptotics

    lim_t→∞1/tloglog<U(t)>=sup_f∈ℓ^2(ℤ^d),f_2=1(γ f(0)^2-1/2∑_x,y∈ℤ^d, x∼ yρ(f(x)-f(y))^2).



Recall the representation (<ref>) as well as the solution v_(0,α) to (<ref>) for a fixed realization of α, which can also be written as

    v_(0,α)(0,t)=𝔼_0^Y[exp(γ∫_0^tα(s)δ_0(Y(s))ṣ)]

in point 0. Let

    v(0,t):=𝔼_0^Y[exp(γ∫_0^tδ_0(Y(s))ṣ)]

which plays the analogous role of the function v in the proof of Proposition <ref>, but now for positive γ. In <cit.> it has been shown that

    lim_t→∞1/tlog v(0,t)=μ

where

    μ:=sup_f∈ℓ^2(ℤ^d),f_2=1(γ f(0)^2-1/2∑_x,y∈ℤ^d, x∼ yρ(f(x)-f(y))^2)

is the largest eigenvalue of the self-adjoint operator ℋ:=ρΔ+γδ_0 on ℓ^2(ℤ^d). Furthermore, it is known from <cit.> that μ is always positive in dimensions d=1,2 and

    μ{[            = 0, 0<γ/ρ≤1/G_d(0),;             >0,   γ/ρ> 1/G_d(0) ].

in dimensions d≥ 3, where G_d is the Green's function of a simple symmetric random walk with jump rate 2d. Next, we aim to compare v(0,t) to v_(0,α)(0,t) by applying lemma <ref> again in the same manner as in the proof of proposition <ref>, which yields

    v_(0,α)(0,t)=v(0,L_t(1)).

Hence, for κ=0, 

    <U(t)>=   𝔼_1^α[exp(νγ∫_0^tα(s)v(0,L_s(1)) ṣ)]=𝔼_1^α[exp(νγ∫_0^L_t(1)v(0,s) ṣ)]
    
    =   𝔼_1^α[exp(νγ∫_0^L_t(1)^μ s(1+o(1)) ṣ)]=𝔼_1^α[exp(νγ/μ^μ L_t(1)(1+o(1)))]

as t→∞. Now, on one hand, we have the upper bound

    <U(t)>≤exp(νγ/μ^μ t(1+o(1))),    t→∞,

and on the other hand,

    𝔼_1^α[exp(νγ/μ^μ L_t(1)(1+o(1)))]≥   exp(νγ/μ^μ t(1+o(1)))ℙ_1^α(L_t(1)=t),

as t→∞. Now, recall the large deviation principle for for the normalized local times with rate function I on scale t, which asserts that

    lim_t→∞1/tlogℙ_1^α(L_t(1)=t)=-I(1).

Hence, combining (<ref>) with (<ref>),

    lim_t→∞1/tloglog𝔼_1^α[exp(νγ/μ^μ L_t(1)(1+o(1)))]≥   lim_t→∞1/tloglog(exp(νγ/μ^μ t)ℙ_1^α(L_t(1)=t))
    
    =   lim_t→∞1/tlog(νγ/μ^μ t+logℙ_1^α(L_t(1)=t))
    
    =   lim_t→∞1/tlog(νγ/μ^μ t)=μ.


The next ingredient for the proof of our main result is the so called Pascal principle which asserts that if we average over the environment, than the best the random walk X can do in order to maximize its mass is to stay still in the starting point, which brings us back to the case κ=0. In the setting of a simple random walk without any switching component, this has been proven in <cit.> and  <cit.> for γ<0 and γ>0 respectively. Therefore, the question arises if the Pascal principle can still provide an upper bound also in our case of a switching random walk, or if there is a better joint strategy of the random walk together with the dormancy component α. The next lemma ensures that the Pascal principle still proves best also in our case.
 Recall v_(X,α) as the solution to (<ref>) and the solution v_(0,α) of (<ref>) for any fixed realization of (X,α). Then, for all γ∈[-∞,∞), y∈ℤ^d and t≥ 0,

    v_(X,α)(y,t)≤ v_(0,α)(0,t).



First, let γ∈[-∞,0) and recall from (<ref>) that v_(0,α)(y,t)=v(y,L_t(1)). Now, <cit.> asserts that for any piecewise constant function X̂:[0,t]→ℤ^d with a finite number of discontinuities,

    v_X̂(y,t)≤ v(0,t),

where v_X̂ is the solution to

    {[                     /ṭv_X̂(y,t)                               = ρΔ v_X̂(y,t)+γδ_X̂(y)v_X̂(y,t),                     y∈ℤ^d, t>0,;                       v_X̂(y,0)                               =                              1,                          y∈ℤ^d. ].

In a similar way as in (<ref>), we see that if v_(X̂,α) is the solution to

    {[                             /ṭv_(X̂,α)(y,t)                                           = ρΔ v_(X̂,α)(y,t)+γα(t)δ_X̂(t)v_(X̂,α)(y,t),                                 y∈ℤ^d, t>0,;                               v_(X̂,α)(y,0)                                           =                                          1,                                      y∈ℤ^d, ].

then v_(X̂,α)(y,t)=v_X̂(y,L_t(1)), as X̂ is independent of α. Hence,

    v_(X̂,α)(y,t)=v_X̂(y,L_t(1))≤ v(0,L_t(1))=v_(0,α)(0,t).

Finally, conditioned on α, the random walk X is a piecewise constant function with a finite number of discontinuities, such that we can replace X̂ with X.

Next, let γ>0. The proof works along the same lines as in <cit.>. From this proof we already know that if p_ρ(x,y) denotes the probability density function of a random walk with generator ρΔ and start in 0, then

    max_x∈ℤ^dp_ρ(x,y)=p_ρ(0,t)

for all t≥ 0. Further, in <cit.> has been shown that h^*n→ 0, n→∞, uniformly on compact intervals, where h^*n denotes the n-fold convolution of the function

    h(t):=γ p_ρ(0,t).

Since

    v_(X,α)(X(t),t)=   1+γ∫_0^tp_ρ(X(t)-X(s),t-s)α(s)v_(X,α)(X(s),s) ṣ
    ≤   1+γ∫_0^tp_ρ(0,t-s)α(s)v_(X,α)(X(s),s) ṣ

and

    v_(0,α)(0,t)=   1+γ∫_0^tp_ρ(0,t-s)α(s)v_(0,α)(0,s) ṣ,

we have

    v_(X,α)(X(·),·)≤ 1+h^*(α v_(X,α)(X(·),·))

as well as

    v_(0,α)(0,·)= 1+h^*(α v_(0,α)(0,·)).

Hence,

    v_(0,α)(0,·)-v_(X,α)(X(·),·)≥ h^*n(α(v_(0,α)(0,·)-v_(X,α)(X(·),·))

by iteration and substraction of (<ref>) and  (<ref>). As h^*n→ 0 for  n→∞, we can deduce

    v_(X,α)(X(·),·)≤ v_(0,α)(0,·)

for all realizations of (X,α). Altogether, we obtain

    v_(X,α)(y,t)=   1+γ∫_0^tp_ρ(y-X(s),t-s)α(s)v_(X,α)(X(s),s) ṣ
    ≤   1+γ∫_0^tp_ρ(0,t-s)α(s)v_(X,α)(X(s),s) ṣ
    ≤   1+γ∫_0^tp_ρ(0,t-s)α(s)v_(0,α)(0,s)ṣ=v_(0,α)(0,t),

as desired. 

We are now ready to prove Theorem 1.3.


  §.§.§ Proof of Theorem <ref>

The proof can be summarized as follows: We will first show that the case κ=0 considered in the Proposition <ref> and <ref> provides a lower bound for the general case κ≥ 0 in dimensions d=1,2. This gives us together with the upper bound asserted in Lemma <ref> the desired asymptotics. 

Let us start with the case γ<0 of traps and show that the asymptotics of <U(t)> are lower-bounded by (<ref>) in dimensions d∈{1,2}. We adapt the notations from the proof of <cit.>: Let E_t be the event that none of the traps starts in a ball B_R_t of radius R_t around 0, where we choose R_t to be t/ln(t) for d=1 resp. ln(t) for d=2. This event has the probability ^-ν(R_t+1)^d. Further, let G_t be the event that X with start in 0 stays in B_R_t up to time t. Analogously, we define G̃_t to be the event that a simple symmetric random walk X̃ without switching and with jump rate 2dκ stays in B_R_t up to time t. Then, using time-change again,

    ℙ(G_t)   =ℙ_(0,1)^(X,α)(X(s)∈ B_R_t∀ s≤ t)=ℙ_1^αℙ_0^X̃(X̃(s)∈ B_R_t∀ s≤ L_t(1))
    
       ≥ℙ_1^αℙ_0^X̃(X̃(s)∈ B_R_t∀ s≤ t)=ℙ(G̃(t))≥exp(ln(β)t/R_t^2),

where the last inequality is known from <cit.> for some β>0. Moreover, let F_t be the event that each trap which starts outside B_R_t only intersects B_R_t during time periods where α takes the value 0, i.e. , where X is dormant. Then, we can lower-bound

    <U(t)>≥ℙ(E_t)ℙ(F_t)ℙ(G_t).

Note that the event F_t differs from the analogous event appearing in the proof of <cit.>, which was the event that each trap which starts outside B_R_t never enters B_R_t up to time t. Here, making use of the protection provided by the dormancy mechanism, we can relax this condition and only require that the traps stay outside of B_R_t whenever X is active. In order to compute ℙ(F_t) we first look at F̃_t which shall denote the event that no trap Y which starts in y≠ 0 ever hits 0 at time points s with α(s)=1. The probability of this event is nothing but the survival probability of X in case κ=0, which has been studied in Proposition <ref>. More precisely,

    ℙ(F_t)=𝔼_1^α[exp(-ν∑_y∈ℤ^dw_(0,α)(y,t))],

for X≡ 0, as seen in (<ref>). Thus, ℙ(F̃_t) is asymptotically equal to (<ref>). Comparing ℙ(F_t) to ℙ(F̃_t) exactly in the same way as in <cit.>, we find that these are asymptotically equal. Finally, we compare the decay rates of all probabilities ℙ(E_t),ℙ(F_t),ℙ(G_t) for t→∞ to conclude that the annealed survival probability is asymptotically lower bounded by ℙ(F_t) and hence (<ref>). 

Note, that we did not prove a lower bound in dimension d=3, so that we can only deduce the existence of a constant λ_d,γ,ρ,κ,s_0,s_1≥λ̃_d,γ which may depend on all the parameters.

We continue with the case γ>0. The upper bound again follows from the Pascal principle stated in Lemma <ref>. For the lower bound, we force the random walk X to stay in the starting point 0 up to time t and use the fact that, for a simple symmetric random walk X̃ without switching and with jump rate 2dκ, 

    ℙ_0^X̃(X̃(s)=0∀ s∈[0,t])=^-2dκ t

to deduce that

    lim_t→∞1/tlogℙ_(0,1)^(X,α)(X(s)=0∀ s∈[0,t])=   lim_t→∞logℙ_1^α(X̃(s)=0∀ s∈[0,L_t(1)])
    
    =   lim_t→∞1/tlog𝔼_1^α(^-2dκ L_t(1))
    
    =   -inf_a∈[0,1](2dκ a+I(a)),

where we used Varadhan's Lemma together with the large deviation principle for 1/tL_t(1) with rate function I. Moreover, recall from the proof of <ref> that

    <U(t)_0>≤exp(νγ/μ^μ t(1+o(1))),    t→∞,

where U(t)_0 shall denote the number of particles up to time t in the case κ=0. Hence, 

    lim_t→∞1/tloglog<U(t)>   =
    lim_t→∞1/tloglog(exp(νγ/μ^μ t(1+o(1)))·ℙ_(0,1)^(X,α)(X(s)=0∀ s∈[0,t]))
    
       =lim_t→∞1/tlog(νγ/μ^μ t(1+o(1))+log𝔼_1^α(^-2dκ L_t(1)))
    
       =μ.




 §.§ Acknowledgement

The author would like to thank Professor Wolfgang König for his invaluable support and many helpful suggestions. 
WK2020	


[AH79]doubleI
E. Aifantis, J. Hill,
On the Theory of Diffusion in Media with Double Diffusivity I. Basic Mathematical Results,
The Quarterly Journal of Mechanics and Applied Mathematics 33, No. 1, 1–21 (February 1980).



[A95]antal
P. Antal,
Enlargement of obstacles for the simple random walk,
Annals of probability 23, No. 3, 1061-1101 (July 1995).






[BYZ13]baran
N. A. Baran, G. Yin, Chao Zhu,
Feynman-Kac formula for switching
diffusions: connections of systems of partial
differential equations and stochastic
differential equations,
Advances in Difference Equations, No. 315 (2013).



[BK01]bounded
M. Biskup, W. König,
Long-time Tails in the Parabolic Anderson Model with Bounded Potential,
The Annals of Probability 29, No. 2, 636–682 (2001).



[BHS21]blath
J. Blath, F. Hermann, M. Slowik,
A branching process model for dormancy and seed banks in
randomly fluctuating environments,
Journal of Mathematical Biology 83, No. 17 (2021).




[BHLWB21]dormancy
J. Blath, J. T. Lennon, F. den Hollander, M. Wilke Berenguer,
Principles of seed banks and the emergence of
complexity from dormancy,
Nature Communications 12, No. 4807 (2021).



[C66]cohen
D. Cohen,
Optimizing reproduction in a randomly varying environment,
Journal of Theoretical Biology 16, 267–282 (1966).



[DGRS11]drewitz
A. Drewitz, J. Gärtner, A. Ramirez, R. Sun,
Survival probability of a random walk among a poisson system of moving traps,
Conference Paper. In Probability in complex physical systems, 119–158, Springer (2012)



[EdHM16]dyn
D. Erhard, F. den Hollander, G. Maillard,
Parabolic Anderson Model in a Dynamic Random Environment: Random Conductances,
Mathematical Physics, Analysis and Geometry 19, No. 5 (2016).



[GdHM07]exclusion
J. Gärtner, F. den Hollander, G. Maillard,
Intermittency on catalysts: symmetric exclusion,
Electronic Journal of Probability 12, No. 12, 516-573 (2007).



[GdH06]intermittency
J. Gärtner, F. den Hollander,
Intermittency in a catalytic random medium,
Annals of probability 6, 2219-2287 (2006).




[GdHO22]spatialseedbanks
A. Greven, F. den Hollander, M. Oomen,
Spatial populations with seed-bank: well-posedness, duality and equilibrium,
Electronic Journal of Probability 27, 1-88 (2022).





[GH06]movingcat
J. Gärtner, M. Heydenreich,
Annealed asymptotics for the parabolic Anderson model with a moving catalyst,
Stochastic Processes and their Applications 116, 1511–1529 (2006).




[GM90]garmol
J. Gärtner, S. Molchanov,
Parabolic problems for the Anderson
model I. Intermittency and related topics,
Communications in Mathematical Physics 132, 613–
655 (1990).



[dHN]nandan
F. den Hollander, S. Nandan,
Spatially Inhomogeneous Populations with Seed-Banks: I. Duality, Existence and Clustering,
Journal of Theoretical Probability 35, 1795–1841 (2022).




[HJV07]bookhaccou
P. Jagers, P. Haccou, V. A. Vatutin,
Branching processes: variation, growth, and extinction of populations,
Biometrics 62, No. 4, 1269–1270 (December 2006).




[H04]heyd
M. Heydenreich,
Intermittency in the Parabolic Anderson Model with Catalyst,
Master's thesis, LMU München (2004).





[JL11]jl11
S. E. Jones, J. T. Lennon,
Microbial seed banks: the ecological and evolutionary implications of dormancy,
Nature Reviews Microbiology 9, 119–130 (2011).





[K16]PAM
W. König, 
The Parabolic Anderson Model: Random Walk in Random Potential, 
Birkhäuser (2016).



[K20]koenigldp
W. König, 
Große Abweichungen: Techniken und Anwendungen, 
Birkhäuser (2020).



[K20a]koenigsurvey
W. König, 
Branching random walks in random environment, 
in Probabilistic Structures in Evolution, 23-41, 
European Mathematical Society (2020).




[L92]landim
C. Landim,
Occupation time large deviations for the symmetric simple exclusion process,
Annals of Probability 20, 206–231, (1992).




[L96]lawler
G. F. Lawler,
Intersections of random walks,
Birkhäuser Boston (1996).





[LS18]ls18
J. T. Lennon, W. TR. Shoemaker,
Evolution with a seed bank: the population genetic consequences of
microbial dormancy,
Evolutionary Applications 11, No. 1, 60–75 (2018).





[PR02]girsanovmarkov
Z. Palmowski, T. Rolski,
A technique for exponential change of
measure for Markov processes,
Bernoulli 8, No. 6, 767–785 (2002).




[SW11]sw
A. Schnitzler, T. Wolff,
Precise asymptotics for the parabolic Anderson
model with a moving catalyst or trap,
Conference Paper. In Probability in complex physical systems, 69-89, Springer (2012)

[YZ10]switching
G. George Yin, Chao Zhu,
Hybrid Switching Diffusions: Properties and Applications
Springer New York, NY (2010)


	
	



