Time Reversal of Some Stationary Jump-Diffusion Processes from Population Genetics
TTime Reversal of Some StationaryJump-Diffusion Processesfrom Population Genetics
Martin Hutzenthaler ∗† Goethe-University Frankfurt
Jesse E. Taylor
University of Oxford
Abstract
We describe the processes obtained by time reversal of a class of stationary jump-diffusionprocesses that model the dynamics of genetic variation in populations subject to repeatedbottlenecks. Assuming that only one lineage survives each bottleneck, the forward process isa diffusion on [0 ,
1] that jumps to the boundary before diffusing back into the interior. Weshow that the behavior of the time-reversed process depends on whether the boundaries areaccessible to the diffusive motion of the forward process. If a boundary point is inaccessibleto the forward diffusion, then time reversal leads to a jump-diffusion that jumps immediatelyinto the interior whenever it arrives at that point. If, instead, a boundary point is accessible,then the jumps off of that point are governed by a weighted local time of the time-reversedprocess.
Kingman’s observation that the genealogy of a random sample of individuals from a panmictic,neutrally-evolving population can be represented as a Markov process [16, 17] ranks as one ofthe most influential contributions of mathematical population genetics. Not only has the coa-lescent led to a deeper understanding of evolution in neutral populations, but it also plays acentral role in statistical genetics where it facilitates the efficient simulation of sample genealogies.Unfortunately, the Markov property that makes Kingman’s coalescent both mathematically andcomputationally tractable is usually not shared by genealogical processes in populations composedof non-exchangeable individuals. In particular, this is true when there are fitness differences be-tween individuals, since then the selective interactions between individuals cause genealogies todepend on the history of lineages that are non-ancestral to the sample. The key to overcomingthis difficulty is to extend the genealogy to a higher-dimensional process that does satisfy theMarkov property. This has been done in two ways. One approach is to embed the genealogicaltree within a graphical process called the ancestral selection graph [18, 24, 6] in which lineages canboth branch and coalesce. The intuition behind this construction is that the effects of selectionon the genealogy can be accounted for by keeping track of a pool of potential ancestors whichincludes lineages that have failed to persist due to being out-competed by individuals of higherfitness.An alternative approach was proposed by Kaplan et al. (1988) [12], who showed that the ge-nealogical history of a sample of genes under selection can be represented as a structured coalescent ∗ Research supported by the DFG in the Dutch German Bilateral Research Group ”Mathematics of RandomSpatial Models from Physics and Biology” (FOR 498) † Research supported by EPSRC Grant no GR/T19537/01
AMS 2010 subject classifications:
Primary 60J60; secondary 60J55, 92D10.
Keywords and phrases:
Time Reversal, Jump-Diffusions, Local Time, Coalescents, Population Bottlenecks,Selective Sweeps a r X i v : . [ m a t h . P R ] J a n rocess. Here we think of the population as being divided into several panmictic subpopulations(called genetic backgrounds) which consist of individuals that share the same genotype at theselected locus. Because individuals with the same genotype are exchangeable (i.e., they have thesame fitness), the rate of coalescence within a background depends only on the size of the back-ground and the number of ancestral lineages sharing that genotype. Thus, to obtain a Markovprocess, we need to keep track of two kinds of information: (i) the types of the ancestral lineages,and (ii) the frequencies of the alleles segregating at the selected locus, followed backwards in time.For many applications it is assumed that the population is at equilibrium and that the forwardsin time dynamics of the allele frequencies are described by a stationary diffusion process. In thiscase, the ancestral process of allele frequencies can be identified by time reversal of the diffusionprocess. In particular, if the diffusion process is one-dimensional, then the time-reversed processconveniently has the same law as the forward process. A formal derivation of the structured co-alescent process for such an equilibrium population is given in [2] and various applications arediscussed in [1, 5, 30].The focus of this article is on the time reversal of a population genetical model that incorporatesmutation, selection, genetic drift and population bottlenecks. To be concrete, consider a locus withtwo alleles, A and A , and let p N ( t ) denote the frequency of A at time t in a population of size N .In the absence of bottlenecks, we will suppose that the jump process p N ( · ) can be approximatedby the Wright-Fisher diffusion p ( · ) with generator Aφ ( p ) = 12 p (1 − p ) φ (cid:48)(cid:48) ( p ) + ( µ (1 − p ) − µ p + s ( p ) p (1 − p )) φ (cid:48) ( p ) ≡ v ( p ) φ (cid:48)(cid:48) ( p ) + µ ( p ) φ (cid:48) ( p ) , (1)where µ and µ are the scaled mutation rates from A to A and from A to A , respectively,and s ( p ) is the scaled and possibly frequency-dependent selection coefficient of A relative to A .In using the diffusion approximation, we assume that N is large, that time is measured in unitsof N generations, and the unscaled mutation rates and selection coefficient are of order N − .Convergence results justifying the passage to the diffusion limit can be found in [7].Population bottlenecks are transient events during which most of the population is descendedfrom a small number of individuals. On the diffusive time scale, these can be modeled as instanta-neous jumps in the allele frequencies, and in this article we will be concerned with a class of modelsin which the bottlenecks always result in the temporary fixation of one of the two alleles, i.e., p ( · )always jumps to 0 or 1. We have two scenarios in mind. In the first, we consider a locus that ispart of a non-recombining segment of DNA (e.g., a mammalian mitochondrial genome) subject tostrong selective sweeps which occur at rate λ . During each sweep, a unique copy of a favorablemutation arises at some linked site and rises rapidly to fixation. Depending on whether the new,strongly-selected mutation occurs on a chromosome carrying an A or A allele, the frequency of A will either increase from p to 1 with probability p or decrease from p to 0 with probability 1 − p .Here we imagine that the selective advantage of the favored mutation is so strong that this changecan be treated as a jump. The pseudohitchhiking model introduced by Gillespie [10] belongs tothis class, as does a related, more general model studied by Kim [15].The second scenario concerns demographic bottlenecks that occur during transmission of par-asites from infected to uninfected hosts. Here we will let p denote the frequency of A in achronological series of infected hosts linked by a transmission chain, and we will assume that p ( · )can be modeled by a diffusion process from the time when one of these hosts is first infected tothe time when that host first transmits the infection to the next host in the transmission chain.Suppose that transmissions occur at rate λ , and that each new infection is founded by a singleparasite, as has been proposed for HIV-1 [31] and for some bacterial pathogens [29]. In this case, p will jump to 0 or 1 following each transmission depending on the type of the transmitted parasite.Also, to allow for the possibility that transmission itself might be selective (e.g., [27]), we will let w ( p ) denote the probability that the transmitted parasite is of type A given that the frequencyof this allele in the transmitting host is p . In general, we stipulate that w (0) = 0, w (1) = 1, andthat w ( p ) is monotonically increasing. If transmission is unbiased, then w ( p ) = p , as in the pseu-2ohitchhiking model. A particular case of this transmission chain model was studied by Rouzineand Coffin [28] to understand the effects of selection and transmission bottlenecks on antigenicvariation in HIV-1.Both of these scenarios can be modeled by a jump-diffusion process with infinitesimal generator Gφ ( p ) = 12 p (1 − p ) φ (cid:48)(cid:48) ( p ) + ( µ (1 − p ) − µ p + s ( p ) p (1 − p )) φ (cid:48) ( p )+ λw ( p ) (cid:0) φ (1) − φ ( p ) (cid:1) + λ (1 − w ( p )) (cid:0) φ (0) − φ ( p ) (cid:1) , (2)where for technical reasons we will assume that s ( p ) and w ( p ) are smooth functions on [0 , µ and µ , are positive. Under these conditions, it can be shown (cf.Lemma 3.1) that the process p ( · ) has a unique stationary distribution, π ( p ) dp , which has a densityon [0 , p ( · ). Formally, this can be done by solvingthe following adjoint problem for the operator ˜ G : (cid:90) ψ ( p ) Gφ ( p ) π ( p ) dp = (cid:90) φ ( p ) ˜ Gψ ( p ) π ( p ) dp, (3)where φ is in the domain of G . If ˜ G generates a Markov process ˜ p ( · ), then this process will havethe same law as the stationary time reversal of p ( · ) [23]. When λ = 0, p ( · ) is a diffusion processand a simple calculation using integration-by-parts shows that ˜ G = G , demonstrating that thelaw of the diffusion is invariant under time-reversal, as remarked above. However, if λ >
0, thenfor the adjoint condition (3) to be satisfied for all φ ∈ C ( R ) ∩ C [0 , Gψ ( p ) = 12 p (1 − p ) ψ (cid:48)(cid:48) ( p ) + ˜ µ ( p ) ψ (cid:48) ( p ) , (4)where ˜ µ ( p ) = 1 π ( p ) (cid:0) p (1 − p ) π (cid:48) ( p ) + (1 − p − µ ( p )) π ( p ) (cid:1) (5)and ψ ∈ C ( R ) ∩ C [0 ,
1] satisfies ψ (1) = (cid:90) ψ ( p ) (cid:18) w ( p ) π ( p ) κ (cid:19) dp and ψ (0) = (cid:90) ψ ( p ) (cid:18) (1 − w ( p )) π ( p )1 − κ (cid:19) dp with κ = (cid:82) w ( p ) π ( p ) dp . Although it is not immediately clear that the operator defined by (4)is the generator of a Markov process, this calculation does show that the process incorporatingbottlenecks is not invariant under time reversal.To gain some insight into the qualitative behavior of the time-reversed process, it is usefulto consider two heuristic descriptions. We begin by observing that the behavior of ˜ p ( · ) dependsstrongly on whether the boundary points { , } are accessible or inaccessible to the diffusive motionof the forward process. Recall that for the Wright-Fisher diffusion corresponding to A (which wecall the diffusive motion of the jump-diffusion process), Feller’s boundary classification conditionsshow that 0 (resp. 1) is accessible if and only if u < / u < / ,
1) whenever it arrives at a boundary that is inaccessible to the forward diffusion. Thebehavior of the time-reversed process at a boundary that is accessible to the forward diffusion isvery different. In this case, when the sample path of the time-reversed jump diffusion hits that3oundary, the forward process may have arrived there either diffusively or via a jump from theinterior (Figure 1B). Accordingly, the time-reversed process need not immediately jump into theinterior (0 ,
1) when it visits the boundary, although jumps can only occur when the process is onthe boundary and are certain to occur at some such times if λ > p t i m e bottleneckA: Inaccessible Boundaries pbottleneckB: Accessible Boundaries Figure 1: Sample paths of the jump-diffusion process (2) with either inaccessible (A) or accessibleboundaries (B). The forward diffusion is a neutral Wright-Fisher process with symmetric mutation: µ = µ = 1 in A and 0 . (cid:15) ∈ (0 , / p (cid:15) ( · ) = (cid:0) p (cid:15) ( t ) : t ≥ (cid:1) be a perturbation of a Wright-Fisher diffusion which at rate λ jumpsto a point chosen uniformly at random from an interval of width (cid:15) adjacent to one of the twoboundaries. More precisely, let p (cid:15) ( · ) be the Markov process with generator G (cid:15) φ ( p ) = 12 p (1 − p ) φ (cid:48)(cid:48) ( p ) + (cid:0) µ (1 − p ) − µ p + s ( p ) p (1 − p ) (cid:1) φ (cid:48) ( p ) + λ (cid:18) w ( p ) 1 (cid:15) (cid:90) − (cid:15) ( φ ( q ) − φ ( p )) dq + (1 − w ( p )) 1 (cid:15) (cid:90) (cid:15) ( φ ( q ) − φ ( p )) dq (cid:19) . Writing π (cid:15) ( p ) for the density of the stationary distribution of this process, a simple calculation4sing (3) shows that the stationary time reversal of p (cid:15) ( · ), denoted ˜ p (cid:15) ( · ), is also a jump diffusionprocess with generator˜ G (cid:15) ψ ( p ) = 12 p (1 − p ) ψ (cid:48)(cid:48) ( p ) + 1 π (cid:15) ( p ) (cid:0) p (1 − p ) π (cid:48) (cid:15) ( p ) + (1 − p − µ ( p )) π (cid:15) ( p ) (cid:1) ψ (cid:48) ( p ) + λκ (cid:15) (cid:18) (cid:15)π (cid:15) ( p ) 1 (1 − (cid:15), ( p ) (cid:19) (cid:90) (cid:18) w ( q ) π (cid:15) ( q ) κ (cid:15) (cid:19) ( ψ ( q ) − ψ ( p )) dq + λ (1 − κ (cid:15) ) (cid:18) (cid:15)π (cid:15) ( p ) 1 [0 ,(cid:15) ) ( p ) (cid:19) (cid:90) (cid:18) (1 − w ( q )) π (cid:15) ( q )1 − κ (cid:15) (cid:19) ( ψ ( q ) − ψ ( p )) dq, where ψ ∈ C ([0 , κ (cid:15) = (cid:82) w ( p ) π (cid:15) ( p ) dp . It is easy to read off the behavior of this processfrom its generator. In particular, we see that ˜ p (cid:15) ( · ) can only jump when it is present in the region[0 , (cid:15) ) ∪ (1 − (cid:15),
1] and that the rate at which jumps occur out of this region is equal to λκ (cid:15) / ( (cid:15)π (cid:15) ( p ))when p ∈ (1 − (cid:15),
1] and λ (1 − κ (cid:15) ) / ( (cid:15)π (cid:15) ( p )) when p ∈ [0 , (cid:15) ).To relate these observations to the process ˜ p ( · ), let T > (cid:15) tends to 0,the sequence of processes ( p (cid:15) ( · )) converges in distribution on D [0 , ([0 , T ]) to p ( · ). Furthermore,because time reversal is a continuous mapping on D [0 , ([0 , T ]), the continuous mapping theorem[7] implies that the sequence of processes (˜ p (cid:15) ( · )) converges in distribution to the process ˜ p ( · ). Inparticular, this suggests that ˜ p ( · ) has the following behavior. For each (cid:15) ∈ (0 , / L ,(cid:15) ( t ) ≡ (cid:15) (cid:90) t (cid:18) π (˜ p ( s )) (cid:19) (1 − (cid:15), (˜ p ( s )) dsL ,(cid:15) ( t ) ≡ (cid:15) (cid:90) t (cid:18) π (˜ p ( s )) (cid:19) [0 ,(cid:15) ) (˜ p ( s )) ds, and suppose that for i = 0 ,
1, the limits L i ( t ) = lim (cid:15) → L i,(cid:15) ( t ) exist for all t ≥
0. Here we wouldlike to interpret L i ( t ) as the local time of the process ˜ p ( · ) at i ∈ { , } . Then, by comparisonwith the jump-diffusion processes ˜ p (cid:15) ( · ), we expect that if both boundaries are accessible, then˜ p ( · ) is a jump-diffusion with diffusive motion in (0 ,
1) governed by (4) which jumps from theboundary point 0 to a random point in the interval (0 ,
1) distributed as κ w ( q ) π ( q ) dq as soonas L ( · ) exceeds an exponential random variable with parameter λκ and which jumps from theboundary point 1 to a random point distributed as − κ (cid:0) − w ( q ) (cid:1) π ( q ) dq on (0 ,
1) as soon as L ( · )exceeds an exponential random variable with parameter λ (1 − κ ). Although these remarks arepurely heuristic, we show below that they correctly describe the stationary time reversal of thepseudo-hitchhiking model and other jump-diffusions with generators of the form (2). Although our principle concern is with the modified Wright-Fisher process corresponding to (2),we state our results for a more general class of jump-diffusion processes, which we now introduce.Let the forward process (cid:0) p ( t ) : t ≥ (cid:1) be the jump-diffusion process on [0 ,
1] corresponding to thegenerator Gφ ( p ) = v ( p ) φ (cid:48)(cid:48) ( p ) + µ ( p ) φ (cid:48) ( p ) + λw ( p ) (cid:0) φ (0) − φ ( p ) (cid:1) + λw ( p ) (cid:0) φ (1) − φ ( p ) (cid:1) , (6)for φ ∈ C (cid:0) [0 , (cid:1) . In other words, the diffusive motion of p ( · ) is governed by the generator Aφ ( p ) = v ( p ) φ (cid:48)(cid:48) ( p ) + µ ( p ) φ (cid:48) ( p ) , φ ∈ C (cid:0) [0 , (cid:1) , (7)with infinitesimal drift and variance coefficients, µ ( · ) and v ( · ), respectively, while jumps occurat constant rate λ ≥ p ∈ [0 ,
1] either to 0 with probability w ( p ) ∈ [0 ,
1] or to 1 with probability w ( p ) := 1 − w ( p ). Throughout this article, we will assumethat the following conditions are satisfied. 5 ssumption 2.1. The infinitesimal mean and variance satisfy µ (0) > > µ (1) and v (0) = v (1) =0 < v ( p ) for all p ∈ (0 , , respectively. Furthermore, v ( · ) , µ ( · ) and w ( · ) are analytic functions ina neighborhood of [0 , , and the infinitesimal variance has non-zero derivatives v (cid:48) (0) > > v (cid:48) (1) at the boundaries. For example, if A is the generator of a neutral Wright-Fisher diffusion (1) (with s ( p ) ≡ µ (0) = µ > µ (1) = − µ <
0, and v (cid:48) (0) = 1 = − v (cid:48) (1).We also remark that when Assumption 2.1 is satisfied, Lemma 3.1 shows that (cid:0) p ( t ) : t ≥ (cid:1) hasa unique stationary distribution π ( p ) dp with a density π ( · ) that satisfies a second order ordinarydifferential equation with non-local boundary conditions.In Theorem 2.1, we characterize the time-reversed process (cid:0) ˜ p ( t ) : t ≥ (cid:1) of the forward process (cid:0) p ( t ) : t ≥ (cid:1) . In keeping with the heuristic description given in the Introduction, (cid:0) ˜ p ( t ) : t ≥ (cid:1) isalso a jump-diffusion process on [0 ,
1] but now with jumps from the boundary { , } to the interior(0 , Aψ ( p ) = v ( p ) ψ (cid:48)(cid:48) ( p ) + ˜ µ ( p ) ψ (cid:48) ( p ) where ˜ µ ( p ) := − µ ( p ) + ( vπ ) (cid:48) ( p ) π ( p ) (8)and ψ ∈ C (cid:0) [0 , (cid:1) . Notice that this diffusion has the same infinitesimal variance as the forwarddiffusion, but has a different infinitesimal drift that depends on the jump events via the stationarydensity π ( · ). Also, the jump rates of the time-reversed process depend on a local time processwhich is described in the following way. Recall that the scale function and the speed measureassociated with ˜ A are˜ S ( p ) := (cid:90) p exp (cid:16) − (cid:90) x µ ( z ) v ( z ) dz (cid:17) dx and ˜ m ( dp ) := 1 v ( p ) ˜ S (cid:48) ( p ) dp, p ∈ [0 , , (9)respectively. The scale function will be identified with the associated measure ˜ S ( dp ) := ˜ S (cid:48) ( p ) dp on [0 ,
1] and the speed measure ˜ m ( dp ) will be identified with its density function.We define the local time process of the jump-diffusion ˜ p ( · ) such that it agrees with the localtime process of the diffusive motion until the first jump. More formally, we will introduce a non-negative process (cid:0) ˜ L p ( t ) : t ≥ , p ∈ [0 , (cid:1) which is almost surely continuous in ( t, p ) and whichsatisfies (cid:90) t f (cid:0) ˜ p ( u ) (cid:1) du = (cid:90) f ( p ) ˜ L p ( t ) ˜ m ( dp ) a.s. t ≥ f : [0 , → [0 , ∞ ). We remark that the local time process satisfying (10) differsfrom the semi-martingale local time of the diffusive motion of the time-reversed process by a scalarfactor (see Eq. (77)), i.e., ˜ L is a weighted semi-martingale local time. That this process is well-defined is shown below in Lemma 6.1. The last ingredient needed in our construction is a pair ofindependent, exponentially-distributed random variables, R and R , with parameters r i := lim p → i (cid:16) ˜ m ( p ) π ( p ) λκ i (cid:17) ∈ [0 , ∞ ] (11)where κ i := (cid:82) w i ( p ) π ( p ) dp , i ∈ { , } . The existence of the limit displayed in (11) is guaranteedby Lemma 4.3. By convention, R i := 0 if r i = ∞ and R i := ∞ if r i = 0.With these definitions, we now describe the dynamics of the time-reversed process (cid:0) ˜ p ( t ) : t ≥ (cid:1) .Between jump times, (cid:0) ˜ p ( t ) : t ≥ (cid:1) evolves according to the law of the diffusion governed by ˜ A . Ifthis diffusion hits a boundary i ∈ { , } at a time t ≥ R i , that is, if ˜ L i ( t ) ≥ R i , then ˜ p ( · ) jumps from i to a random pointchosen from (0 ,
1) according to the distribution κ i (cid:82) w i ( p ) π ( p ) dp . From this point, ˜ p ( · ) restartsindependently of the sample path up to that time.To better understand how the dynamics of ˜ p ( · ) are influenced by the boundary behavior of theforward process, we take a closer look at the jump times. Because the coefficients v ( · ) and µ ( · ) are6mooth on an interval containing [0 , i ∈ { , } is accessible to the forward diffusive motion if and onlyif 2 | µ ( i ) | < | v (cid:48) ( i ) | . Then, in conjunction with Lemma 3.2, which describes the asymptotics of thedensity π ( p ) near the boundaries, Lemma 4.3 implies that r i := lim p → i (cid:16) ˜ m ( p ) π ( p ) λκ i (cid:17) ∈ (0 , ∞ ) if 2 | µ ( i ) | < | v (cid:48) ( i ) | and λw i ( · ) (cid:54)≡ ∞ if 2 | µ ( i ) | ≥ | v (cid:48) ( i ) | and λw i ( · ) (cid:54)≡
0= 0 if λw i ( · ) ≡ i ∈ { , } . Thus, provided that λw i ( · ) (cid:54)≡
0, the time-reversed process immediately jumps intothe interior (0 ,
1) if the boundary point is inaccessible to the forward diffusive motion, that is,if 2 | µ ( i ) | ≥ | v (cid:48) ( i ) | . In this case, the state space of ˜ p ( · ) is in fact [0 , \ { i } . In contrast, if i isaccessible to the forward diffusion and λw i ( · ) >
0, then the exponential random variable R i isalmost surely positive and so a positive amount of local time will have to be accrued at i beforea jump occurs off of this boundary point.Notice that, in either case, we expect that both boundary points are accessible to the backwarddiffusive motion. According to Lemma 4.1˜ µ ( i ) = (cid:26) µ ( i ) if 2 | µ ( i ) | ≤ | v (cid:48) ( i ) | v (cid:48) ( i ) − µ ( i ) if 2 | µ ( i ) | ≥ | v (cid:48) ( i ) | , i ∈ { , } , (13)and again an application of Feller’s boundary criteria shows that the boundary point i is acces-sible to the backward diffusive motion whenever 2 µ ( i ) (cid:54) = v (cid:48) ( i ). The critical case is more subtle.Then, 2˜ µ ( i ) = v (cid:48) ( i ), and so i would be inaccessible if the drift coefficient ˜ µ ( · ) were analytic in aneighborhood of i . However, we show in Lemma 4.1 that˜ µ ( p ) = µ ( i ) + v (cid:48) ( i )ln (cid:0) | p − i | (cid:1) + O (cid:0) | p − i | ln | p − i | (cid:1) , (14)and then Feller’s criteria reveal that the logarithmic singularity is just sufficient to render thepoint i accessible to the backward diffusive motion when 2˜ µ ( i ) = v (cid:48) ( i ).Our main result states that the process ˜ p ( · ) has the same law as the stationary time reversalof the jump-diffusion p ( · ). Theorem 2.1.
Assume 2.1. Let p ( · ) be the jump-diffusion on [0 , with generator G as definedin (6) . Then the process (cid:0) ˜ p ( t ) : t ≥ (cid:1) is a version of the stationary time reversal of (cid:0) p ( t ) : t ≥ (cid:1) ,that is, (cid:0) ˜ p ( t ) : t ≤ T (cid:1) d = (cid:0) p ( T − t ) : t ≤ T (cid:1) ∀ T ≥ if the distribution of p (0) is the stationary distribution π ( p ) dp . The proof of Theorem 2.1 is deferred to Section 7.Theorem 2.1 establishes the time reversal of the stationary process over a fixed time interval[0 , T ], T < ∞ fixed and non-random. Readers being interested in other pathwise time reversalsare referred to the literature. It has been shown that processes which are in ’Hunt duality’ (see[4, Chapter VI]) are time reversals of each other. Reversing time at the end point of an excursionfrom an accessible boundary point results in the dual process being started at this boundary point,see [9, 21]. The paper of Mitro [22] reverses time at inverse local time points.The remainder of the paper is organized as follows. The next section collects some resultsconcerning the stationary distribution of the jump-diffusion process (2). Section 4 describes theboundary behavior of ˜ p ( · ). In particular we show that the time-reversed process jumps immediatelyoff of any boundary that is inaccessible to the forward diffusion. In Section 5 we identify a core forthe generator ˜ G satisfying the adjoint condition (3). The local time process of ˜ p ( · ) is introducedand studied in Section 6. Finally, Section 7 shows that ˜ p ( · ) has generator ˜ G . The proof of thisresult depends on an application of the Itˆo-Tanaka formula.7 The stationary distribution
The following lemma asserts that, if the conditions of Assumption 2.1 are satisfied, then the jump-diffusion process p ( · ) has a unique stationary distribution on [0 , π ( · ) with respect to Lebesgue measure which satisfies a second-orderordinary differential equation (ODE) subject to boundary conditions that are non-local whenever λ >
0. If λ = 0, then this equation can be solved explicitly, leading to the familiar expression π ( p ) = C − v ( p ) exp (cid:16) (cid:90) p µ ( q ) /v ( q ) dq (cid:17) , (16)where C is a normalizing constant, e.g., see Section 4.5 in [8]. Although a general closed-formexpression for π ( · ) apparently does not exist when λ > π ( · ) can be calculated by numericallysolving (17) using a modification of the shooting method [25]. In addition, below we give anexplicit formula for the stationary density in the important special case of a neutral Wright-Fisherdiffusion subject to recurrent bottlenecks. Lemma 3.1.
Assume 2.1. Then there exists a unique stationary distribution for the process (cid:0) p ( t ) : t ≥ (cid:1) . This distribution is given by (0 , ( p ) π ( p ) dp where π : (0 , → (0 , ∞ ) is the uniquesolution of the non-local boundary value problem (cid:0) ( vπ ) (cid:48)(cid:48) − ( µπ ) (cid:48) − λπ (cid:1) ( p ) = 0 ∀ p ∈ (0 , p → (cid:0) µπ − ( vπ ) (cid:48) (cid:1) ( p ) = λκ lim p → (cid:0) µπ − ( vπ ) (cid:48) (cid:1) ( p ) = − λκ lim p → ( vπ )( p ) = 0 = lim p → ( vπ )( p ) (cid:90) π ( p ) dp = 1 , (17) where κ i := (cid:82) w i ( p ) π ( p ) dp for i ∈ { , } . Furthermore p ( t ) converges in distribution to thestationary distribution as t → ∞ for every initial distribution of p (0) .Proof. Existence and uniqueness of a stationary distribution ¯ π ( dp ) follow from standard argu-ments, so we only give a sketch. Couple two versions of (cid:0) p ( t ) : t ≥ (cid:1) with different initial distri-butions through the same jump times such that the diffusive motions in between jumps are inde-pendent until they first meet and are identical thereafter. Due to the assumption µ (0) > > µ (1),the coupling is successful if there are no jumps, that is, if λ = 0, see Theorem V.54.5 in [26]. In thepresence of jumps ( λ > , p ( t ) converges in distribution to a probabilitymeasure ¯ π ( dp ) as t → ∞ and ¯ π ( dp ) is an invariant distribution.Next we prove that ¯ π ( · ) has a smooth density π ( · ). Denote by ( X ( t )) t ≥ the diffusion governedby A (see (1)). The scale function and the speed measure associated with A are S ( p ) := (cid:90) p exp (cid:16) − (cid:90) x µ ( z ) v ( z ) dz (cid:17) dx and m ( p ) dp := 1 v ( p ) S (cid:48) ( p ) dp, p ∈ [0 , , (18)respectively. Existence and smoothness of the density π ( · ) will be derived from existence anduniqueness of the transition density Q ( t ; p, q ) of ( X ( t )) t ≥ with respect to the speed measure.Existence of Q ( t ; p, q ) is established in Itˆo and McKean (1974) [11] ([20] is more detailed in aspecial case) via an eigen-differential expansion. To state this result more formally, we introducethe following notation. The interval defined in [11] – here denoted by I • – is the unit interval closedat 0 if 0 is accessible, closed at 1 if 1 is accessible and open otherwise. For this, note that whenever( X ( t )) t ≥ hits a boundary point, it immediately returns to the interior (0 ,
1) because of the8ssumption µ (0) > > µ (1). Moreover note that the stopping time min { t ≥ X ( t ) (cid:54)∈ I • } = ∞ is infinity almost surely. The generator of ( X ( t )) t ≥ is defined in [11] via right derivatives. As( X ( t )) t ≥ is a regular diffusion, this generator coincides with Af ( p ) = 1 m ( p ) ddp (cid:16) S (cid:48) ( p ) f (cid:48) ( p ) (cid:17) p ∈ I • (19)for f ∈ C ( I • ). There exists a solution e ( γ, · ) = (cid:0) e ( γ, · ) , e ( γ, · ) (cid:1) of (cid:16) A e (cid:0) γ, · (cid:1)(cid:17) ( p ) = γ e (cid:0) γ, p (cid:1) ∀ < p < e (cid:0) γ, (cid:1) = (1 ,
0) 1 m ( ) e (cid:48) (cid:0) γ, (cid:1) = (0 ,
1) (20)for every γ ∈ ( −∞ ,
0] such that γ (cid:55)→ e ( γ, p ) is continuous for every p ∈ I • . Based on theseeigenfunctions, it is shown in [11] that there exists a Borel measure s ( dγ ) from ( −∞ ,
0] to 2 × s ( dγ ) = (cid:18) s ( dγ ) s ( dγ ) s ( dγ ) s ( dγ ) (cid:19) (21)such that Q ( t ; p, q ) = (cid:90) −∞ e γt e T ( γ, p ) · s ( dγ ) · e ( γ, p ) , ( t, p, q ) ∈ (0 , ∞ ) × I • × I • , (22)is the transition density of ( X ( t )) t ≥ with respect to the speed measure m ( · ). Now as our jumpdiffusion p ( · ) could also jump to an inaccessible boundary, we need to extend p (cid:55)→ Q ( t ; p, q ) onto[0 , i ∈ { , } is inaccessible, then i is an entrance boundary due to the assumption( − i µ ( i ) >
0. As in Problem 3.6.3 in [11], one uses the Markov property to extend ( X ( t )) t ≥ tothe state space [0 , Q ( t ; p, q ) to be defined on (0 , ∞ ) × [0 , × I • .With these results on the transition density of ( X ( t )) t ≥ , we now establish existence of asmooth density of ¯ π ( · ). Define κ ∈ [0 , κ := 1 − κ by κ i := (cid:90) w i ( p )¯ π ( dp ) for i ∈ { , } (23)and observe that κ i is the probability that a stationary version of the process jumps to theboundary point i when it jumps. Recall that the jump times of p ( · ) form a Poisson process withrate λ and that in between jumps, p ( · ) evolves according to A . If U is any Borel measurable setin [0 , π ( U ) = κ (cid:90) ∞ λe − λt (cid:90) U Q ( t ; 0 , q ) m ( q ) dq dt + κ (cid:90) ∞ λe − λt (cid:90) U Q ( t ; 1 , q ) m ( q ) dq dt. Interchanging integrals, we infer that ¯ π ( · ) has a density with respect to Lebesgue measure and weset π ( q ) dq := ¯ π ( dq ) where π : (0 , → [0 , ∞ ) satisfies π ( q ) = (cid:88) j =0 κ j m ( q ) (cid:90) ∞ λe − λt (cid:90) −∞ e γt e T ( γ, j ) s ( dγ ) e ( γ, q ) dt = (cid:88) j =0 κ j m ( q ) (cid:90) −∞ λλ − γ e T ( γ, j ) s ( dγ ) e ( γ, q ) dt = λm ( q ) (cid:16) κ G λ (0 , q ) + κ G λ (1 , q ) (cid:17) . (24)9he function G λ ( p, q ) is the Green’s function and is C in the second variable for every p ∈ [0 , m ( · ) is also C in (0 ,
1) due to Assumption 2.1, we conclude that thestationary density π ( · ) is twice continuously differentiable.The main step of the proof is to show that π ( · ) satisfies (17). By Proposition 4.9.2 of Ethierand Kurtz [7], the stationary distribution π ( p ) dp satisfies (cid:90) Gφ ( p ) π ( p ) dp = 0 (25)for all φ ∈ C (cid:0) [0 , (cid:1) . Let 0 < ε < . The functions v, µ, φ and π are C in [ ε, − ε ]. Integrationby parts yields (cid:90) − εε Gφ · π dp − φ (0) (cid:90) − εε λw · π dp − φ (1) (cid:90) − εε λw · π dp = (cid:90) − εε φ (cid:48)(cid:48) · ( vπ ) + φ (cid:48) · ( µπ ) − λφπ dp = [ φ (cid:48) vπ ] − εε + [ φ · (cid:0) µπ − ( vπ ) (cid:48) (cid:1) ] − εε − (cid:90) − εε φ · (cid:104) ( µπ ) (cid:48) − ( vπ ) (cid:48)(cid:48) + λπ (cid:105) dp. (26)By considering all functions φ ∈ C with support in ( ε, − ε ) and then letting ε →
0, we concludethat π ( · ) satisfies the second-order ODE in (17). Furthermore, because the functions Gφ , w , w are bounded and π is integrable, we may apply the dominated convergence theorem to theintegralson the left-hand side of (26) as ε →
0. Together with (25) this shows thatlim ε → [ φ (cid:48) vπ ] − εε + φ (1) · lim ε → (cid:0) µπ − ( vπ ) (cid:48) (cid:1) (1 − ε ) − φ (0) · lim ε → (cid:0) µπ − ( vπ ) (cid:48) (cid:1) ( ε )= − φ (1) λκ − φ (0) λκ . (27)As φ was arbitrary this implies the non-local boundary conditions in (17).If ˆ π ( · ) is another normalized solution of (17), then reversing the previous arguments showsthat (25) holds with π replaced by ˆ π ( · ). This in turn implies that ˆ π ( p ) dp is another stationarydistribution and we conclude that ˆ π = π . It remains to show that π ( · ) is strictly positive. Assuming π ( p ) = 0 for some p ∈ (0 , π (cid:48) ( p ) = 0 from p being necessarily a global minimum.However, the only solution of the second-order ODE in (17) satisfying π ( p ) = 0 = π (cid:48) ( p ) is the zerofunction, which contradicts the assumption that π ( · ) is a probability density. Remark 3.1.
Lemma 3.1 can be used to find an explicit formula for π ( · ) when the jump-diffusionprocess is a model of a neutrally-evolving population subject to recurrent bottlenecks, i.e., when p ( · ) has generator Gφ ( p ) = 12 p (1 − p ) φ (cid:48)(cid:48) ( p ) + ( µ (1 − p ) − µ p ) φ (cid:48) ( p ) + λ (cid:16) pφ (1) + (1 − p ) φ (0) − φ ( p ) (cid:17) . In this case, (17) is a hypergeometric equation and, using the fact that the mean frequency of allele A in a stationary population is µ / ( µ + µ ) , we find that the density π ( p ) is equal to π ( p ) = C − p µ − (1 − p ) µ − (cid:104) µ F (1 − a, − b, µ , p ) + µ F (1 − a, − b, µ , − p ) (cid:105) , where C is a normalizing constant, F ( a, b, c ; z ) is Gauss’ hypergeometric function, and the con-stants a and b are determined (up to interchange) by the equations a + b = 3 − µ + µ ) and ab = 2( λ + 1 − µ − µ ) . The second lemma of this section provides information on the boundary behavior of the densityof the stationary distribution. This information is derived using results on second-order ODEswith regular singular points.We adopt the Landau big-O and little-o notation. In addition, for two functions ψ ( · ) and ψ ( · ), we write ψ ( p ) ∼ ψ ( p ) as p → i if both ψ ( p ) = O (cid:0) ψ ( p ) (cid:1) and ψ ( p ) = O (cid:0) ψ ( p ) (cid:1) as p → i .10 emma 3.2. Assume 2.1. Let π ( · ) be the density of the stationary distribution of the jump-diffusion p ( · ) corresponding to the generator (6) . Then, for i ∈ { , } , π ( · ) is equal to π ( p ) = C i | p − i | µ ( i ) − v (cid:48) ( i ) v (cid:48) ( i ) + O (1) if | µ ( i ) | < | v (cid:48) ( i ) | λκ i | v (cid:48) ( i ) | ln (cid:0) | p − i | (cid:1) + O (1) if µ ( i ) = v (cid:48) ( i ) λκ i | µ ( i ) − v (cid:48) ( i ) | + O (cid:0) | p − i | + | p − i | µ ( i ) − v (cid:48) ( i ) v (cid:48) ( i ) (cid:1) if | µ ( i ) | > | v (cid:48) ( i ) | and λw i (cid:54)≡ C i | p − i | µ ( i ) − v (cid:48) ( i ) v (cid:48) ( i ) + O (cid:0) | p − i | µ ( i ) v (cid:48) ( i ) (cid:1) if | µ ( i ) | > | v (cid:48) ( i ) | and λw i ≡ as p → i where C i ∈ (0 , ∞ ) . In addition if µ ( i ) = v (cid:48) ( i ) and λw i ≡ , then π ( i ) > .Proof. We only consider i = 0 as the case i = 1 is analogous. We begin by observing that i = 0is a regular singular point for the differential equation in (17), see e.g. Section 9.6 in [3] for thisconcept. The associated indicial equation for 0 is ν ( ν −
1) + 2 v (cid:48) (0) − µ (0) v (cid:48) (0) · ν = 0 (29)and has roots α := 0 and β := µ (0) − v (cid:48) (0) v (cid:48) (0) . Note that β > −
1. If α − β (cid:54)∈ Z , then Theorem IX.7in [3] tells us that π ( · ) is equal to a linear combination of b ( p ) := p α (cid:0) h ( p ) (cid:1) and b ( p ) := p β (cid:0) h ( p ) (cid:1) in a neighborhood of 0 where h and h are suitable analytic functions satisfying h (0) = 0 = h (0).If α − β ∈ Z , then Theorem IX.8 in [3] shows that π ( · ) is equal to a linear combinationof b ( p ), b ( p ) and ln( p ) b ( p ) in a neighborhood of 0. If 2 µ (0) /v (cid:48) (0) ∈ N ≥ , then assuming π ( p ) = − c ln( p ) + O (1), π (cid:48) ( p ) = − c p + O (1) for some constant c > ∞ > λκ = (cid:0) µ (0) − v (cid:48) (0) (cid:1) lim p → π ( p ) − v (cid:48) (0) lim p → pπ (cid:48) ( p ) = ∞ (30)where we have used (17). Therefore ln( p ) does not contribute to π ( · ) if 2 µ (0) > v (cid:48) (0).It remains to calculate the coefficients. In the case 2 µ (0) (cid:54) = v (cid:48) (0), insert π ( b ) = c b ( p )+ c b ( p )into (17) to obtain the coefficient c λκ = lim p → (cid:104)(cid:0) µ ( p ) − v (cid:48) ( p ) (cid:1) π ( p ) − v ( p ) π (cid:48) ( p ) (cid:105) = lim p → (cid:104)(cid:0) µ (0) − v (cid:48) (0) (cid:1)(cid:0) c + c p β (cid:1) − v (cid:48) (0) pc βp β − (cid:105) = (cid:0) µ (0) − v (cid:48) (0) (cid:1) c . (31)Of course if λκ = 0, then π ( · ) (cid:54)≡ c >
0. Next we show that λκ > µ (0) < v (cid:48) (0) implies c > c = 0 implies that π (0) = λκ µ (0) − v (cid:48) (0) < π ( · ) being a densityfunction. In the critical case 2 µ (0) = v (cid:48) (0), (17) implies that λκ = − v (cid:48) (0) lim p → (cid:0) pπ (cid:48) ( p ) (cid:1) . (32)Therefore the coefficient of − ln( p ) is λκ v (cid:48) (0) . If 2 µ (0) = v (cid:48) (0) and λκ = 0, then assuming π (0) = 0implies π ( p ) = cp n + O ( p n +1 ) with c (cid:54) = 0 and n ≥
1. Inserting this into the ODE in (17) leads to0 = 12 v (cid:48) (0) c (cid:0) p n +1 (cid:1) (cid:48)(cid:48) − µ (0) c (cid:0) p n (cid:1) (cid:48) + O (cid:0) p n (cid:1) = ( n + 1) n v (cid:48) (0) cp n − − n v (cid:48) (0)2 p n − + O (cid:0) p n (cid:1) (33)11s p →
0. Dividing by nv (cid:48) (0) cp n − / p → n + 1 =1. We begin this section by characterizing the boundary behavior of the infinitesimal drift coefficientof the time-reversed process. This information is of interest for two reasons. First, it will beused to establish that any boundary point that is accessible to the forwards-in-time process,either diffusively or via jumps, is accessible to the diffusive motion of the time-reversed process.Secondly, we also expect the time-reversed process to have the same state space, [0 , i is inaccessible to the forward diffusive motion, thensubsequent results will show that the time-reversed process jumps back into the interior as soonas it hits a boundary. If, however, i is accessible to the forward diffusive motion, then becausethe time-reversed process may visit i without jumping, we need to confirm that ˜ p ( · ) does not thenwander outside of [0 , µ (0) ≥ µ (1) ≤ Lemma 4.1.
Assume 2.1. Then the drift function ˜ µ ( · ) of the backward diffusive motion definedin (8) satisfies ˜ µ ( p ) = µ ( i ) + O (cid:0) | p − i | (cid:1) if | µ ( i ) | < | v (cid:48) ( i ) | or λκ i = 0 µ ( i ) + v (cid:48) ( i )ln (cid:0) | p − i | (cid:1) + O (cid:0) | p − i | ln | p − i | (cid:1) if µ ( i ) = v (cid:48) ( i ) and λκ i (cid:54) = 0 v (cid:48) ( i ) − µ ( i ) + O (cid:0) | p − i | (cid:1) if | µ ( i ) | > | v (cid:48) ( i ) | and λκ i (cid:54) = 0 (34) as p → i for i ∈ { , } .Proof. Recall that lemma 3.2 describes the asymptotic behavior of π ( · ) as p → i . From this weobtain v ( p ) π (cid:48) ( p ) π ( p ) = v (cid:48) ( i ) µ ( i ) − v (cid:48) ( i ) v (cid:48) ( i ) + O (cid:0) | p − i | (cid:1) if 2 | µ ( i ) | < | v (cid:48) ( i ) | or λκ i = 0 v (cid:48) ( i )ln | p − i | + O (cid:0) | p − i | ln | p − i | (cid:1) if 2 µ ( i ) = v (cid:48) ( i ) and λκ i (cid:54) = 00 + O (cid:0) | p − i | (cid:1) if 2 | µ ( i ) | > | v (cid:48) ( i ) | and λκ i (cid:54) = 0 (35)as p → i for i ∈ { , } . Inserting this into˜ µ ( p ) := − µ ( p ) + ( vπ ) (cid:48) ( p ) π ( p ) = − µ ( i ) + v (cid:48) ( i ) + O (cid:0) | p − i | (cid:1) + v ( p ) π (cid:48) ( p ) π ( p ) (36)results in assertion (34). Remark 4.1.
Notice that ˜ µ (0) < if µ (0) > v (cid:48) (0) , so that the diffusive motion of the time-reversedprocess need not be confined to [0 , . Nonetheless, because the boundary p = 0 is inaccessible tothe forward diffusion in this case (i.e., µ (0) > v (cid:48) (0) ), the fact that the process ˜ p ( · ) immediatelyjumps back into (0 , upon hitting will ensure that the jump-diffusion is confined to [0 , . We next show that if a boundary point is accessible to the forward jump-diffusion p ( · ), eitherdiffusively or via a jump, then it must be accessible to the backward diffusive motion governed by˜ A . Recall the scale function ˜ S from (9). Lemma 4.2.
Assume 2.1. The boundary point i ∈ { , } is accessible to the diffusive motiongoverned by ˜ A , that is ˜ S ( i ) ∈ R , if and only if i is accessible to the forward jump-diffusion p ( · ) ,that is, if λw i (cid:54)≡ or | µ ( i ) | < | v (cid:48) ( i ) | . roof. W.l.o.g. we only prove the case i = 0. According to Lemma 15.6.1 in [14], the boundarypoint 0 is accessible if and only if ˜ S (0+) is finite. (This is a special case of Feller’s boundaryclassification criteria.) Substituting the asymptotic expression for ˜ µ near p = 0 (see Lemma 4.1)into the definition of ˜ S , we obtain in the case λκ > S (cid:48) ( p ) := exp (cid:16) − (cid:90) p µ ( z ) v ( z ) dz (cid:17) ∼ p − µ (0) v (cid:48) (0) if 2 | µ ( i ) | < | v (cid:48) ( i ) | p (cid:0) ln( p ) (cid:1) if 2 µ ( i ) = v (cid:48) ( i ) p − v (cid:48) (0) − µ (0) v (cid:48) (0) if 2 | µ ( i ) | > | v (cid:48) ( i ) | (37)as p →
0. In all three cases, ˜ S (cid:48) ( · ) is integrable over (0 , ]. The case λκ = 0 follows from similararguments.The following lemma shows that the rate constant r i (defined in (11)) is equal to infinity ifthe boundary point i ∈ { , } is inaccessible to the forward diffusive motion. Therefore ˜ p ( · ) jumpswhenever it hits i as ˜ L i ( t ) ≥ R i . In addition, if there are no jumps in the forward process,then r i = 0, and ˜ p ( · ) never jumps as ˜ L i ( t ) < ∞ = R i . Lemma 4.3.
Assume 2.1. Then the rate constant r i used to define the jump times of ˜ p ( · ) satisfies r i := lim p → i (cid:16) ˜ m ( p ) π ( p ) λκ i (cid:17) ∈ (0 , ∞ ) if | µ ( i ) | < | v (cid:48) ( i ) | and λw i ( · ) (cid:54)≡ ∞ if | µ ( i ) | ≥ | v (cid:48) ( i ) | and λw i ( · ) (cid:54)≡
0= 0 if λw i ( · ) ≡ for i ∈ { , } .Proof. W.l.o.g. we assume i = 0 as the case i = 1 is similar. If λκ = 0, then r = 0 is triviallycorrect. Assume λκ > S (cid:48) ( · ) is given in (37). From this we derive the asymptotic behavior of the speed density ˜ m ( p )(defined in (9)) as p → m ( p ) = 2 v ( p ) ˜ S (cid:48) ( p ) ∼ p µ (0) − v (cid:48) (0) v (cid:48) (0) if 2 µ (0) < v (cid:48) (0) (cid:0) ln( p ) (cid:1) if 2 µ (0) = v (cid:48) (0) p v (cid:48) (0) − µ (0) v (cid:48) (0) if 2 µ (0) > v (cid:48) (0) . (39)Compare (39) with the boundary behavior of π ( · ) (see Lemma 3.2) to obtain (38). In this section we identify the generator of the time-reversed process and show that this operatorsatisfies the duality condition given in (2). That this operator is also the generator of the jump-diffusion process ˜ p ( · ) described in Section 2 will be established in the final two sections of thepaper.The following notation will be needed to formulate the generator of the time-reversed process.If ν ( dp ) = f ( p ) dp is a measure on [0 ,
1] with continuous density f : (0 , → (0 , ∞ ) with respect toLebesgue measure, then we write (cid:16) D ν ψ (cid:17) ( p ) := lim y → p ψ ( y ) − ψ ( p ) (cid:82) yp f ( q ) dq ∀ p ∈ (0 ,
1) (40)whenever this limit exists in R and denote by D ( D ν ) := (cid:110) ψ ∈ C (cid:0) [0 , (cid:1) : D ν ψ ( p ) exists ∀ p ∈ (0 , , D ν ψ is continuousin (0 , , D ν ψ (0+) and D ν ψ (1 − ) exist in R (cid:111) (41)13he subset of functions which are mapped to continuous functions on [0 , ψ ∈ C (cid:0) (0 , (cid:1) and D ν ψ ( p ) = ψ (cid:48) ( p ) f ( p ) , p ∈ (0 , ψ ∈ D ( D ν ). For ψ ∈ D ( D ν ) the defini-tion of D ν extends to the boundary via D ν ψ (0) := D ν ψ (0+) and D ν ψ (1) := D ν ψ (1 − ). In thisnotation, the generator ˜ A of the backward diffusive motion reads as (cid:0) ˜ Aψ (cid:1) ( p ) = v ( p ) ψ (cid:48)(cid:48) ( p ) + ˜ µ ( p ) ψ (cid:48) ( p ) = v ( p ) ˜ S (cid:48) ( p ) (cid:16) S (cid:48) ψ (cid:48)(cid:48) + 1˜ S (cid:48) µv ψ (cid:48) (cid:17) ( p ) = D ˜ m D ˜ S ψ ( p )for every ψ ∈ C (cid:0) [0 , (cid:1) . The following set will be a core for the generator of the time-reversedprocess H := (cid:26) ψ ∈ D (cid:0) D ˜ S (cid:1) : D ˜ S ψ ∈ D (cid:0) D ˜ m (cid:1) and for i ∈ { , } lim p → i (cid:0) vπψ (cid:48) (cid:1) ( p ) = ( − i +1 (cid:90) (cid:0) ψ ( p ) − ψ ( i ) (cid:1) λw i ( p ) π ( p ) dp (cid:27) . (42)The following lemma asserts that the restriction of D ˜ m D ˜ S to H extends to a strong generator ofa Markov process. Indeed, this can be deduced from Theorem II.4 of Mandl (1968) which showsthat the restriction of D ˜ m D ˜ S to (cid:26) ψ ∈ D (cid:0) D ˜ S (cid:1) : D ˜ S ψ ∈ D ( D ˜ m ) and if i ∈ { , } is accessible: χ i ψ ( i ) + (cid:90) ψ ( i ) − ψ ( p ) | p i ( i ) − p i ( p ) | dq i ( p ) + η i D ˜ m D ˜ S ψ ( i ) = 0 (cid:27) (43)is the strong generator of a Feller semigroup if χ i and η i are both non-negative, if q i is a non-decreasing function on [0 , p i is continuous, non-decreasing and equal to ˜ S in a neigh-borhood of i , i = 0 ,
1. Only the case 0 = χ = χ = η = η = q (1 − ) − q (0+) (cid:54) = q (1) − q (0) isexcluded. The quotient in (43) is to be interpreted as ( − i +1 D ˜ S ψ ( i ) for p = i ∈ { , } and theintegral with respect to dq i ( p ) denotes the Lebesgue-Stiltjes integral with respect to q i . Lemma 5.1.
Assume 2.1. The restriction of D ˜ m D ˜ S to the set H extends to a strong generatorof a Markov process.Proof. By Theorem II.4 in [19], it suffices to prove that H is of the form (43). According toLemma 4.2, the boundary point i ∈ { , } is accessible to the diffusion governed by D ˜ m D ˜ S if andonly if λw i ( · ) (cid:54)≡ | µ ( i ) | < | v (cid:48) ( i ) | . First we show that the condition in (42) is trivial if i isinaccessible, that is, we show lim p → i (cid:0) vπψ (cid:48) (cid:1) ( p ) = 0 for every ψ ∈ D ( D ˜ S ). Suppose that C := lim p → i (cid:0) vπψ (cid:48) (cid:1) ( p ) > ψ ∈ D (cid:0) D ˜ S (cid:1) . By Lemma 3.2, π ( p ) is bounded above by − ¯ C ln (cid:0) | p − i | (cid:1) in aneighborhood of i for some ¯ C >
0. Thus, in a neighborhood of i , ψ (cid:48) ( p ) ≥ − C v ( p ) π ( p ) ≥ C C | v (cid:48) ( i ) || p − i | ln (cid:0) | p − i | (cid:1) . (45)Integrating over [ , p ] implies that ψ ( p ) is bounded below by C C | v (cid:48) ( i ) | ln (cid:0) − ln( | p − i | ) (cid:1) as p → i which contradicts ψ ∈ C ( I ). An analogous argument applies to the case C < i be accessible to the forward process, that is, λκ i > | µ ( i ) | < | v (cid:48) ( i ) | . Note that˜ S ( · ) is a bounded continuous non-decreasing function in a neighborhood of i . Choose χ i , η i = 0and p i ( p ) := ˜ S ( p ). Furthermore, let q i ( p ) := (cid:90) p | p i ( i ) − p i ( y ) | w i ( y ) π ( y ) dy + c (0 , ( p ) i =0 + c ( p ) i =1 (46)14here c , c ∈ [0 , ∞ ) are to be chosen later. Note that q i is bounded and that dq i puts mass c i onthe point i . With these definitions, the condition in (43) takes the form (cid:90) (cid:0) ψ ( i ) − ψ ( p ) (cid:1) w i ( p ) π ( p ) dp = ( − i D ˜ S ψ ( i ) c i . (47)It remains to choose c i ∈ [0 , ∞ ) such that D ˜ S ψ ( i ) c i = lim p → i ( vπψ (cid:48) )( p ) (48)for every ψ ∈ H . Using the boundary behavior (37) of ˜ S ( · ) and the asymptotic behavior (28) of˜ π ( · ), we arrive at (cid:0) vπψ (cid:48) (cid:1) ( p ) ∼ | p − i |· π ( p ) · ˜ S (cid:48) ( p ) · D ˜ S ψ ( p ) ∼ D ˜ S ψ ( p ) if 2 | µ ( i ) | < | v (cid:48) ( i ) | D ˜ S ψ ( p ) − (cid:0) | p − i | (cid:1) if 2 µ ( i ) = v (cid:48) ( i ) D ˜ S ψ ( p ) | p − i | µ ( i ) v (cid:48) ( i ) − if 2 | µ ( i ) | > | v (cid:48) ( i ) | as p → i . This shows that (48) holds with some constant c i ∈ [0 , ∞ ). Lemma 5.2.
Assume 2.1. Let the process (cid:0) p ( t ) : t ≥ (cid:1) be in equilibrium. Then the time-reversedprocess (cid:0) ¯ p t : t ≥ (cid:1) exists, that is, there exists a process (cid:0) ¯ p t : t ≥ (cid:1) satisfying (cid:0) ¯ p ( t ) : t ≤ T (cid:1) d = (cid:0) p ( T − t ) : t ≤ T (cid:1) ∀ T ≥ . (49) In addition, H is a core for the generator ˜ G of (cid:0) ¯ p t : t ≥ (cid:1) and ˜ Gψ = vψ (cid:48)(cid:48) + ˜ µψ (cid:48) = D ˜ m D ˜ S ψ ∀ ψ ∈ H . (50) Proof.
Let ˜ G be the closure of the operator defined in (50). By Lemma 5.1, ˜ G is the stronggenerator of a Markov process (cid:0) ¯ p t : t ≥ (cid:1) . Recall the generator G of p ( · ) from (6). We will provethat ˜ G is the adjoint operator of G with respect to the invariant measure π ( p ) dp . Let 0 < ε < , φ ∈ C (cid:0) [0 , (cid:1) and ψ ∈ H . The functions v, µ, φ, ψ and π are C in [ ε, − ε ]. Integration by partsyields (cid:90) − εε Gφ · ψ · π dp − φ (0) (cid:90) − εε λw · ψ · π dp − φ (1) (cid:90) − εε λw · ψ · π dp = (cid:90) − εε φ (cid:48)(cid:48) · ( vψπ ) + φ (cid:48) · ( µψπ ) − λφψπ dp = [ φ (cid:48) vψπ ] − εε + [ φ · (cid:0) µψπ − ( vψπ ) (cid:48) (cid:1) ] − εε + (cid:90) − εε φ · (cid:104) ( vψπ ) (cid:48)(cid:48) − ( µψπ ) (cid:48) − λψπ (cid:105) dp. (51)As π satisfies (17), we see that( vψπ ) (cid:48)(cid:48) − ( µψπ ) (cid:48) − λψπ = vψ (cid:48)(cid:48) π + (cid:0) ( vπ ) (cid:48) π − µ (cid:1) ψ (cid:48) π = D ˜ m D ˜ S ψ · π. (52)The functions Gφ , ψ , w , w , φ and D ˜ m D ˜ S ψ are bounded and π is integrable. Hence, we mayapply the dominated convergence theorem to the integrals in (51) as ε →
0. This also proves that15he limits of the boundary terms in (51) exist as ε →
0. Thus letting ε → (cid:90) Gφ · ψπ dp − (cid:90) φ · D ˜ m D ˜ S ψ · π dp = φ (1) (cid:18) ψ (1) (cid:0) µπ − ( vπ ) (cid:48) (cid:1) (1 − ) − lim p → (cid:0) vπψ (cid:48) (cid:1) ( p ) + (cid:90) λw · ψ · π dp (cid:19) − φ (0) (cid:18) ψ (0) (cid:0) µπ − ( vπ ) (cid:48) (cid:1) (0+) − lim p → (cid:0) vπψ (cid:48) (cid:1) ( p ) − (cid:90) λw · ψ · π dp (cid:19) = 0 (53)for all φ ∈ C and ψ ∈ H . The last equality follows from (17) and from ψ ∈ H . This proves that G and ˜ G are adjoint to each other. Consequently, the semigroups of (cid:0) p ( t ) : t ≥ (cid:1) and of (cid:0) ¯ p t : t ≥ (cid:1) are adjoint to each other. According to [23], this implies that the Markov process (cid:0) ¯ p ( t ) : t ≥ (cid:1) associated with ˜ G has the same law as the time-reversed process of (cid:0) p ( t ) : t ≥ (cid:1) . This section describes some properties of the local time process of ˜ p ( · ). First we show existence.Recall the scale function ˜ S and the speed measure ˜ m from (9). Lemma 6.1.
Assume 2.1. Then there exists a unique, non-negative process (cid:0) ˜ L p ( t ) : t ≥ , p ∈ [0 , (cid:1) (54) which is almost surely continuous in ( t, p ) and which satisfies (cid:90) t f (cid:0) ˜ p ( u ) (cid:1) du = (cid:90) f ( p ) ˜ L p ( t ) ˜ m ( dp ) ∀ t ≥ for all measurable f : [0 , → [0 , ∞ ) almost surely. In addition, if | µ ( i ) | ≥ | v (cid:48) ( i ) | for i ∈ { , } ,then ˜ L i ( · ) ≡ almost surely.Proof. Let 0 =: τ ≤ τ < τ < · · · be the jump times of ˜ p ( · ). Then, by construction of˜ p ( · ), (cid:0) ˜ p ( t + τ n − ) : 0 ≤ t < τ n − τ n − (cid:1) , n ∈ N ≥ , are independent diffusions governed by ˜ A . Itis well-known that ˜ p ( · + τ n − ) can be written in terms of a Brownian motion as follows. Let (cid:8) B y,n · : y ∈ R , n ∈ N (cid:9) be a family of independent standard Brownian motions with B y,n = y .Denote by (cid:0) L B y,n x ( t ) : t ≥ , x ∈ R (cid:1) the local time process of B y,n · , see e.g. Section 2.8 in [11], anddefine ξ ( n ) t := (cid:90) L B ˜ S (˜ p ( τn − ,n ˜ S ( p ) ( t ) ˜ m ( dp ) t ≥ . (56)Then a version of ˜ p ( · + τ n − ) is given by˜ p ( t + τ n − ) = ˜ S − (cid:18) B ˜ S (cid:0) ˜ p ( τ n − ) (cid:1) ,n (cid:0) ξ ( n ) t (cid:1) − (cid:19) ≤ t < τ n − τ n − . (57)Inserting this into the occupation time formula of the Brownian motion, a short calculation (seee.g. Section 5.4 in [11]) shows that (cid:90) t f (cid:0) ˜ p ( r + τ n − ) (cid:1) dr = (cid:90) f ( p ) ˆ L ( n ) p ( t ) ˜ m ( dp ) 0 ≤ t < τ n − τ n − , (58)where the local time process of ˜ p ( · + τ n − ) with respect to the speed measure isˆ L ( n ) p ( t ) := L B ˜ S (˜ p ( τn − ,n ˜ S ( p ) (cid:16) ( ξ ( n ) t ) − (cid:17) ≤ t < τ n − τ n − , p ∈ [0 , , n ∈ N ≥ . (59)16ow we put the independent path segments together by defining˜ L p ( t ) := ˆ L ( n ) p (cid:0) t − τ n − (cid:1) + n − (cid:88) k =1 ˆ L ( k ) p (cid:0) τ k − τ k − (cid:1) if τ n − ≤ t < τ n . (60)It is easy to use (58) to show that (cid:0) ˜ L p ( t ) : t ≥ , p ∈ [0 , (cid:1) satisfies (55). Uniqueness follows fromstandard arguments.If 2 | µ ( i ) | ≥ | v (cid:48) ( i ) | , then ˜ p ( · ) jumps into (0 ,
1) as soon as it hits the boundary and we concludethat ˜ p ( t ) (cid:54) = i for all t ≥
0. Thus the local time at this boundary point is identically zero.In the next section, we will need to be able control the second moment of the local time of thetime-reversed jump-diffusion at a boundary point. We first prove the following estimate concerningthe local time of a standard Brownian motion.
Lemma 6.2.
Let ε, δ > and let (cid:0) B t : t ≥ (cid:1) be a standard Brownian motion with local time L Bx ( · ) at x ∈ R . Suppose that the function ¯ S : [0 , δ ] → R is non-decreasing. Define ∆ := ¯ S ( δ ) − ¯ S (0) and ζ t := (cid:90) δ ε L B ¯ S ( y ) ( t ) dy t ≥ . (61) Then, for each m > , there exists a constant C m independent of ε and of δ such that E ¯ S ( p ) (cid:20)(cid:16) L B ¯ S ( p ) (cid:0) ζ − t (cid:1)(cid:17) m (cid:21) ≤ C m (cid:16) εt (cid:17) m (cid:104)(cid:16) εt ∆ (cid:17) m + 1 (cid:105) (62) for all p ∈ [0 , δ ] and t ≥ .Proof. Inequality (62) is trivial if ∆ = 0 or t = 0, so we may and will assume that ∆ > t > p ∈ [0 , δ ]. The left-hand side of (62) does not depend on the value of ¯ S ( p ), so we will alsoassume w.l.o.g. that ¯ S ( p ) = 0. Denote by B ∗ t := max { B s : s ≤ t } and by | B | ∗ t := max {| B s | : s ≤ t } the process of the maximum and the process of the absolute maximum, respectively. Define Z t := ¯ S − (cid:0) B ζ − t (cid:1) for t ≥
0. According to Section 5.4 in [11], L Zp ( t ) := L B (cid:0) ζ − t (cid:1) is the local time of (cid:0) Z t : t ≥ (cid:1) at p . The process (cid:0) Z t : t ≥ (cid:1) is equal in distribution to the process (cid:0) ¯ S − (cid:0) B εt (cid:1) : t ≥ (cid:1) reflected at 0 and at δ . Another way to construct (cid:0) ¯ S ( Z t ) : t ≥ (cid:1) is to take the path of (cid:0) B εt : t ≥ (cid:1) and to identify each x ∈ [ ¯ S (0) , ¯ S ( δ )] with the set { x +2∆ z, S ( δ ) − x +2∆ z : z ∈ Z } . Thus the localtime L Zp ( t ) of (cid:0) Z t : t ≥ (cid:1) in p is equal in distribution to the sum of L B z (cid:0) εt (cid:1) + L B z +2 ¯ S ( δ ) (cid:0) εt (cid:1) over z ∈ Z . Note that L Bx (cid:0) εt (cid:1) = 0 almost surely on the event {| B | ∗ εt < | x |} , x ∈ R . In addition, notethat convexity of 0 ≤ x (cid:55)→ x m implies k m − ( a + · · · + a k ) m ≤ ( a m + · · · + a mk ) for a , . . . , a k ≥ k ∈ N . Therefore, E p (cid:104)(cid:0) L Zp ( t ) (cid:1) m (cid:105) = E (cid:20) (cid:88) k ∈ N ≥ [ k,k +2∆) (cid:0) | B | ∗ εt (cid:1)(cid:18) k (cid:88) z = − k (cid:88) x ∈{ , S ( δ ) } L B z + x (cid:0) εt (cid:1)(cid:19) m (cid:21) ≤ E (cid:20) (cid:88) k ∈ N ≥ [ k,k +2∆) (cid:0) | B | ∗ εt (cid:1) · (cid:16) k ∆ + 2 (cid:17) m − k (cid:88) z = − k (cid:88) x ∈{ , S ( δ ) } (cid:16) L B z + x (cid:0) εt (cid:1)(cid:17) m (cid:21) . (63)Use the strong Markov property (e.g. Proposition 2.6.17 in [13]) to restart the Brownian motionat the first hitting time of 2∆ z and of 2∆ z + 2 ¯ S ( δ ), respectively. Thus the left-hand side of (63)is bounded above by (cid:88) k ∈ N ≥ P (cid:16) | B | ∗ εt ∈ [ k, k + 2∆) (cid:17) · (cid:16) k ∆ + 2 (cid:17) m − k (cid:88) i = − k E (cid:104)(cid:16) L B ( εt ) (cid:17) m (cid:105) ≤ E (cid:20)(cid:16) | B | ∗ εt ∆ + 2 (cid:17) m (cid:21) E (cid:104)(cid:16) L B ( εt ) (cid:17) m (cid:105) . (64)17ote that 2 L B ( t ) and B ∗ t are equal in distribution, see e.g. Theorem 3.6.17 in [13]. Therefore theleft-hand side of (63) is bounded above by2 m (cid:20) E (cid:104)(cid:16) | B | ∗ εt ∆ (cid:17) m (cid:105) + 1 (cid:21) E (cid:104)(cid:16) B ∗ εt (cid:17) m (cid:105) ≤ (cid:104) K m (cid:16) εt ∆ (cid:17) m + 1 (cid:105) K m (cid:16) εt (cid:17) m (65)where K m/ ≥ , ε and t . The last step followsfrom the Burkholder-Davis-Gundy inequality, see e.g. Theorem 3.3.28 in [13]. Therefore (62) holdswith C m := K m .In the proof of Theorem 2.1, we will need to exploit the fact that, in the L sense, the localtime (cid:0) ˜ L i ( t ) : t ≥ (cid:1) at a boundary point i ∈ { , } of the backwards process started at i decreasesto zero faster than √ t as t →
0. This might be surprising as one can show that E (cid:104)(cid:0) L B ( t ) (cid:1) (cid:105) ∼ t as t → . (66)However, the infinitesimal variance v ( · ) is zero in i . Thus, informally speaking, the diffusiongoverned by ˜ A is pushed away from zero almost deterministically at rate ˜ µ ( i ) > Lemma 6.3.
Assume 2.1. Then the local time at the boundary satisfies lim t → t E i (cid:20)(cid:16) ˜ L i ( t ) (cid:17) (cid:21) = 0 (67) for i ∈ { , } .Proof. If 2 | µ ( i ) | ≥ | v (cid:48) ( i ) | , then Lemma 6.1 tells us that ˜ L i ( t ) = 0, which implies the assertion inthis case. For the rest of the proof assume that 2 | µ ( i ) | < | v (cid:48) ( i ) | . W.l.o.g. we assume that i = 0 asthe case i = 1 is similar. To begin, we prove that (67) holds with ˜ L ( · ) replaced by ˆ L (1)0 ( · ). For δ ∈ (0 ,
1) define ˜ m ( δ ) = inf x ≤ δ ˜ m ( x ) . (68)The asymptotic behavior (39) of ˜ m ( · ) implies lim δ → ˜ m ( δ ) = lim p → ˜ m ( p ) = ∞ . Recall that B y, · is a standard Brownian motion started at B y, = y . Observe that ξ (1) t := (cid:90) L B ˜ S (0) , ˜ S ( y ) ( t ) ˜ m ( p ) dp ≥ ˜ m ( δ ) (cid:90) δ L B ˜ S (0) , ˜ S ( y ) ( t ) dp =: ζ t ∀ t ≥ . (69)Using (cid:0) ξ (1) t (cid:1) − ≤ ζ − we obtain an upper bound for ˆ L (1)0 ( · ) as follows1 t E (cid:104)(cid:0) ˆ L (1)0 ( t ) (cid:1) (cid:105) = 1 t E (cid:104)(cid:16) L B ˜ S (0) , ˜ S (0) (cid:16)(cid:0) ξ (1) t (cid:1) − (cid:17)(cid:17) (cid:105) ≤ t E (cid:104)(cid:16) L B ˜ S (0) , ˜ S (0) (cid:0) ζ − t (cid:1)(cid:17) (cid:105) ≤ C ˜ m ( δ ) (cid:20) t ˜ m ( δ ) (cid:0) ˜ S ( δ ) − ˜ S (0) (cid:1) + 1 (cid:21) t → −−−→ C ˜ m ( δ ) δ → −−−→ C which is independent of t and δ . The last inequality is Lemma 6.2.Now we come to the local time process ˜ L ( · ). Recall r , R from Section 2 and let τ be thefirst jump time of ˜ p ( · ) from the boundary point 0. The local time ˆ L ( t ) converges to zero almostsurely as t →
0. By the theorem of dominated convergence, this implies E ˆ L ( t ) → t → t ≥ r E (cid:0) ˆ L (1)0 ( t ) (cid:1) ≤ / t ≤ t . Then we obtain from thedefinition (60) of ˜ L ( · ) and from the Markov property E (cid:20)(cid:16) ˜ L ( t ) (cid:17) τ Assume 2.1. Let (cid:0) ˜ Y ( t ) : t ≥ (cid:1) be a diffusion corresponding to the generator ˜ A defined in (8). Then for each ψ ∈ H ψ ( ˜ Y ( t )) − ψ ( ˜ Y (0)) = (cid:90) t D ˜ m D ˜ S ψ ( ˜ Y ( u )) du + (cid:90) t ψ (cid:48) ( ˜ Y u ) (cid:113) v ( ˜ Y u ) dB u + 12 ˜ L ( t ) · D ˜ S ψ (0) − 12 ˜ L ( t ) · D ˜ S ψ (1) (73) for all t ≥ almost surely.Proof. Fix ψ ∈ H . We approximate ψ with suitable functions and apply the semimartingale Itˆo-Tanaka formula. Denote by (cid:0) ¯ L p ( t ) (cid:1) t ≥ ,p ∈ [0 , the semimartingale local time process of ( ˜ Y ( t )) t ≥ .We remark that, in general, this local time process is distinct from the local time, ˜ L p ( · ), introducedin the preceding section. By Theorem 3.7.1 in [13] we may and we will assume that ¯ L p ( t ) iscontinuous in t and c`adl`ag in p . The occupation time formula (Theorem 3.7.1 in [13]) states that (cid:90) t ψ (cid:0) ˜ Y ( u ) (cid:1) v (cid:0) ˜ Y ( u ) (cid:1) du = 2 (cid:90) ψ ( p ) ¯ L p ( t ) dp, t ≥ , (74)19lmost surely. Let f be a continuous function which is C except in { a , . . . , a n } ⊂ [0 , 1] andwhich admits finite limits f (cid:48) ( a k +) and f (cid:48) ( a k − ), k = 1 , . . . , n . Then the Itˆo-Tanaka formula forcontinuous semimartingales (see Theorem 3.7.1 and Problem 3.6.24 in [13]) states that f ( ˜ Y t ) − f ( ˜ Y )= (cid:90) t f (cid:48) ( ˜ Y u )˜ µ ( ˜ Y u ) du + (cid:90) t f (cid:48)(cid:48) ( ˜ Y u ) v ( ˜ Y u ) du + (cid:90) t f (cid:48) ( ˜ Y u ) (cid:113) v ( ˜ Y u ) dB u + n (cid:88) k =1 ¯ L a k ( t ) (cid:104) f (cid:48) (cid:0) a k + (cid:1) − f (cid:48) (cid:0) a k − (cid:1)(cid:105) (75)almost surely.For every n ∈ N , let ψ n be a continuous function which is equal to ψ in ( n , − n ) and whichis constant both in [0 , n ] and in [1 − n , ψ n ) n ∈ N approximates ψ uniformly in [0 , 1] and that ( D ˜ m D ˜ S ψ n ) n ∈ N approximates D ˜ m D ˜ S ψ pointwise and boundedly in(0 , D ˜ S ψ n (cid:0) n − (cid:1) = 0 and D ˜ S ψ n (cid:0) n + (cid:1) = D ˜ S ψ (cid:0) n (cid:1) . (76)Comparing the occupation time formula (74) of ¯ L p ( · ) with the occupation time formula (10) of˜ L p ( · ) we see that ¯ L p ( t ) = ˜ L p ( t ) 12 v ( p ) ˜ m ( p ) = ˜ L p ( t ) 12 ˜ S (cid:48) ( p ) ∀ p ∈ (0 , . (77)Now applying the Itˆo-Tanaka formula (75) to ψ n ( · ) and inserting (77), we arrive at ψ n ( ˜ Y t ) − ψ n ( ˜ Y ) = (cid:90) t D ˜ m D ˜ S ψ n ( ˜ Y u ) du + (cid:90) t ψ (cid:48) n ( ˜ Y u ) (cid:113) v ( ˜ Y u ) dB u + 12 ˜ L n ( t ) D ˜ S ψ n (cid:0) n (cid:1) − 12 ˜ L − n ( t ) D ˜ S ψ n (cid:0) − n (cid:1) . (78)Note that the Lebesgue measure of (cid:8) u ≤ t : ˜ Y u ∈ { , } (cid:9) is equal to zero almost surely. Letting n → ∞ in (78) completes the proof. Proof of Theorem 2.1. Recall ˜ G , D ˜ m D ˜ S and H from Section 5. Lemma 5.2 shows that the gen-erator of the time-reversed process is the closure of ˜ G . Therefore it remains to be shown that thegenerator of the Markov process (cid:0) ˜ p ( t ) : t ≥ (cid:1) restricted to the set H coincides with ˜ G , that is,that E p ψ (cid:0) ˜ p ( t ) (cid:1) − ψ ( p ) t t → −−−→ ˜ Gψ ( p ) = D ˜ m D ˜ S ψ ( p ) (79)holds for all p ∈ [0 , 1] and every ψ ∈ H .Recall R i , r i , κ i for i ∈ { , } from Section 2. Fix ψ ∈ H and note that ψ is C in (0 , p ∈ (0 , λκ i (cid:54) = 0 and holds for every p ∈ [0 , 1] if λκ i = 0. It remains to prove (79) for i ∈ { , } if λκ i (cid:54) = 0. Starting at i ∈ { , } , (cid:0) ˜ p ( t ) : t ≥ (cid:1) evolves according to a diffusion (cid:0) ˜ Y ( t ) : t ≥ (cid:1) whichis governed by ˜ A until the first time t such that ˜ L i ( t ) ≥ R i . At that time, the process restarts20rom an independent random point J i in (0 , 1) with distribution κ i w i ( p ) π ( p ) dp . Thus E i ψ (˜ p ( t )) − ψ ( i ) − E i (cid:104) ˜ L i ( t ) 0. In the last step we used the inequality 1 − e − x − x ≤ x for x ≥ ψ ∈ H . The local time ˜ L i ( t ) converges to zero a.s. as t → 0. By the dominatedconvergence theorem, this implies that E i ˜ L i ( t ) converges to zero as t → 0. Thus the last summandon the right-hand side of (80) is of order o ( t ). Furthermore Lemma 4.3 implies r i lim p → i (cid:0) vπψ (cid:48) (cid:1) ( p ) 1 λκ i = lim p → i (cid:0) v ˜ mψ (cid:48) (cid:1) ( p ) = D ˜ S ψ ( i ) . (81)Next we consider the second expectation on the left-hand side of (80). Using H¨older’s inequalitywe see that E i (cid:104) ˜ L i ( t ) ≥ R i (cid:0) ψ ( ˜ Y ( t )) − ψ ( i ) (cid:1)(cid:105) = E i (cid:104)(cid:0) − e − r i ˜ L i ( t ) (cid:1)(cid:0) ψ ( ˜ Y ( t )) − ψ ( i ) (cid:1)(cid:105) ≤ (cid:114) E i (cid:104)(cid:0) ˜ L i ( t ) (cid:1) (cid:105)(cid:114) E i (cid:104)(cid:0) ψ ( ˜ Y ( t )) − ψ ( i ) (cid:1) (cid:105) = (cid:112) o ( t ) (cid:112) O ( t ) = o ( t ) (82)where we have applied Lemma 6.3. Thus we obtain from the Itˆo-Tanaka formula (73) E i (cid:104) ˜ L i ( t ) 0. 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