Feynman-Kac Particle Integration with Geometric Interacting Jumps

Home , Time series

arXiv:1211.7191v1 [math.PR] 30 Nov 2012 ok 3 0 1 7 4 n,mr eety n[,1,26]. 16, [4, in recently, more and, 34] 27, do 21, application 10, their [3, and books models functional cont these stochastic of estimation, account parameter filte nonlinear chain in Markov problems complex hidden of variety a solve these to of physic used Applications numerical sciences. probability, engineering applied and interacting learning, in their role and significant models a these mathema precisely, integration More path theory. central are formulae Feynman-Kac Introduction 1 ireDlMoral Del Pierre ena-a atceitgainwt emti interact geometric with integration particle Feynman-Kac ess o smttcba n aineestimates. variance and bias asymptotic non cesses, original an on based errors. is fluctuation approach the Our of d horizon. decompositions also b We time to the models. appear what integration w.r.t. yielding particle system, Feynman-Kac the asymptotic of of non class size provide the we and situations, step sequen time time both a integer of In at terms integers. in observed the jumps is interacting geometric process with discrete models tim continuous the with that de models assume models Feynman-Kac in particle the is generation of proximations process discrete reference consider the we proces whether former, jump to interacting corresponding geometric model, with models integration rmr:6L0 50;6G5 eodr:6G7 10;82 81Q05; 60G57, Secondary: ; 60G35. 65C05; 62L20; Primary: hsatcei ocre ihtedsg n nlsso dis of analysis and design the with concerned is article This ahmtc ujc Classification Subject Mathematics Keywords 4 1 SR ahmtc,IfraisadSaitc,Prh Aus Perth, Statistics, and Informatics Mathematics, CSIRO NI odaxSdOetadUiest fBreu,Franc Bordeaux, of University and Sud-Ouest Bordeaux INRIA 1 ena-a omle neatn uppril systems particle jump interacting formulae, Feynman-Kac : ireE Jacob E. Pierre , 2 ainlUiest fSnaoe Singapore Singapore, of University National 5 nvriyCleeLno,UK London, College University 3 nvriyo awc,UK Warwick, of University 2 nhn Lee Anthony , : Abstract 1 3 arneMurray Lawrence , atceitgaintcnqe r increasingly are techniques integration particle ,Bysa ttsis rbblsi machine probabilistic statistics, Bayesian s, ig aaasmlto,rr vn sampling, event rare assimilation, data ring, an a efudi h eiso research of series the in found be can mains atceitrrttoshv oet play to come have interpretations particle o n nnilmteais detailed A mathematics. financial and rol ndb rirrl n iems ap- mesh time fine arbitrarily by fined e.W nlz w eea ye of types general two analyze We ses. isadvrac hoeswrt the w.r.t. theorems variance and bias eo nemdaetm tp between steps time intermediate of ce susuiomcnegneestimates convergence uniform iscuss otnoso iceetm.Frthe For time. discrete or continuous n edsg e approximation new design we and s ia oesi hsc n probability and physics in models tical h rtrslso hstp o this for type this of results first the e ahitgas o h atr we latter, the For integrals. path e eiru nlsswt rtorder first with analysis semigroup C22. rt ieFymnKcparticle Feynman-Kac time crete 4 esr audpro- valued measure , n aehW Peters W. Gareth and , tralia e n jumps ing 5 In computational physics, these techniques are used for free energy computations, specifically in estimating ground states of Schrödinger operators. In this context, these particle models are often referred as quantum or diffusion Monte Carlo methods [1, 2, 7, 39]. We also refer the reader to the series of articles [11, 24, 40, 41, 46]. In advanced signal processing, they are known as particle filters or sequential Monte Carlo methods, and were introduced in three independent works in the 90’s [9, 32, 37]. These stochastic particle algorithms are now routinely used to compute sequentially the flow of conditional distributions of the random states of a signal process given some noisy and partial observations [3, 10, 12, 21, 27, 28, 35, 38]. Feynman-Kac formulae and their particle interpretations are also commonly used in financial mathematics to model option prices, futures prices and sensitivity measures, and in insurance and risk models [4, 5, 33, 42, 44, 43]. They are used in rare event analysis to model conditional distributions of stochastic processes evolving in a rare event regime [6, 5, 20]. This article presents geometric interacting jump particle approximations of Feynman-Kac path integrals. It also contains theoretical results related to the practical implementation of these particle algorithms for both discrete and continuous time integration problems. A key result is the presentation of connections between the interacting jump particle interpretations of the continuous time models and their discrete time generation versions. This is motivated by the fact that while the continuous time nature of these models is fundamental to describing certain phenomena, the practical implementation of these models on a computer requires a judicious choice of time discretization. Conversely, as shown in section 2.1 in [25], a discrete time Feynman-Kac model can be encapsulated within a continuous time framework by considering stochastic processes only varying on integer times. Continuous time Feynman-Kac particle models are based on exponential interacting jumps [15, 24, 21, 30, 31, 29, 46], while their discrete time versions are based on geometric type jumps [10, 16, 19]. From a computational perspective, the exponential type interacting jumps thus need to be approximated by geometric type jumps. Incidentally, some of these geometric type interacting jump particle algorithms are better suited to implementation in a parallel computing environment (see section 5.3). Surprisingly, little attention has been paid to analyze the connections between exponential and geometric type jump particle models. There are references dealing with these two models separately [8, 17, 18, 22, 25, 24, 46], but none provide a convergence analysis between the two. In this paper we initiate this study with a non asymptotic bias and variance analysis w.r.t. the time step parameter and the size of the particle population scheme. Special attention is paid to the stochastic modeling of these interacting jump processes, and to a stochastic perturbation analysis of these particle models w.r.t. local sampling random fields. We conclude this section with basic notation used in the article. We let b(E) be the Banach 1 B space of all bounded Borel functions f on some Polish state space E equipped with a Borel σ-field , E equipped with the uniform norm f = sup f(x) . We denote by osc(f) := sup f(x) f(y) the k k x∈E | | x,y | − | oscillation of a function f (E). We let µ(f)= f(x)µ(dx) be the Lebesgue integral of a function ∈Bb f (E) with respect to a finite signed measure´ µ on E. We also equip the set (E) of finite ∈ Bb M signed measures µ with the total variation norm µ = sup µ(f) , where the supremum is taken k ktv | | over all functions f (E) with osc(f) 1. We let (E) (E) be the subset of all probability ∈ Bb ≤ P ⊂ M measures. We recall that any bounded integral operator Q on E is an operator Q from (E) into Bb itself defined by Q(f)(x) = Q(x,dy)f(y), for some measure Q(x, .), indexed by x E, and we set ´ ∈ 1i.e. homeomorphic to a complete separable metric space

2 Q = sup Q(x, .) . These operators generate a dual operator µ µQ on the set of finite k ktv x∈E k ktv 7→ signed measures defined by (µQ)(f) = µ(Q(f)). A Markov kernel is a positive and bounded integral operator Q s.t. Q(1) = 1. The Dobrushin contraction coefficient of a Markov kernel Q is defined by β(Q) := sup osc(Q(f)), where the supremum is taken over all functions f (E) s.t. osc(f) 1. ∈ Bb ≤ Given some positive potential function G on E, we denote by ΨG the Boltzmann-Gibbs transformation µ (E) Ψ (µ) (E) defined by Ψ (µ)(f)= µ(fG)/µ(G). ∈ P 7→ G ∈ P G 2 Description of the models

2.1 Feynman-Kac models We consider an E-valued Markov process , t R = [0, [ defined on a standard filtered probability Xt ∈ + ∞ space (Ω, = ( ) R , P). The set Ω= D(R , E) represents the space of càdlàg paths equipped with F Ft t∈ + + the Skorokhod topology which turn it into a Polish space. A point ω Ω represents a sample path ∈ of the canonical process (ω) = ω . We also let X = σ( , s t) and P be the sigma-field Xt t Ft Xs ≤ and probability measure of the process ( ) R . Finally, we also consider the P-augmentation of Xt t∈ + Ft X so that the resulting filtration satisfies the usual conditions of right continuity and completion Ft by P-negligible sets (see for instance [36, 45], and the references therein). We also consider a time inhomogeneous bounded Borel function on E. Vt We let Qt and Λt be the Feynman-Kac measures on Ωt := D([0,t], E) defined for any bounded measurable function f on Ωt, by the following formulae

t Q (f) := Λ (f)/Λ (1) with Λ (f)= E f( ) exp ( )ds (1) t t t t X[0,t] ˆ Vs Xs 0 and we let νt and µt, respectively, be the t-marginals of Λt and Qt. We consider the mesh sequence t = k/m, k 0, with time step h = t t = 1/m associated k ≥ n − n−1 with some integer m 1, and we let Q(m) and Λ(m) be the Feynman-Kac measures on Ω deﬁned for ≥ tn tn tn any bounded measurable function f on Ωtn , by the following formulae

(m) (m) (m) (m) Q (f) := Λ (f)/Λ (1) with Λ (f)= E f( ) eVtp (Xtp )/m . (2) tn tn tn tn  X[0,tn]  0≤Yp

In this case, the marginal νn = γn and µn = ηn of the Feynman-Kac measures of Λt and Qt on integer times t = n are given for any f (E) by the formula ∈Bb

η (f)= γ (f)/γ (1) with γ (f) := E f(X ) G (X ) . (3) n n n n  n p p  0≤Yp

3 Case (C) : The process is a continuous time Markov process with infinitesimal generators • Xt L : D(L) D(L) defined on some common domain of functions D(L), and 1(R , D(L)). t → V∈C + The set D(L) is a sub-algebra of the Banach space (E) generating the Borel σ-field , and for Bb E any measurable function : t R (L) the Feynman-Kac semigroup , s t, U ∈ + 7→ Ut ∈ D Qs,t ≤ defined by t (f)(x)= E f( ) exp ( )dr = x Qs,t Xt ˆ Ur Xr | Xs s leaves D(L) invariant; that is we have that (D(L)) D(L). For any s t, the mappings Qs,t ⊂ ≤ r [0,s] L (Q (f)2) and r [0,s] L2(Q (f)2) 1([0,s], D(L)), and their norm as well ∈ 7→ r s,t ∈ 7→ r s,t ∈C the norm of the first order derivatives only depend on ( ) and on the norms of the functions Xs s≤t ( ) and their derivatives. Us s≤t The regularity conditions stated in (C) correspond to time inhomogeneous versions of those introduced in [24]. They hold for pure jump processes with bounded jump rates with D(L) = (E), or for Bb Euclidean diffusions on E = Rd with regular and Lipschitz coefficients by taking D(L) as the set of ∞- C functions with derivatives decreasing at infinity faster that any polynomial function. These regularity conditions allow the use of most of the principal theorems of stochastic differential calculus, e.g. the “carré du champ”, or square field, operator that characterizes the predictable quadratic variations of the martingales that appear in Ito’s formulae. These regularity conditions can probably be relaxed using the extended setup developed in [21]. We have already mentioned that the particle interpretations associated with the continuous time models (1) are defined in terms of interacting jump particle systems [21, 22, 24, 25]. The implementation of these continuous time particle algorithms is clearly impractical and we therefore resort to the geometric interacting processes associated with the m-approximation models defined in (2). These discrete generation interacting jumps models provide new and different types of adaptive resampling procedures, which differ from those discussed in the articles [12, 13], and the references therein.

2.2 Mean field particle models In this section, we provide a brief description of the geometric type interacting jump particle models associated with the m-approximation Feynman-Kac model defined in (2). First, if we define

(x,dy)= P dy = x and = exp ( /m), Mtn,tn+1 Xtn+1 ∈ | Xtn Gtn Vtn (m) then it is well known that µtn satisﬁes the following evolution equation

µ(m) =Ψ (µ(m)) . (4) tn+1 Gtn tn Mtn,tn+1 Further details on the derivation of these evolution equations can be found in [10, 21, 16]. The particle interpretation of this model depends on the interpretation of the Boltzmann-Gibbs transformation in terms of a Markov transport equation

Ψ (µ)= µ (5) Gtn Stn,µ for some Markov transitions , that depend on the time parameter t and on the measure µ. The Stn,µ n choice of these Markov operators is not unique; we refer to [10] for a more thorough discussion of

4 these models. In this article, we consider an abstract general model, and illustrate our study with the following three classes of models.

Case 1 : We have = , for some non negative and bounded function . In this situation, • Vt −Ut Ut (5) is satisﬁed by the Markov transition

−Utn (x)/m −Utn (x)/m (x,dy) := e δ (dy)+ 1 e Ψ −U /m (µ)(dy). Stn,µ x − e tn Case 2 : The function is non negative. In this situation, (5) is satisﬁed by the Markov • Vt transition

1 1 tn,µ(x,dy) := δx(dy)+ 1 Ψ Vt /m (µ)(dy). Vt /m Vt /m (e n −1) S µ e n − µ e n ! Case 3: The Markov transport equation (5) is satisﬁed by the Markov transition •

tn,µ(x,dy) := (1 atn,µ(x)) δx(dy)+ atn,µ(x)Ψ eVtn /m−eVtn (x)/m (µ)(dy) S − ( )+

with the rejection rate a (x) := µ eVtn /m eVtn (x)/m /µ eVtn /m [0, 1] tn,µ − + ∈ In these three cases we have the following ﬁrst order expansion 1 1 = Id + L + R (6) Stn,µ m tn,µ m2 tn,µ b b with some jump type generator L and some integral operator R (m) s.t. sup R < , tn,µ t ,µ tn,µ n tn tv ∞ where the supremum is taken over all m 1 and µ (E). The jump generators L corresponding b ≥ ∈ P b tn,µ b to the three cases presented above are described respectively in (12), (13), and (14). The proofs of these expansions is rather elementary, and they are housed in the appendix, on pageb 37. In addition, whenever (5) is satisﬁed, we have the evolution equation

(m) (m) µ = µ (m) with the Markov kernels = . (7) tn+1 tn n+1,µ tn,tn+1,µ tn,µ tn,tn+1 K tn K S M

i The mean ﬁeld N-particle model ξtn := ξtn 1≤i≤N associated with the evolution equation (7) is a N Markov process in E with elementary transitions given by

i i N 1 P ξtn+1 dx ξtn = t ,t ,µN (ξt , dx ) with µt = δξi , (8) ∈ | K n n+1 tn n n N tn 1≤i≤N 1≤i≤N Y X where dx = dx1 ...dxN stands for an inﬁnitesimal neighborhood of the point x = (xi) EN . 1≤i≤N ∈

5 3 Statement of the main results

Our ﬁrst main result relates the Feynman-Kac models (1) and their m-approximation measures (2) in case (D) and (C).

Theorem 3.1 In case (D), we have

(m) (m) νn = νn = γn and µn = µn = ηn with the Feynman-Kac measures γn and ηn defined in (3). In case (C), we have the first order decomposition 1 1 Λ(m) = Λ + r and Q(m) = Q + r tn tn m m,tn tn tn m m,tn with some remainder signed measures r , r s.t. sup r r < . m,tn m,tn m≥1 k m,tn ktv ∨ k m,tn ktv ∞ The proof of the theorem is rather technical and it is postpon ed to the appendix. The first assertion of theorem 3.1 allows us to turn a discrete time Feynman-Kac model (3) into a (m) (m) (m) continuous time model (1). To be more precise, we have that νtn = νtp Qtp,tn with the Feynman-Kac semigroup

(m) Q (f)(x) := E f( ) eVtq (Xtq )/m = x tp,tn  Xtn Xtp  p≤q

Q (f)(x) := E f(X ) G (X ) X = x = Q(m)(f)(x). k,n  n l l | k  k,n k≤Yl

(m) Q (f)(x) = G (x)r/m E f( ) eVtq (Xtq )/m = x . k,n k ×  Xtnm Xk+r/m  k+r/m≤t Markov chain

6 ( ) are generally unknown. To get some feasible Monte Carlo approximation scheme, we need a Xtn n≥0 dedicated technique to sample the transitions of this chain. One natural strategy is to replace in (2), the reference Markov chain ( ) by the Markov chain ( ˆ ) associated with some Euler type Xtn n≥0 Xtn n≥0 discretization model with time step ∆t = 1/m. The stochastic analysis of these models is discussed in some detail in the articles [17, 18, 15], including ﬁrst order expansions in terms of the size of the time mesh sequence. Our second main result is the following non asymptotic bias and variance theorem for the N- approximation mean ﬁeld model introduced in (8).

Theorem 3.2 We assume that the Markov transport equation (5) is satisfied for Markov transitions also satisfying the first order decomposition (6). Stn,µ In case (C), for any function f D(L), and any N m 1 we have the non asymptotic bias and ∈ ≥ ≥ variance estimates 1 1 E µN (f) µ (f) c (f) + tn − tn ≤ tn N m and 2 1 1 E µN (f) µ (f) c (f) + tn − tn ≤ tn N m2 for some finite constant c (f) < that only depends on t and on f. tn ∞ n In case (D), for any f (E) s.t. osc(f) 1, and for any N m 1 we have the non ∈ Bb ≤ ≥ ≥ asymptotic bias and variance estimates

2 1 N E µN (f) η (f) a(n) and N E µN (f) µ (f) a(n) 1+ a(n) n − n ≤ tn − tn ≤ N for some some constant

3 3 2 a(n) c gk,ngk,k+1 ( log Gk 1) β (Pk,n) with gk,n := sup Qk,n(1)(x)/Qk,n(1)(y). ≤ k k∨ x,y 0≤Xk

β(Pk,n)= sup Φk,n(µ1) Φk,n(µ2) tv . µ1,µ2∈P(E) k − k On the other hand, we also have that

Qk,n(1)(x) Qk,n(1)(x)= Φk,l(δx)(Gl) log = (log Φk,l(δx)(Gl) log Φk,l(δy)(Gl)) . ⇒ Qk,n(1)(y) − k≤Yl

7 1 (a−b) Using the fact that log a log b = 0 ta+(1−t)b dt, we ﬁnd that − ´ Q (1)(x) 1 [Φ (δ )(G ) Φ (δ )(G )] log k,n = k,l x l − k,l x l dt. Qk,n(1)(y) ˆ0 tΦk,l(δx)(Gl) + (1 t)Φk,l(δy)(Gl) k≤Xl

−c4(k−l) c1 Gl(x) c2 and sup Φk,l(µ1) Φk,l(µ2) tv c3 e (10) ≤ ≤ µ1,µ2∈P(E) k − k ≤ for some positive and bounded constants c , 1 i 4, we ﬁnd that i ≤ ≤

β(P ) c e−c4(k−n) and log g 2(c c /c ) e−c4(k−l) 2(c c )/(c (1 e−c4 )). k,n ≤ 3 k,n ≤ 2 3 1   ≤ 2 3 1 − k≤Xl

m 1, ρ> 0 : x,y E M m(x, .) ρ M m(y, .). ∃ ≥ ∃ ∀ ∈ ≥ It is well known that this condition is satisfied for any aperiodic and irreducible Markov chains on finite state spaces, as well as for bi-Laplace exponential transitions associated with a bounded drift function, and for Gaussian transitions with a mean drift function that is constant outside some compact domain. The remainder of the article is organized as follows: Section 4 is concerned with continuous time particle interpretations of the Feynman-Kac models (1). By the representation theorem 3.1, these schemes also provide a continuous time particle interpretation of the discrete time models (3) without further work. In section 4.2, we present the McKean interpretation of the Feynman-Kac models in terms of a time inhomogeneous Markov process whose generator depends on the distribution of the random states. The choice of these McKean models is not unique. We discuss the three interpretation models corresponding to the three selection type transitions presented on page 5. The mean field particle interpretation of these McKean models are discussed in section 4.3. Of course, even for discrete time models (3) these continuous time particle interpretations are based on continuous time interacting jump models and they cannot be used in practice without an additional level of approximation. In this context, when using an Euler type approximation these exponential interacting jumps are replaced by geometric type recycling clocks. These interacting geometric jumps particle models are discussed in section 5, which is dedicated to the discrete time particle interpretations of the Feynman-Kac models presented in (2). In section 5.1, we discuss the McKean interpretation of the Feynman-Kac models in terms of a time inhomogeneous Markov chain model whose elementary transitions depends on the distribution of the random states. Again, the choice of these McKean models is not unique. We discuss the three interpretation models corresponding to the three cases presented on page 5. The mean field particle interpretation of these McKean models are discussed on page 20.

8 Once again, using the representation formulae (2) we emphasize that these schemes also provide a discrete generation particle interpretation of the discrete time models (3). In contrast to standard discrete generation particle models associated with (3), these particle schemes are defined on a refined time mesh sequence between integers. This time mesh sequence can be interpreted as a time dilation. Between two integers, the particle evolution undergoes an additional series of intermediate time evolution steps. In each of these time steps, a dedicated Bernoulli acceptance-rejection trial coupled with a recycling scheme is performed. As the time step decreases to 0, the resulting geometric interacting jump processes converge to the exponential interacting jump processes associated with the continuous time particle model. The final section, section 6, is mainly concerned with the proof of theorem 3.2.

4 Continuous time models

4.1 Feynman-Kac semigroups In case (C) the semigroup of the ﬂow of non negative measures ν is given for any s t by the following t ≤ formulae ν = ν Q , with the Feynman-Kac semigroup Q deﬁned for any f (E) by t s s,t s,t ∈B t Q (f)(x)= E f( ) exp ( ) ds X = x . s,t Xt ˆ Vs Xs | s s

This yields µt = Φs,t(µs), with the nonlinear transformation Φs,t on the set of probability measures deﬁned for any f (E) by ∈B

Φs,t(µs)(f) := µs(Qs,t(f))/µs(Qs,t(1)).

Using some stochastic calculus manipulations, we readily prove that µt satisﬁes the following integro- diﬀerential equation d µ (f)= µ (L (f)) + µ ( f) µ ( )µ (f) (11) dt t t t t Vt − t Vt t for any function f D(L). Further details on the derivation of these evolution equations can be found ∈ in the articles [24, 22]. The particle interpretation of this model depends on the interpretation of the correlation term in the r.h.s. of (11) in terms of a jump type generator. The choice of these generators is not unique. Next, we discuss three important classes of models. These three situations are the continuous time versions of the three cases discussed on page 5.

Case 1 : We assume that = , for some non negative function . In this situation, we • Vt −Ut Ut have the formula

µ ( f) µ ( )µ (f)= µ ( [µ (f) f]) = µ L (f) t Vt − t Vt t t Ut t − t t,µt with the interacting jump generator b

L (f)(x)= (x) [f(y) f(x)] µ (dy). (12) t,µt Ut ˆ − t b

9 Case 2 : When is a positive function, then we have the formula • Vt µ ( f) µ ( )µ (f)= µ L (f) t Vt − t Vt t t t,µt with the interacting jump generator b

L (f)(x)= [f(y) f(x)] (y) µ (dy)= µ ( ) [f(y) f(x)] Ψ (µ )(dy). (13) t,µt ˆ − Vt t t Vt ˆ − Vt t

Caseb 3 : For any bounded potential function we have • Vt µ ( f) µ ( )µ (f)= (f(y) f(x)) ( (y) (x)) µ (dx) µ (dy)= µ L (f) t Vt − t Vt t ˆ − Vt −Vt + t t t t,µt with a = a 0, and with the interacting jump generator b + ∨ L (f)(x)= [f(y) f(x)] ( (y) (x)) µ (dy). (14) t,µt ˆ − Vt −Vt + t

4.2 McKean interpretationb models In the three cases discussed above, for any test functions f D(L) we have the evolution equation ∈ d µ (f)= µ (L (f)) with L := L + L . (15) dt t t t,µt t,µt t t,µt

These integro-diﬀerential equations can be interpreted as the evolution ofb the laws, given by Law(Xt)=

µt, of a time inhomogeneous Markov process Xt with inﬁnitesimal generators Lt,µt that depend on the distribution of the random states at the previous time increment. This probabilistic model is called the McKean interpretation of the evolution equation (15) in terms of a time inhomogeneous Markov process. In this framework, using Ito’s formula for any test function f 1([0, [, (L)), we have ∈C ∞ D ∂ df ( )= + L (f )( )+ dM (f) (16) t X t ∂t t,µt t X t t with a martingale term M t(f) with predictable angle bracket

d M(f) t =ΓL (ft,ft)( t)dt. h i t,µt X Using the r.h.s. description of L in (15), for any f D(L) we notice that t,µt ∈

ΓLt,µ (f,f)=ΓLt (f,f)+Γ (f,f). t Lt,µt Next, we provide a description of this Markov process inb the three cases discussed above. Case 1: In this situation, between the jump times the process evolves as the process . The • X t Xt rate of the jumps is given by the function . In other words, the jump times (T ) are given Ut n n≥0 by the following recursive formulae

t T = inf t T : ( ) ds e n+1 ≥ n ˆ Us X s ≥ n Tn 10 where T0 = 0, and (en)n≥0 stands for a sequence of i.i.d. exponential random variables with unit parameter. At the jump time T the process = x jumps to new site = y randomly n X Tn− X Tn chosen with the distribution µTn−(dy). For any f D(L) we also have that ∈

2 2 ΓL (f,f)(x)= Lt,µt (f f(x)) (x)= t(x) [f(y) f(x)] µt(dy). t,µt − U ˆ − b b In this situation, an explicit expression of the time inhomogeneous semigroup , s t, of Ps,t,µs ≤ the process is provided by the following formula X t (f)(x) = E f(X ) X = x Ps,t,µs t s = Qs,t (1)(x)Φ s,t(δx)(f) + (1 Qs,t(1)(x)) Φs,t(µs)(f) − = Q (f)(x) + (1 Q (1)(x)) Φ (µ )(f). s,t − s,t s,t s

(m) We let and Φ(m) be the Markov transition and the transformation of probability Ptn,tn+1,µtn tn,tn+1 measures deﬁned as and Φ replacing Q by the integral operator Ps,t,µs tn,tn+1 tn,tn+1 (m) Q (f)(x) = e−Utn (x)/m E f( ) = x . tn,tn+1 Xtn+1 | Xtn Under the assumptions of theorem 3.1, using elementary calculations we prove that

(m) 1 (m) = t ,t ,µ + (17) Ptn,tn+1,µtn P n n+1 tn m Rtn,tn+1,µtn

(m) (m) with some remainder signed measures t ,t ,µ such that supm≥1 t ,t ,µ ctn , for R n n+1 tn R n n+1 tn tv ≤ some ﬁnite constant whose values only depend on the potential function . Ut

Case 2: In this situation, between jump times the process evolves as the process . The • X t Xt rate of the jumps is given by the parameter µ ( ). In other words, the jump times (T ) are t Vt n n≥0 given by the following recursive formulae

t T = inf t T : µ ( ) ds e n+1 ≥ n ˆ s Vs ≥ n Tn

where T0 = 0, and (en)n≥0 stands for a sequence of i.i.d. exponential random variables with unit parameter. At the jump time T the process = x jumps to new site = y randomly n X Tn− X Tn chosen with the distribution ΨVTn− (µTn−)(dy). For any f D(L) we also have that ∈

2 2 ΓL (f,f)(x)= Lt,µt (f f(x)) (x)= [f(y) f(x)] t(y) µt(dy). t,µt − ˆ − V b b

11 Case 3: In this case, between jump times the process evolves as the process . The rate of • X t Xt the jumps is given by the function

(x) := µ (( (x)) ) Wt,µt t Vt −Vt + = µ ( 1 ) µ ( (x)) (x). t Vt Vt≥Vt(x) − t Vt ≥Vt Vt

In other words, the jump times (Tn)n≥0 are given by the following recursive formulae

t T = inf t T : ( ) ds e n+1 ≥ n ˆ Wt,µt X s ≥ n Tn

where T0 = 0, and (en)n≥0 stands for a sequence of i.i.d. exponential random variables with unit parameter. At the jump time T the process = x jumps to new site = y randomly n X Tn− X Tn chosen with the distribution Ψ (µ ). (VTn −VTn (x))+ Tn− For any f D(L) we also have that ∈

2 2 ΓL (f,f)(x)= Lt,µt (f f(x)) (x)= [f(y) f(x)] ( t(y) t(x))+ µt(dy) t,µt − ˆ − V −V b so that b 2 µt ΓL (f,f) = [f(y) f(x)] ( t(y) t(x))+ µt(dx)µt(dy). (18) t,µt ˆ − V −V h b i We end this section with another McKean interpretation model combining cases 1 and 2, as an alternative to the generator described in the latter case. First, using the fact that

µ [ µ ( )] [ µ ( )] = µ ([ µ ( )]) = 0 t Vt − t Vt + − Vt − t Vt − t Vt − t Vt we prove the following decompositions

µ ( f) µ ( )µ (f) = µ ([ µ ( )] f) t Vt − t Vt t t Vt − t Vt = µ [ µ ( )] f µ [ µ ( )] f t Vt − t Vt + − t Vt − t Vt − = µ [ µ ( )] [f µ (f)] t Vt − t Vt + − t µ [ µ ( )] [f µ (f)] . − t Vt − t Vt − − t Using the same line of arguments as those used in cases 1 and 2, this implies that

µ ( f) µ ( )µ (f)= µ (L (f)) with L = L+ + L− t Vt − t Vt t t t,µt t,µt t,µt t,µt where the pair of interacting jump generatorsb is given by b b b

L− (f)(x) = [ (x) µ ( )] [f(y) f(x)] µ (dy) t,µt Vt − t Vt − ˆ − t and b L+ (f)(x)= [f(y) f(x)] [ (y) µ ( )] µ (dy). t,µt ˆ − Vt − t Vt + t b 12 In this situation, for any f D(L) we also have that ∈

ΓL (f,f)(x) = ΓL+ (f,f)(x)+ΓL− (f,f)(x) t,µt t,µt t,µt b b b = [f(y) f(x)]2 [ (y) µ ( )] + [ (x) µ ( )] µ (dy) ˆ − Vt − t Vt + Vt − t Vt − t so that 2 µt ΓL (f,f) = [f(y) f(x)] t(y) µt( t) µt(dx)µt(dy). t,µt ˆ − |V − V | h b i 4.3 Mean field particle interpretation models i The mean field N-particle model ξt := ξt 1≤i≤N associated with a given collection of generators Lt,µt N satisfying the weak equation (15) is a Markov process in E with infinitesimal generator given by the following formulae

1 N (i) 1 i N 1 (F )(x ,...,x ) := L (F )(x ,...,x ,...,x ) with m(x) := δ i (19) Lt t,m(x) N x 1≤Xi≤N 1≤Xi≤N for suﬃciently regular functions F on EN , and for any x = (xi) EN . In the above formulae, 1≤i≤N ∈ L(i) stands for the operator L acting on the function xi F (x1,...,xi,...,xN ). t,m(x) t,m(x) 7→ Before entering into the description of the particle model associated with the three cases presented in section 4.2, we provide a brief discussion of the convergence analysis of these stochastic models. Firstly, we recall that dF (ξ )= (F )(ξ ) dt + d (F ) t Lt t Mt for some martingale (ϕ) with increasing process given by Mt t

(F ) t := ΓLs (F, F ) (ξs) ds. hM i ˆ0 In the above we denote by Γ the carré du champ operator associated with , and defined by Ls Ls Γ (F, F ) (x) := (F F (x))2 (x)= (F 2)(x) F (x) (F )(x). Ls Ls − Ls − Ls h i For empirical test functions of the following form F (x)= m(x)(f), with f D(L), we find that ∈ 1 (F )(x)= m(x)(L (f)) and Γ (ϕ, ϕ) (x)= m(x) Γ (f,f) . Ls s,m(x) Ls N Ls,m(x) N 1 From this discussion, if we set µ = δ i , then we find that t N 1≤i≤N ξt

N PN 1 N dµt (f)= µt (L N (f)) dt + dMt (f) t,µt √N for any f D(L), with the martingale M N (f)= √N (F ) with angle bracket given by ∈ t Mt t M N (f) := µN Γ (f,f) ds. t s Ls,µN h i ˆ0 s 13 A more explicit description of the r.h.s. terms in the above can be given in the three cases discussed in section 4.2. For instance, in the third case, using formula (18) we ﬁnd that

t N 2 N N M (f) t := [f(y) f(x)] ( s(y) s(x))+ µs (dx)µs (dy) ds. h i ˆ0 ˆ − V −V We conclude that µN “almost solve”, as N , the nonlinear evolution equation (15). For a more t ↑∞ thorough discussion of these continuous time models, we refer to the reader to the review article [15], and the references therein. By construction, the generator t associated with the nonlinear model (15) is decomposed into a L jump mutation generator mut and an interacting jump generator Lt Lt = mut + jump Lt Lt Lt with mut and jump deﬁned by Lt Lt mut(F )(x) = L(i)(F )(x1,...,xi,...,xN ) Lt t 1≤Xi≤N jump(F )(x) = L(i) (F )(x1,...,xi,...,xN ). Lt t,m(x) 1≤Xi≤N b The mutation generator mut describes the evolution of the particles between the jumps. Between Lt jumps, the particles evolve independently with Lt-motions in the sense that they explore the state space as independent copies of the process with generator L . The jump transition depends on the Xt t form of the generator Lt,µt . Case 1: In this situation the jump generator is given by • b jump(F )(x) = (xi) [F (θi (x)) F (x)] m(x)(du) Lt Ut ˆ u − 1≤Xi≤N i with the population mappings θu deﬁned below

θi : x EN θi (x) = (x1,...,xi−1, u ,xi+1,...,xN ) EN . u ∈ 7→ u ∈ i-th

i |{z} i The quantity t(xt) represents the jump rate of the i-th particle ξt. More precisely, if we denote i U i by Tn the n-th jump time of ξt, we have

t i i i i Tn+1 = inf t Tn : s(ξs)ds en (20) ≥ ˆ i U ≥ ( Tn )

i where (en)1≤i≤N,n∈N stands for a sequence of i.i.d. exponential random variables with unit i i i i parameter. At the jump time T the process ξ i = x jumps to new site ξ i = u randomly n Tn− Tn chosen with the distribution m(ξ i )(du). In other words, at the jump time the i-th particle Tn− jumps to a new state randomly chosen in the current population.

14 The probabilistic interpretation of the jump generator is not unique. For instance, it is easily checked that jump can be rewritten in the following form Lt jump(F )(x) = λ (x) [F (y) F (x)] (x,dy) Lt t ˆ − Pt with the population jump rate λ (x) and the Markov transition (x,dy) on EN given below t Pt i t(x ) 1 λt(x) := Nm(x) ( t) and t(x,dy)= U ′ δ i (dy). i θ j (x) U P 1≤i′≤N t(x ) N x 1≤Xi≤N U 1≤Xj≤N P In this interpretation, the individual jumps are replaced by population jumps at rate λt(ξt). More precisely, the jump times Tn of the whole population are deﬁned by

t i Tn+1 = inf t Tn : s(ξs) ds en  ≥ ˆTn  U  ≥   1≤Xi≤N    where (en)n∈N stands for a sequence of i.i.d. exponential random variables with unit parameter.

At the jump time Tn the population ξTn− = x jumps to new population ξTn = y randomly chosen

with the distribution Tn−(ξTn−, dy). In other words, at the jump time Tn, we select randomly a i P i state ξ with a probability proportional to t(ξ ), and we replace this state by a randomly Tn− U Tn− chosen state ξj in the population, with 1 j N. We end this description with an alternative Tn− ≤ ≤ interpretation when C for some ﬁnite constant C < . In this situation, we clearly have kUtk≤ ∞ λ NC and k tk≤ jump(F )(x)= λ′ [F (y) F (x)] ′(x,dy) Lt ˆ − Pt with the jump rate λ′ and the Markov jump transitions ′ deﬁned below Pt λ (x) λ (x) λ′ = NC and ′(x,dy) := t (x,dy)+ 1 t δ (dy). Pt NC Pt − NC x ′ In this interpretation, the population jump times Tn arrive at the higher rate λ = NC. At the

jump time Tn the population ξTn− = x jumps to new population ξTn = y randomly chosen with ′ the distribution (ξT −, dy). PTn− n In the models described above, as usual, between the jump times Tn of the population every particle evolves independently with Lt-motions. Case 2 : In this situation, the jump generator is given by • jump(F )(x) = m(x)( ) [F (θi (x)) F (x)] Ψ (m(x)) (du). Lt Vt ˆ u − Vt 1≤Xi≤N The particles have a common jump rate given by the empirical average m(ξt)( t). In other words, i i V the jump times Tn of a particle ξt are given by the following recursive formulae

t i i i Tn+1 = inf t Tn : m(ξs)( s) ds en ≥ ˆ i V ≥ ( Tn )

15 i where (en)1≤i≤N,n≥0 stands for a sequence of i.i.d. exponential random variables with unit i i i i parameter. At the jump time Tn the process ξTn− = x jumps to new site ξTn = u randomly

chosen with the weighted distribution ΨVTn− (m(ξTn−))(du). As mentioned in the ﬁrst case, the probabilistic interpretation of the jump generator is not unique. In this situation, it is easily checked that jump can be rewritten in the following form Lt

jump(F )(x) = λ (x) [F (y) F (x)] (x,dy) Lt t ˆ − Pt

with the population jump rate λ and the Markov transition (x,dy) on EN deﬁned below t Pt j 1 t(x ) λt(x) := Nm(x)( t) and t(x,dy)= V δ i (dy). j′ θ j (x) V P N 1≤j′≤N t(x ) x 1≤Xi≤N 1≤Xj≤N V P The description of the evolution of the population model follows the same lines as the ones given in case 1.

Case 3 : In this situation, the jump generator is given by • jump(F )(x) = [F (θi (x)) F (x)] ( (u) V (xi)) m(x)(du) Lt ˆ u − Vt − t + 1≤Xi≤N i i = m(x)(( (x )) ) [F (θ (x)) F (x)] Ψ i (m(x))(du). Vt −Vt + ˆ u − (Vt−Vt(x ))+ 1≤Xi≤N In this interpretation, the jump rate of the i-th particle is given by the average potential variation of the particle with higher values

i 1 j i m(x)(( (x )) )= 1 j i (x ) (x ) . Vt −Vt + N {Vt(x )>Vt(x )} Vt −Vt 1≤j≤N X i i More precisely, if we denote by Tn the n-th jump time of ξt, we have

t i i i i Tn+1 = inf t Tn : m(ξs)(( s s(ξs))+)ds en ≥ ˆ i V −V ≥ ( Tn )

i where (en)1≤i≤N,n∈N stands for a sequence of i.i.d. exponential random variables with unit parameter. i i i i At the jump time T the particle ξ i = x jumps to new site ξ i = u randomly chosen with the n Tn− Tn distribution

j i Ψ i (m(ξ i ))(du) 1 i (ξ i ) i (x ) δ j (du). (VT i −−VT i −(x ))+ Tn− j i Tn− T − Tn− ξ n n ∝ V i (ξ )>V i (x ) V n −V T i − T − i − T − n 1≤Xj≤N n Tn n i j In other words, we choose randomly a new site ξ i = ξ i , among the ones with higher potential value Tn Tn− j i with a probability proportional to the diﬀerence of potential T i −(ξ i ) T i −(ξ i ) . V n Tn− −V n Tn− 16 As the ﬁrst two cases discussed above, we can also interpret this jump generator at the level of the population. In this interpretation we have

jump(F )(x)= λ (x) [F (y) F (x)] (x,dy) Lt t ˆ − Pt with the population jump rate

λ (x)= N m(x)(du) m(x)(dv) ( (u) (v)) t ˆ Vt −Vt + and the population jump transition

j i ( t(x ) Vt(x ))+ t(x,dy) = V − ′ ′ δ i (dy) . j i θ j (x) P 1≤i′,j′≤N ( t(x ) t(x ))+ x 1≤Xi,j≤N V −V P Remark 4.1: In case (D), the reference Markov process = X has deterministic and fixed Xt ⌊t⌋ time jumps on integer times so that the generator approach developed above does not apply directly. Nevertheless their probabilistic interpretation is defined in the same way: Between the jumps, the process X evolves as , and the N particles explore the state space as t Xt independent copies of the process . The rate of the jumps and their random spatial location are Xt defined using the same interpretations as the ones given above. The stochastic modeling and the analysis of these continuous time models and their particle interpretations can be developed using the semigroup techniques provided in [25].

5 Discrete time models

5.1 McKean models and Feynman-Kac semigroups As in the continuous time case, these discrete time evolution equations (7) can be interpreted as the (m) evolution of the laws deﬁned by Law Xtn = µtn of a time inhomogeneous Markov process Xtn with Markov transitions generators (m) that depend on the distribution of the random states at t ,t ,µ K n n+1 tn the previous sub-integer mesh time increment. This probabilistic model is also called the McKean interpretation of the evolution equation (7) in terms of a time inhomogeneous Markov chain. By construction, the elementary transitions of the Markov chain Xtn Xtn+1 are decomposed into two separate transitions Xtn Xtn Xtn+1 .

First, the state Xtn = x jumps to a new location Xtn = y randomly chosen with the Markov transition (m) (x,dy), givenb by one of the three cases presented in section 2.2 when applied to t ,µ S n tn (m) b the particle approximation of the measure µtn at mesh time increment tn. Then, the selected state

Xtn = y evolves to a new site Xtn+1 = z according to the Markov transition tn,tn+1 (y, dz). (m) M (m) Next, we recall some basic properties of the semigroup Φtp,tn of the ﬂow of measures µtn . By construction,b we have (m) (m) (m) (m) (m) (m) Φtp,tn (µtp )(f)= µtp Qtp,tn (f)/µtp Qtp,tn (1)

17 (m) with Feynman-Kac semigroup Qtp,tn deﬁned by

(m) Q (f)(x) = E f( ) eVtq (Xtq )/m = x . (21) tp,tn  Xtn Xtp  p≤q

Deﬁnition 5.1 We consider the integral operators

L(m) := Id, L(m) := Id and L(m) := Id. tn,µ Ktn,tn+1,µ − tn Mtn,tn+1 − tn,µ Stn,µ − Lemma 5.2 We have the decomposition b (m) (m) (m) (m) (m) Ltn,µ = Ltn + Ltn,µ + Ltn,µLtn . In addition, for any f (E) µ (E), and any x E, we have ∈Bb ∈ P b ∈ b 2 (m) 2 f (f)(x) (x)=Γ (m) (f,f)(x) L (f)(x) . tn,tn+1,µ tn,tn+1,µ L tn,µ K − K tn,µ − Proof: Using the decomposition

L(m) = Id + ( Id) + ( Id) Id tn,µ Mtn,tn+1 − Stn,µ − Stn,µ − Mtn,tn+1 − we readily check the ﬁrst assertion. We prove the second decomposition using the fact that

2 (f) 2 = L(m) (f) + f 2 2fL(m) (f). Ktn,tn+1,µ tn,µ − tn,µ This ends the proof of the lemma.

In the further development of this section c < stands for some generic ﬁnite constant whose tn ∞ values may vary from line to line. For any function f D(L), such that L (f) 1([t ,t ], D(L)), ∈ t ∈C n n+1 we also deﬁne 2 f tn := f + sup ∂Lt(f)/∂t + Lt(f) + Lt (f) . (23) k k k k tn≤t≤tn+1 k k k k

Proposition 5.3 In case (D), we have

(m) (m) (m) L = 1N(t ) M Id and L = L + 1N(t ) M Id . tn n+1 tn+1 − tn,µ tn,µ n+1 Stn,µ tn+1 − In case (C), we have the ﬁrst order expansion b 1 1 L(m)(f)= L (f) + R (f) . (24) tn tn m tn m2

18 for any function f D(L), such that L (f) 1([t ,t ], D(L)), with some remainder operator R ∈ t ∈ C n n+1 tn such that R (f) c f . Furthermore, we have the first order expansion k tn k≤ tn k ktn 1 1 L(m) (f)= L (f)+ R (f) (25) tn,µ m tn,µ m2 tn,µ with some second order remainder term R (f) such that sup R (f) c f . In addi- tn,µ µ∈P(E) k tn,µ k≤ tn k ktn tion, we have 1 1 f (f)(x) 2 (x)=Γ (f,f)(x) + (f,f) (x) (26) Ktn,tn+1,µ − Ktn,tn+1,µ Ltn,µ m Rtn,µ m2 with some remainder operator s.t. sup (f,f) c f 2 . µ∈P(E) kRtn,µ k≤ tn k ktn The proof of proposition 5.3 is provided in the appendix, on page 36. The first order expansions stated in the proposition 5.3 can be used to develop a stochastic per- (m) turbation approach to estimate the deviations of the measures µtn around their limiting values µtn . Next, we provide an alternative approach based on the explicit representation (17) of the time inhomogeneous transition of the limiting process on the time mesh sequence t . In the first case discussed X tn n on page 10, we have

(m) (m) (m) (m) (m) (m) µ =Φ µ := Ψ −U /m µ = µ (27) tn+1 tn,tn+1 tn e tn tn tn,tn+1 tn (m) M Ptn,tn+1,µtn with the Markov transition (m) (m) (x,dy) Ptn,tn+1,µtn (28)

−Ut (x)/m −Ut (x)/m (m) = e n (x,dy)+ 1 e n Ψ −U /m µ (dy). Mtn,tn+1 − e tn tn Mtn,tn+1 Using (17) we readily ﬁnd that

(m) (m) 1 (m) (m) 1 (m) µ = µ (m) + =Φ µ + tn+1 tn t ,t ,µ tn,tn+1 tn,tn+1 tn tn,tn+1 P n n+1 tn m W m W with the signed measure

(m) (m) (m) (m) tn,tn+1 := µtn tn,tn+1,µt s.t. sup tn,tn+1 ctn W R n m≥1 W tv ≤

for some ﬁnite constant whose values only depend on the poten tial function . In summary, we have Ut proven the following ﬁrst order local perturbation decompositions 1 µ(m) = Φ µ(m) + (m) tn+1 tn,tn+1 tn m Wtn,tn+1 µtn+1 = Φtn,tn+1 (µtn ) These local expansions allow the use of perturbation theory developed in section 7.1 of [10] to derive (m) several qualitative estimates between µtn and µtn in terms of the stability properties of the Feynman-

Kac semigroup Φtn,tn+1 .

19 5.2 Mean ﬁeld particle interpretation models N 1 If we set µ = δ i , then we have the decomposition tn N 1≤i≤N ξtn

P N N 1 N µt = µt t ,t ,µN + Wt ,t n+1 n K n n+1 tn √N n n+1 N with the sequence of empirical random ﬁelds Wtn,tn+1 such that

E W N (f) ξ = 0 tn,tn+1 | tn and 2 N 2 N E Wt ,t (f) ξtn = µt (du) t ,t ,µN (u, dv) f(v) t ,t ,µN (f)(u) . n n+1 | ˆ n K n n+1 tn − K n n+1 tn As for the continuous time models, we conclude that µN “almost solve”, as N , the nonlinear tn ↑ ∞ evolution equation (7). For a more thorough discussion of these local sampling random ﬁeld models, we refer the reader to [10, 15, 16], and references therein.

By construction, the elementary transitions of the Markov chain ξtn ξtn+1 are decomposed into two separate transitions: i ξtn ξtn = ξt ξtn+1 . (29) n 1≤i≤N i i i i First, every particle ξtn = x jumpsb independentlyb to a new location ξtn = y randomly chosen with the Markov transition (xi, dyi), with 1 i N. Following this, each particle ξi = yi Stn,m(ξtn ) ≤ ≤ tn evolves independently to a new site ξi = zi according to the Markov transitionb (yi, dzi), tn+1 Mtn,tn+1 with 1 i N. b ≤ ≤ In other words, the mutation transition describes the evolution of the particles between the jumps. Between the jumps, the particles evolve independently with -motions in the sense that they Mtn,tn+1 explore the state space as independent copies of the process with Markov transition . The Xtn Mtn,tn+1 jump transition can also be interpreted as an acceptance-rejection transition equipped with a recycling mechanism. In this interpretation, the mutation transition can be interpreted as a proposal transition. Notice that the selection type transition is dictated by the choice of the transition . Stn,µtn We illustrate these jump type transitions in the ﬁrst case presented on page 5. In this situation, we i i recall that the selection transition of the i-th particle ξtn ξtn is given by the following distribution

i i i −Ut (ξ )/m −Ut (ξ )/m N n tn i n tn t ,µN (ξt , dy) := e δξ (dy)+ 1 e b Ψ −Utn /m (µt )(dy). (30) S n tn n tn − e n Next, we provide an interpretation of this transition as an acceptance-rejection scheme with a i recycling mechanism. We let ξtn = ξt be a sequence of conditionally independent random n 1≤i≤N variables with common law

e e i −Utn (ξt )/m N e n i Ψ −Utn /m (µt )= j δξ . e n −U (ξ )/m tn e tn tn 1≤Xi≤N 1≤j≤N We also consider a sequence of conditionally independentP Bernoulli random variables with distribution

i i −U (ξi )/m P ǫ = 1 ξ = 1 P ǫ = 0 ξ = e tn tn . tn | tn − tn | tn 20 In this notation, we have that ξi = ǫi ξi + 1 ǫi ξi . tn tn tn − tn tn i i In in other words, the particle ξtn is accepted whenǫtn = 1; otherwise, it is rejected and replaced by i b e N −U a particle ξtn randomly chosen with the updated weighted distribution Ψe tn /m (µtn ). The pool of particles that have been accepted from the start provide a sequence of exact samples. More precisely, it can be easilye shown that

Law( ξi p : 0 p

T i = inf t > T i : (ξi )/m ei n+1  p n tk tk n i U ≥  TnX≤k≤tp 

i  i −Ut (ξ )/m i = inf t > T : e k tk u  p n n i ≤  Tn≤Yk≤tp  i i where (un)1≤i≤N,n∈N stands for a sequence of i.i.d.uniform random variables on ]0, 1], and en := log ui , with 1 i N,n N, is the corresponding sequence of i.i.d. exponential random variables − n ≤ ≤ ∈ with unit parameter. We check this claim using the following observations

i i i i P T = tp T , T k tp ξ n+1 | n ∀ n ≤ ≤ tk = P T i k

5.3 An mean ﬁeld model with uniform recycling When m is large enough, the recycling distribution (31) in the Markov transition (30) is almost equal N to µ i . For instance, we have the total variation estimate Tn

−U i /m N N Tn Ψ −U i /m (µT i ) µT i 1 e T i /m. e Tn n − n tv ≤ − ≤ kU n k

21 We prove these inequalities using the decomposition

−U/m Ψ −U/m (µ)(f) µ(f)= µ 1 e [Ψ −U/m (µ)(f) f] e − − e − h i which is valid for any bounded functions and f. Hence, to save computational time we can replace U N N the recycling weighted measure Ψ −U i /m (µ i ) by µ i . In this situation, the selection transition takes e Tn Tn Tn the following form 1+⌊Nτ i ⌋ ξi = ǫi ξi + 1 ǫi ξ tn tn tn tn − tn tn where τ i stands for a sequence of i.i.d. uniform random variables on ]0, 1]. tn b Next, we detail some analysis of the diﬀerences between the McKean models with recycling Boltzmann- Gibbs transitions, and the models discussed which utilise uniform recycling.

We let the collection of selection transitions deﬁned as in (30), by replacing Ψ −U /m (µ) by Stn,µ e tn the measure µ. Furthermore, we denote by Φ(m) , resp. Φ(m) , with p n, the semigroups associated tp,tn tp,tn ≤ with these ﬂowse (m) (m) (m) (m) (m) (m) Φtp,tn µtp = µtn and Φtep,tn µtp = µtn . Then, using the decomposition e e e

−Utn (x)/m (x,dy)= 1 e (Ψ −U/m (µ) µ) (dy) Stn,µ − Stn,µ − e − h i we ﬁnd that e 2 2 sup tn,µ(x, .) tn,µ(x, .) tn /m . x∈E S − S tv ≤ kU k

Replacing Ψe−Utn /m (µ) by the measure µ in (28),e the evolution equation (27) takes the following form

(m) (m) (m) (m) (m) (m) µ = Φ µ := µ (m) with (m) = (m) tn,tn+1 . tn+1 tn,tn+1 tn tn tn,µ Ptn,tn+1,µtn Ptn,tn+1,µtn S tn M e From previouse estimates,e wee ﬁnd thate e e e e e 1 Φ(m) (µ)=Φ(m) (µ)+ (m) (µ) tn,tn+1 tn,tn+1 m2 Rtn,tn+1 with some measures (m) e(µ) s.t. Rtn,tn+1 (m) 2 sup tn,tn+1 (µ) tn . µ∈P(E) R tv ≤ kU k

(m) (m) We end this section with an estimate of the difference between the flow of measures µtn+1 , and µtn+1 . We are now in position to state, and to prove the following theorem. e Theorem 5.4 For any n 0, we have ≥ (m) (m) µt µt ctn /m n − n tv ≤ for some finite constant c < , whose values don’t depend on the parameter m. tn ∞ e

22 Proof: We use the stochastic perturbation analysis developed in section 6.3 in [16] (see also chapter 7 in [10]). We denote by Φ(m) , resp. Φ(m) , with p n, the semigroups associated with these ﬂows tp,tn tp,tn ≤ (m) (m) e (m) (m) (m) (m) Φtp,tn µtp = µtn and Φtp,tn µtp = µtn Using the interpolating sequencee ofe measurese

0 p n Φ(m) Φ(m) (µ(m)) =Φ(m) µ(m) ≤ ≤ 7→ tp,tn t0,tp t0 tp,tn tp from the distribution e e e (m) (m) (m) (m) (m) (m) (m) (m) (m) (m) Φt0,tn Φt0,t0 (µt0 )) =Φt0,tn µt0 ) = µtn to the measure Φtn,tn Φt0,tn (µt0 )) = µtn e (me) (m) e e e e Recalling that µt0 = µt0 , we ﬁnd that

n µ(m) µ(m) =e Φ(m) Φ(m) (µ(m))) Φ(m) Φ(m) (µ(m))) tn − tn tq,tn t0,tq t0 − tq−1,tn t0,tq−1 t0 q=1 X h i n e e e 1 e e = Φ(m) Φ(m) µ(m) + (m) µ(m) Φ(m) Φ(m) µ(m) tq,tn tq−1,tq tq−1 m2 Rtq−1,tq tq−1 − tq,tn tq−1,tq tq−1 q=1 X e e e By (22), we prove that

(m) (m) (m) Φ (µ) Φ (ν) (f) = Ψ (m) (µ) Ψ (m) (ν) P (f) tp,tn tp,tn G G tp,tn − tp,tn − tp,tn h i ν G(m) (m) tp,tn Gtp,tn (m) (m) = [µ ν] P (f) Ψ (m) (ν)P (f) tp,tn G tp,tn (m) −  (m) − tp,tn  µ Gtp,tn ν Gtp,tn   This implies that (m) (m) (m) (m) Φt ,t (µ) Φt ,t (ν) 2 gt ,t β Pt ,t µ ν tv p n − p n tv ≤ p n p n k − k with the Dobrushin contraction coeﬃcient β P (m) ( 1), and the parameters tp,tn ≤ (m) Gt ,t (x) g(m) := sup p n exp (2t sup ) . tp,tn (m) n t x,y ≤ t∈[0,tn] kU k Gtp,tn (y)

This yields the rather crude estimates

n (m) (m) −1 (m) (m) 2 m µt µt 2m gt ,t β Pt ,t tq−1 n − n tv ≤ p n p n kU k p=1 X 2 e 2tn exp (2tn sup t ) sup t ≤ t∈[0,tn] kU k t∈[0,tn] kU k

23 This ends the proof of the theorem.

Working a little harder, under some regularity conditions, the estimates developed in the proof of the theorem can be used to obtain uniform estimates w.r.t. the time parameter. For instance, in case (D), under the stability conditions (10), the constant ctn in theorem 5.4 can be chosen so that sup c < . We can extend these uniform results to continuous time models, using the stability n tn ∞ analysis of continuous Feynman-Kac semigroups developed in [23].

6 First order decompositions

The main objective of this section is to prove theorem 3.2. In the further development of this section we let c, cn, ctn , and ctn (f) be, respectively, some universal constant, and some ﬁnite constants that depend on the parameters n, tn, and the pair (tn,f), with values that may vary from line to line but do not depend on the parameters m and N. We also assume that m is chosen so that c m, kVtn k≤ tn for any n 0, and N m. ≥ ≥ 6.1 Continuous time models We start with the continuous time case (C) presented on page 4. By lemma 5.2 we ﬁnd that

2 N 2 N N (m) E (m) Wtn,tn+1 (f) ξtn = µtn Γ (f,f) µtn L N (f) . (32) L N tn,µt | " tn,µ # − n ! tn Using (26), for any function f D(L), such that L (f),L (f 2) 1([t ,t ], D(L)), we find that ∈ t t ∈C n n+1 1 1 E W N (f)2 ξ = µN Γ (f,f) + RN (f) tn,tn+1 tn tn Lt ,µN tn 2 | n tn m m N with some remainder term Rtn (f) such that N 2 sup Rtn (f) ctn f tn N≥1 ≤ k k with the norm f of a function f D (L), such that L (f) 1([t ,t ], D(L)) defined in (23) By k ktn ∈ t ∈C n n+1 (21), we have tn Q(m) (f)(x)= E f( ) exp ( )ds = x tp,tn tn ˆ τ(s) τ(s) tp X " tp V X # X ! with τ(s)= 1 (s) t . Thus, in case (C), for any function f D(L), the mappings n≥0 [tn,tn+1[ n ∈ P t [t ,t ] L Q(m) (f)2 and t [t ,t ] L2 Q(m) (f)2 ∈ p−1 p 7→ t tp,tn ∈ p−1 p 7→ t tp,tn are uniformly bounded w.r.t. the parameter m, and differentiable with uniformly bounded derivatives w.r.t. the parameter m. The first assertion of theorem 3.2 is based on the first order decompositions of the fluctuation of N (m) µtn around its limiting value µtn developed in [16]. Using theorem 6.2 in [16], we prove the following proposition.

24 Proposition 6.1 For any N 1 and any n N, we have ≥ ∈ n 1 √N µN µ(m) = W N D(m,N)(f) tn tn (m,N) tp−1,tp tp,tn − µN G h i Xp=0 tp tp,tn n N (m,N) 1 (m,N) = Wt ,t D (f) + (f) p−1 p tp,tn √N Rtn Xp=0 with the ﬁrst order integral operator

(m,N) (m,N) D(m,N)(f) := G P (m) (f) Φ (µN )(f) with G = G(m) /Φ(m)(µN ) G(m) tp,tn tp,tn tp,tn − tp,tn tp−1 tp,tn tp,tn tp tp−1 tp,tn and the remainder second order term n (m,N) 1 N (m,N) N (m,N) t (f)= Wt ,t Gt ,t Wt ,t Dt ,t (f) R n − N (m,N) p−1 p p n p−1 p p n p=0 µ G X tp tp,tn In the above, we have used the convention W N = √N[µN µ ], for p = 0. t−1,t0 t0 − t0 We are now in position to prove the bias and the variance estimates stated in theorem 3.2 for the continuous time models. Proof of theorem 3.2 - case (C): Firstly, the ﬁrst order decomposition stated above clearly implies that

N E µN µ(m) = E (m,N)(f) . tn − tn Rtn h i On the other hand, we have

n 1/2 1/2 (m,N) 2 2 E (m,N)(f) c1 E W N G E W N D(m,N)(f) . Rtn ≤ tn tp−1,tp tp,tn tp−1,tp tp,tn p=0 X To get one step further, we use the fact that

2 E N (m,N) Wtp−1,tp Dtp,tn (f) 1 1 E N (m,N) (m,N) E N (m,N) = µt − ΓL N (Dt ,t (f), Dt ,t (f)) + Rt − Dt ,t (f) . p 1 t − ,µ p n p n p 1 p n 2 p 1 tp−1 m m After some elementary manipulations we prove that

2 2 E N (m,N) E N (m,N) sup Wtp−1,tp Dtp,tn (f) ctn (f)/m and sup Wtp−1,tp Gtp,tn ctn /m. (33) 0≤p≤n ≤ 0≤p≤n ≤ This ends the proof of the bias estimate. The proof of the variance estimates is based on the following technical lemma.

25 Lemma 6.2 For any f with osc(f) 1, we have the fourth conditional moment estimate ≤ 1 2 E W N (f)4 ξ E W N (f)2 ξ + 6 E W N (f)2 ξ . tn,tn+1 | tn ≤ N tn,tn+1 | tn tn,tn+1 | tn Proof: With some elementary computation, we have that

E W N (f)4 ξ tn,tn+1 | tn 4 1 N N N = N µtn (du) tn,tn+1,µ (u, dv) f(v) tn,tn+1,µ (f)(u) K tn − K tn ´ 2 1 N N ′ +6 1 ′ µ (du)µ (du ) N (u, dv) f(v) N (f)(u) N u6=u tn tn tn,tn+1,µt tn,tn+1,µt − ´ K n − K n 2 ′ ′ ′ ′ t ,t ,µN (u , dv ) f(v ) t ,t ,µN (f)(u ) . K n n+1 tn − K n n+1 tn This implies that

4 N 4 1 N E Wt ,t (f) ξtn µt (du) t ,t ,µN (u, dv) f(v) t ,t ,µN (f)(u) n n+1 | ≤ N ˆ n K n n+1 tn − K n n+1 tn 2 2 N +6 µt (du) t ,t ,µN (u, dv) f(v) t ,t ,µN (f)(u) . ˆ n K n n+1 tn − K n n+1 tn The end of the proof is based on the fact that

4 2 f(v) t ,t ,µN (f)(u) f(v) t ,t ,µN (f)(u) − K n n+1 tn ≤ − K n n+1 tn as soon as osc(f) 1. This ends the proof of the lemma. ≤

Combining this lemma with (33), we prove that

4 1 1 1 E N (m,N) sup Wtp−1,tp Dtp,tn (f) ctn (f) + 0≤p≤n ≤ m N m and 4 1 1 1 E N (m,N) sup Wtp−1,tp Gtp,tn ctn (f) + . 0≤p≤n ≤ m N m This implies that

n 1/4 1/4 1/2 (m,N) 4 4 E (m,N)(f)2 c E W N G E W N D(m,N)(f) Rtn ≤ tn tp−1,tp tp,tn tp−1,tp tp,tn p=0 X and therefore E (m,N)(f)2 c (f) (1+ m/N). Rtn ≤ tn 26 Using the fact that

2 n 2 1 N E µN (f) µ(m)(f) 2 E W N D(m,N)(f) + E (m,N)(f)2 tn − tn ≤   tp−1,tp tp,tn   N Rtn  Xp=0      with

2 n n 2 E W N D(m,N)(f) = E W N D(m,N)(f) c (f)  tp−1,tp tp,tn   tp−1,tp tp,tn ≤ tn p=0 p=0 X X    we conclude that 2 1 m N E µN (f) µ(m)(f) c (f) 1+ + . tn − tn ≤ tn N N 2 This ends the proof of theorem 3.2.

6.2 Discrete time models The main objective of this section is to prove theorem 3.2 for the discrete time models related to case

(D). In this situation, we recall that µtnm = ηn, for any integer parameter n N. As usual, the N ∈ approximation measures µtn are defined as the occupation measures of the mean field particle model of the McKean distribution flow µ . By lemma 5.3, for any k N we have tn ∈ (m) L 1 := 1 Id = 1 Mk Id k− ,µ 1 k− m , k, µk− 1 k− m ,µk− 1 m k− m K m − S m − and for any (k 1)m

2 N 2 N N (m) E W (f) ξt = µ Γ (m) (f,f) µ L N (f) tp−1,tp n tp−1 L tp−1 t − ,µ  N  p 1 t − | tp−1,µ − p 1 ! tp−1   1 1 = µN Γ (f,f) + RN (f) tp−1 L N tp−1 2 t − ,µ m m " p 1 tp−1 # b

27 N with some remainder term Rtp−1 (f) such that

N 2 2 sup Rtp−1 (f) c tp−1 osc(f) . N≥1 ≤ U

This clearly implies that

N 2 2 2 1 N 2 2 E W (f) ξ c osc(f) and E W 1 (f) ξ 1 osc(f) . (34) tp−1,tp tn tp−1 k− ,k k− | ≤ U m m | m ≤ As in proposition 6.1, we prove the following decomposition. Proposition 6.3 For any N 1, f (E), and any n N we have the decomposition ≥ ∈Bb ∈ [µN η ](f)= AN + BN n − n n n with (k−1)m+(m−1) 1 AN := W N D(m,N)(f) n (m,N) tp−1,tp tp,n µN G 1≤Xk≤n p=(kX−1)m+1 tp tp,n and N 1 N (m,N) B = W 1 D (f) . n (m,N) k− ,k k,n µN G m 0≤Xk≤n k k,n (m,N) (m,N) In the above, the function Gtp,n and the integral operator Dtp,n (f) are defined in proposition 6.1, N N and for k = 0, we have used the convention W 1 = W0 . − m ,0 To analyze the bias and the variance, we also need to consider the first order decompositions presented in the following corollary. Corollary 6.4 For any N 1 and any n N we have the decomposition ≥ ∈ N (N,1) 1 (N,2) N (N,1) 1 (N,2) An = An + An and Bn = Bn + Bn √N √N with the first order terms

(k−1)m+(m−1) (N,1) N (m,N) An = Wtp−1,tp Dtp,n (f) 1≤Xk≤n p=(kX−1)m+1 (N,1) N (m,N) Bn = W 1 Dk,n (f) k− m ,k 0≤Xk≤n and the remainder second order terms (k−1)m+(m−1) 1 (m,N) A(N,2) = W N G W N D(m,N)(f) n (m,N) tp−1,tp tp,n tp−1,tp tp,n − µN G 1≤Xk≤n p=(kX−1)m+1 tp tp,n (N,2) 1 N (m,N) N (m,N) B = W 1 G W 1 D (f) n (m,N) k− ,k k,n k− ,k k,n − µN G m m 0≤Xk≤n k k,n 28 Now we come to the proof of the bias and the variance estimates presented in theorem 3.2. In the further development of this section f stands for some bounded function s.t. osc(f) 1. ≤ By construction, the random ﬁelds W N and W N are uncorrelated for any p = q. Combining tp−1,tp tq−1,tq 6 this property with the estimates (34) we prove that

(k−1)m+(m−1) 2 2 E (N,1) E N (m,N) An = Wtp−1,tp Dtp,n (f) . 1≤Xk≤n p=(kX−1)m+1 h i On the other hand, we have

2 c 2 E W N D(m,N)(f) 2 g(m) osc P (m)(f) tp−1,tp tp,n ≤ m Utp−1 tp,n tp,n h i with (m) (m) (m) gtp,n := sup Gtp,n(x)/Gtp,n(y) . x,y h i We prove the last assertion using the fact that

osc D(m,N)(f) /2 D(m,N)(f) g(m) osc P (m)(f) . tp,n ≤ tp,n ≤ tp,n tp,n By the semigroup formulae (9), for any p = (k 1)m + r, with r

2 2 E A(N,1) c log G 2 g g1/m osc P (f) . n ≤ k k−1k k−1,n × k−1,k (k−1),n 1≤k≤n X In the same way, we prove that

2 2 (N,1) N (m,N) E Bn = E W 1 Dk,n (f) k− m ,k 0≤Xk≤n h i 2 c g(m) osc P (m)(f) = c [g osc (P (f))]2 . ≤ k,n k,n k,n k,n 0≤Xk≤n h i 0≤Xk≤n

29 On the other hand, we have

(N,2) E An

(k−1)m+(m−1) g g1/m ≤ k−1,n k−1,k 1≤Xk≤n p=(kX−1)m+1 (35) 2 1/2 2 1/2 E N (m,N) E N (m,N) Wtp−1,tp Gtp,n Wtp−1,tp Dtp,n (f) log G 2 (g g )3 osc (P (f)). ≤ k kk k,n k,k+1 k,n 0≤Xk

1/2 (N,2) 2 E An (k−1)m+(m−1) 1/2 (m,N) 2 g g1/m E W N G W N D(m,N)(f) ≤ k−1,n k−1,k tp−1,tp tp,n tp−1,tp tp,n 1≤Xk≤n p=(kX−1)m+1

(k−1)m+(m−1) g g1/m ≤ k−1,n k−1,k 1≤Xk≤n p=(kX−1)m+1

4 1/4 4 1/4 E N (m,N) E N (m,N) Wtp−1,tp Gtp,n Wtp−1,tp Dtp,n (f) . On the other hand, using lemma 6.2 we have

4 E N (m,N) Wtp−1,tp Gtp,n h i 2 (m,N) 2 (m,N) 2 1 (2g(m) )2 E W N G + 6 E W N G ≤ N tp,n tp−1,tp tp,n tp−1,tp tp,n h i h i c (g(m) )4 1 2 1 + 4 1 ≤ tp,n N kUtp−1 k m kUtp−1 k m2 from which we ﬁnd the crude upper bound

(m,N) 4 E W N G c g4 g4 ( log G 1)4 /m2. tp−1,tp tp,n ≤ k−1,n k−1,k k k−1k∨ h i

30 for any N m, and for any p = (k 1)m + r, with r

2 1/2 4 1/4 4 1/4 E (N,2) E N (m,N) E N (m,N) Bn gk,n Wk− 1 ,k Gk,n Wk− 1 ,k Dk,n (f) . ≤ m m 0≤Xk≤n h i h i Using the fact that 2 E N (m,N) 2 Wk− 1 ,k Gk,n c gk,n m ≤ h i and 2 E N (m,N) 2 2 Wk− 1 ,k Dk,n (f) c gk,n osc (Pk,n(f)) m ≤ h i we check that

2 1/2 2 1/2 E (N,2) E N (m,N) E N (m,N) Bn gk,n Wk− 1 ,k Gk,n Wk− 1 ,k Dk,n (f) ≤ m m 0≤k≤n h i h i X c g3 osc (P (f)) . (36) ≤ k,n k,n 0≤Xk≤n

31 Combining (35) with (36), we obtain the bias estimate N E µN (f) η (f) n − n

c g3 osc (P (f)) + log G 2 (g g )3 osc (P (f)) ≤  0≤k≤n k,n k,n k kk k,n k,k+1 k,n  0≤k

4 N (m,N) E W 1 Gk,n k− m ,k h i g2 2 2 2 k,n E N (m,N) E N (m,N) 4 c Wk− 1 ,k Gk,n + Wk− 1 ,k Gk,n c gk,n ≤ " N m m # ≤ h i h i and 4 N (m,N) E W 1 Dk,n (f) k− m ,k h i g2 2 2 2 k,n 2 E N (m,N) E N (m,N) c osc (Pk,n(f)) Wk− 1 ,k Dk,n (f) + Wk− 1 ,k Dk,n (f) ≤ " N m m # h i h i c g4 osc (P (f))4 . ≤ k,n k,n This implies that 2 1/2 E B(N,2) c g3 osc (P (f)) . n ≤ k,n k,n 0≤Xk≤n We conclude that 2 1 1 E BN c b + b2 c b 1+ b n ≤ 1,n N 2,n ≤ 2,n N 2,n with

b := [g osc (P (f))]2 b a and b := g3 osc (P (f)) . 1,n k,n k,n ≤ 2,n ≤ 2,n 2,n k,n k,n 0≤Xk≤n 0≤Xk≤n This yields the variance estimate

2 1 N E µN (f) η (f) c a 1+ a . n − n ≤ 2,n N 2,n This ends the proof of the theorem.

32 7 Appendix

7.1 Proof of theorem 3.1

We let τ and τ be the mappings on R+ deﬁned by

τ(s)= 1[tn,tn+1[(s) tn and τ(s)= 1[tn,tn+1[(s) tn+1. Xn≥0 nX≥0 With this notation, we clearly have that

tn tp+1

τ(s)( τ (s))ds = τ(s)( τ (s))ds = tp ( tp )/m. ˆ0 V X ˆtp V X V X 0≤Xp

s( s)ds = tp ( tp )/m ˆ0 V X V X 0≤Xp

ν (f)= E f( ) eVtp (Xtp )/m ν = γ and µ = η . tn  Xtn  tnm n tnm n 0≤Yp

∂ d ( )= + L ( )( )+ dM ( ) Vt Xt ∂t t Vt Xt t V with a martingale term M ( ) with predictable angle bracket t V d M( ) =Γ ( , )( )dt h V it Lt Vt Vt Xt deﬁned in terms of the carré du champ ΓLt operator associated with the generator Lt and deﬁned for any f D(L) by the following formula ∈ Γ (f,f)(x)= L ((f f(x))2)(x)= L (f 2)(x) 2f(x)L (f)(x). Lt t − t − t

33 We recall that the predictable process M( ) is the unique right-continuous and increasing predictable h V it such that the random process M ( )2 M( ) is again a martingale. We also recall that M ( )2 t V − h V it t V − [M( )] is also a martingale, for the quadratic variation [M( )] of the process M( ) deﬁned as V t V V n 2 [M( )]t = lim Mtk ( ) Mtk−1 ( ) V kπk→0 V − V k=1 X where π ranges over partitions of the interval [0,t], and the norm π of the partition π is the size of k k the mesh. For continuous martingales M ( ), it is well known that [M( )] = M( ) . t V V h V i Using an elementary integration by part formula, for any p 0 we have ≥ tp+1 tp+1

s( s) tp ( tp ) ds = (tp+1 s) d s( s) ˆtp V X −V X ˆtp − V X from which we conclude that

tn tn tn s( s)ds = τ(s)( τ(s))ds + (τ(s) s) d s( s). ˆ0 V X ˆ0 V X ˆ0 − V X We set tn V (X )ds tn V (X )ds R (f)= E f( ) e 0 s s e 0 τ(s) τ (s) . tn X[0,tn] ´ − ´ Using the fact that ex ey x y ex + ehy , we prove that i | − | ≤ | − | | | tn tn V (X )ds tn V (X )ds R (f) E (τ(s) s) d ( ) e 0 s s + e 0 τ(s) τ (s) . tn s s ´ ´ | |≤ ˆ0 − V X h i Using Hölder’s inequality we have

p 1/p ′ tn p′ 1/p tn V (X )ds tn V (X )ds R (f) E (τ(s) s) d ( ) E e 0 s s + e 0 τ(s) τ(s) . tn s s ´ ´ | |≤ ˆ0 − V X h i To estimate the ﬁrst term in the r.h.s. of the above estimate, we use the inequality

p 1/p E tn (τ(s) s) d ( ) 0 − Vs Xs h´ i p 1/p p 1/p E tn (τ(s) s) ∂ + L ( )( )ds + E tn (τ(s) s) dM ( ) . ≤ 0 − ∂s s Vs Xs 0 − s V h´ i h´ i Using the generalized Minkowski inequality we prove that

p 1/p tn ∂ E 0 (τ(s) s) ∂s + Ls ( s)( s) ds ´ − V X h i 1/p tn ∂ p 0 (τ(s) s) E ∂s + Ls ( s)( s) ds ≤ ´ − V X 1/p 1 tn ∂ p m 0 E ∂s + Ls ( s)( s) ds. ≤ ´ V X 34 On the other hand, by the Burkholder-Davis-Gundy inequality, for any p> 0 we have that

1/p tn p 1/p tn p/2 E (τ(s) s) dM ( ) c E (τ(s) s)2 d [M( )] ˆ − s V ≤ p ˆ − V s 0 0 ! c 1/p p E [M( )]p/2 ≤ m V tn for some ﬁnite constant c < whose values only depends on the parameter p. The end of the proof p ∞ of the ﬁrst assertion. The second one is proved using the decomposition

(m) 1 (m) Qt Qtn (f)= Λt (fn) Λtn (fn) n − Λ(m)(1) n − h i tn h i with the centered function f = (f Λ (f)). n − tn 2 Q(m)(f) Q (f) sup Λ(m) (f) Λ (f) . tn tn (m) tn tn − ≤ Λ (1) Λ (1) f∈Bb(En): kfk≤1 − tn tn ∨ From these estimates, we ﬁnd that 2 rm,tn tv rm,tn tv k k ≤ Λ(m)(1) Λ (1) k k tn ∨ tn and (p′) (p) (p) sup rm,tn tv ap btn ctn + dtn m≥1 k k ≤ 1 1 with for any p + p′ = 1

1/p′ 1/p′ (p′) p′ t V (X )ds p′ t V (X )ds b = E e 0 s s E e 0 τ(s) τ (s) < t ´ ∨ ´ ∞ as well as

t p 1/p (p) ∂ (p) p/2 1/p ct := E + Ls ( s)( s) ds < and dt := E [M( )]t < . ˆ0 ∂s V X ∞ V ∞ Whenever 1([0, [, (L)), we have V∈C ∞ D ′ ∂ b(p ) exp t sup and c(p) t sup + L ( ) . t ≤ kVsk t ≤ ∂s s Vs 0≤s≤t 0≤s≤t

(p) Notice that for stochastic processes with continuous paths, the constant dt is given by

p/2 1/p 1/p t (p) p/2 √ 1/2 dt := E M( ) t = E ΓLs ( s, s)( s)ds t sup ΓLs ( s, s) . h V i ˆ0 V V X ! ≤ 0≤s≤t k V V k This ends the proof of the theorem.

35 7.2 Proof of proposition 5.3 In case (D), the mutation transition only occurs at integer times. More formally, we have = X , Xtn ⌊tn⌋ so that for any n = km + r with r + 1

n = km + m 1 t = (k + 1) 1/m with t = (k + 1) − ⇒ n − n+1 and (y, dz)= (y, dz)= M (y, dz). Mtn,tn+1 M(k+1)−1/m,(k+1) k+1 This ends the proof of the ﬁrst assertion. Now, we come to the proof of (24). Firstly, we recall that

tn+1 ∂ f(t , X )= f(t , X )+ + L (f)(s, X ) ds + M (f) M (f) n+1 tn+1 n tn ˆ ∂s s s tn+1 − tn tn for any f 1([t ,t ], D(L)), with some martingale M (f). This implies that ∈C n n+1 t tn+1 ∂ E f(t , X ) = f(t ,x)+ E + L (f)(s, X ) ds tn,x n+1 tn+1 n ˆ tn,x ∂s s s tn where Etn,x(.) stands for the conditional expectation operator given that Xtn = x. Iterating this formula, we ﬁnd that

∂ ∂ s ∂ 2 E + L (f)(s, X ) = + L (f)(t ,x)+ E + L (f)(r, X ) dr tn,x ∂s s s ∂t tn n ˆ tn,x ∂r r r n tn " # as soon as ∂ + L (f) 1([t ,t ], D(L)). Under this condition, we ﬁnd the ﬁrst order decompo- ∂t t ∈C n n+1 sition ∂ 1 1 E f(t , X ) = f(t ,x)+ + L (f)(t ,x) + R (f) tn,x n+1 tn+1 n ∂t tn n m tn m2 n with some remainder operator

tn+1 s ∂ 2 R (f)= m2 E + L (f)(r, X ) dr tn ˆ ˆ tn,x ∂r r r tn tn " # such that ∂ 2 Rtn (f) sup sup Etn,x + Lt (f)(t, Xt) . k k≤ x∈E tn≤t

The end of the proof of (24) is now clear. Using lemma 5.2, and the ﬁrst order decompositions (37) and (24) we have 1 1 L(m) = L + R tn,µ tn,µ m tn,µ m2

36 with 1 1 1 1 R = R + R + m2 L + R L + R . tn,µ tn tn,µ tn,µ m tn,µ m2 tn m tn m2 On the other hand, we have theb estimates b b 1 1 1 1 1 m2 L + R L + R = L L + R + R L(m) tn,µ m tn,µ m2 tn m tn m2 tn,µ tn tn m tn,µ tn with b b b b R L(m)(f) c f tn,µ tn ≤ tn k k and b 1 1 L L + R (f) c L (f)+ R (f) tn,µ tn tn m ≤ tn tn tn m 2 b ctn sup ∂Lt(f)/∂t + Lt(f) + Lt (f) ≤ tn≤t≤tn+1 k k k k for some ﬁnite constant c < . This ends the proof of (25). tn ∞ The proof of the last assertion is based on the decomposition

2 (m) 2 f (f)(x) (x)=Γ (m) (f,f)(x) L (f)(x) tn,tn+1,µ tn,tn+1,µ L tn,µ K − K tn,µ − and the fact that

1 2 1 Γ (m) (f,f)(x)=Γ (f,f)(x) + R (f f(x)) (x) . L Ltn,µ tn,µ 2 tn,µ m − m This ends the proof of the proposition.

7.3 Proof of the ﬁrst order decompositions (6) Case 1 : We assume that = , for some non negative and bounded function . In this • Vt −Ut Ut situation, (5) is satisﬁed by the Markov transition

−Utn (x)/m −Utn (x)/m (m) (m) (x,dy) := e δ (dy)+ 1 e Ψ −U /m (µ )(dy). t ,µ x e tn tn S n tn − In this situation, we notice that

−Utn (x)/m (m) m (m) (f)(x) f(x) = m 1 e Ψ −U /m (µ )(f) f(x) t ,µ e tn tn S n tn − − − h i (m) = (x) (f(y) f(x)) µ (dy)+ O(1) = L (m) (f)+ O(1). tn tn t ,µ U ˆ − n tn b

37 Much more is true; if we set

R (m) (f) := m m (m) (f) f L (m) (f) t ,µ t ,µ t ,µ n tn S n tn − − n tn −Utn /m (m) (m) b = m m 1 e Ψ −Ub /m (µ )(f) f µ (f) f − e tn tn − −Utn tn − h h i h ii then we ﬁnd that

R (m) (f) tn,µ tn

b −Ut /m (m) (m) (m) m m 1 e n Ψ −U /m (µ )(f) f + m [Ψ −U /m (µ )(f) µ (f) ≤ − −Utn × e tn tn − Utn e tn tn − tn

Using the fact that 1 (m) (m) (m) −Utn /m (m) m Ψ −U /m (µ ) µ (f)= µ m e 1 f µ (f) e tn tn tn (m) tn tn − µ e−Utn /m − − h i tn h ih i we prove the following ﬁrst order expansion 1 1 (m) (f) f = L (m) (f) + R (m) (f) (37) t ,µ t ,µ t ,µ 2 S n tn − n tn m n tn m b b with some integral operator R (m) such that tn,µtn

b 2 sup R (m) (f) c osc(f). t ,µ tn n tn ≤ kU k m≥1

b Case 2 : We assume that is a positive and bounded function. In this situation, (5) is satisﬁed • Vt by the Markov transition

1 1 (m) (m) (x,dy) := δx(dy)+ 1 Ψ Vt /m (µ )(dy). tn,µ (m) (m) (e n −1) tn S tn Vtn /m − Vtn /m µtn e µtn e ! In this situation, we notice that

1 (m) m (m) (f)(x) f(x) = m 1 Ψ Vt /m (µ )(f) f(x) tn,µ (m) (e n −1) tn S tn − − µ eVtn /m ! − tn h i = (f(y) f(x)) (y) µ(m)(dy)+ O(1). ˆ − Vtn tn Using some elementary calculations, we also prove a ﬁrst order expansion of the same form as in (37).

38 Case 3: The Markov transport equation (5) is satisﬁed by the transitions • (m) µ eVtn /m−eVtn (x)/m tn ( )+ (m) (x,dy) := 1 (m) δx(dy) tn,µ µ eVtn /m S tn − tn ( ) !

(m) µ eVtn /m eVtn (x)/m tn − + (m) + Ψ V /m V (x)/m (µ )(dy). (m) (e tn −e tn ) tn Vtn /m + µtn e

The above Markov transition is well deﬁned since we have

(m) Vt /m Vt (x)/m µ e n e n 1V >V (x) tn − tn tn 1 (m) (m) Vtn /m Vtn /m ≤ µtn e 1Vtn >Vtn (x) + µtn e 1Vtn ≤V tn (x) as soon as

(m) Vtn /m (m) Vtn (x)/m (m) [Vtn ∧Vtn (x)]/m µtn e 1Vtn ≤Vtn (x) + µtn e 1Vtn >Vtn (x) = µtn e > 0. Also notice that

(m) Vtn /m Vtn (x)/m (m) (m) Vtn /m Vtn (x)/m µt e e = µt ( tn > tn (x)) µt e tn > tn (x) e n − + n V V × n | V V − h i so that a particle in some state x is more likely to be recycled when the potential values ( ) Vtn X tn of random states with law µ(m) are more likely to be larger that (x). X tn tn Vtn Finally, for bounded potential functions we observe that

m (m) (f)(x) f(x) t ,µ S n tn − (m) µ eVtn /m eVtn (x)/m tn − + (m) = m Ψ V /m V (x)/m (µ )(f) f(x) (m) (e tn −e tn ) tn µ eVtn /m + − tn = [f(y) f(x)] ( (y) (x)) µ(m)(dy)+ O(1). ˆ − Vtn −Vtn + tn Using some elementary calculations, we also prove a ﬁrst order expansion of the same form as in (37).

References

[1] R. Assaraf and M. Caﬀarel. Computing forces with quantum monte carlo. The Journal of Chemical Physics, 113:4028, 2000.

[2] R. Assaraf, M. Caffarel, and A. Khelif. Diffusion monte carlo methods with a fixed number of walkers. Physical Review E, 61(4):4566, 2000.

39 [3] O. Cappé, E. Moulines, and T. Ryden. Inference in Hidden Markov Models. Statistics. Springer, 2005.

[4] R. Carmona, P. Del Moral, P. Hu, and N. Oudjane, editors. Numerical Methods in Finance, volume 12 of Proceedings in Mathematics. Springer, 2012.

[5] R. Carmona, J.P. Fouque, and D. Vestal. Interacting particle systems for the computation of rare credit portfolio losses. Finance and Stochastics, 13(4):613–633, 2009.

[6] F. Cérou, P. Del Moral, F. Le Gland, and P. Lezaud. Genetic genealogical models in rare event analysis. Latin American Journal of Probability and Mathematical Statistics, 1:181–203, 2006.

[7] M.H. Chung and D.P. Landau. Diﬀusion monte carlo for fermions with replica reduction. Phys. Rev. B, 85:115115, Mar 2012.

[8] D. Crisan, P. Del Moral, and T. Lyons. Interacting particle systems approximations of the kushner stratonovitch equation. Advances in Applied Probability, 31(3):819–838, 1999.

[9] P. Del Moral. Non linear ﬁltering: Interacting particle solution. Markov Processes and Related Fields, 4(2):555–580, 1996.

[10] P. Del Moral. Feynman-Kac formulae. Genealogical and interacting particle approximations. Prob- ability and Applications. Springer, New York, 2004.

[11] P. Del Moral and A. Doucet. Particle motions in absorbing medium with hard and soft obstacles. Stochastic Analysis and Applications, 22(5), 2004.

[12] P. Del Moral, A. Doucet, and A. Jasra. Sequential Monte Carlo samplers. Journal of the Royal Statistical Society, Series B, 68(3):411–436, 2006.

[13] P. Del Moral, A. Doucet, and A. Jasra. On adaptive resampling procedures for sequential Monte Carlo methods. Bernoulli, 18(1):252–278, 2012.

[14] P. Del Moral and A. Guionnet. On the stability of interacting processes with applications to ﬁltering and genetic algorithms. Annales de l’Institut Henri Poincaré, 37(2):155–194, 2001.

[15] P. Del Moral and N. Hadjiconstantinou. An introduction to probabilistic methods, with applications. M2AN, 44(5):805–830, 2010.

[16] P. Del Moral, P. Hu, and L. Wu. On the concentration properties of interacting particle processes. Foundations and Trends in Machine Learning, 3(3-4):225–389, 2012.

[17] P. Del Moral and J. Jacod. The monte-carlo method for ﬁltering with discrete time observations: Central limit theorems. In T. J. Lyons and T. S. Salisbury, editors, Numerical Methods and Stochastics, Fields Institute Communications. American Mathematical Society, 2002.

[18] P. Del Moral, J. Jacod, and Ph. Protter. The Monte-Carlo method for ﬁltering with discrete-time observations. Probability Theory and Related Fields, 120:346–368, 2001.

40 [19] P. Del Moral, M. A. Kouritzin, and L. Miclo. On a class of discrete generation interacting particle systems. Electronic Journal of Probability, 6(16):1–26, 2001.

[20] P. Del Moral and P. Lezaud. Branching and interacting particle interpretation of rare event probabilities. In H. Blom and J. Lygeros, editors, Stochastic Hybrid Systems: Theory and Safety Critical Applications. Springer-Verlag, Heidelberg, 2006.

[21] P. Del Moral and L. Miclo. Branching and interacting particle systems approximations of Feynman-Kac formulae with applications to non-linear ﬁltering. In J. Azéma, M. Emery, M. Ledoux, and M. Yor, editors, Séminaire de Probabilités XXXIV, volume 1729 of Lecture Notes in Mathematics, pages 1–145. Springer-Verlag Berlin, 2000.

[22] P. Del Moral and L. Miclo. A Moran particle system approximation of Feynman-Kac formulae. Stochastic Processes and their Applications, 86:193–216, 2000.

[23] P. Del Moral and L. Miclo. On the stability of non linear semigroup of Feynman-Kac type. Annales de la Faculté des Sciences de Toulouse, 11(2), 2002.

[24] P. Del Moral and L. Miclo.. Particle approximations of Lyapunov exponents connected to Schrödinger operators and Feynman-Kac semigroups. ESAIM: Probabilités et Statistiques, 7:171– 208, 2003.

[25] P. Del Moral and L. Miclo. Strong propagations of chaos in Moran’s type particle interpretations of Feynman-Kac measures. Stochastic Analysis and Applications, 25(3):519–575, 2007.

[26] P. Del Moral, GW Peters, and C. Vergé. An introduction to particle integration methods: with applications to risk and insurance. arXiv preprint arXiv:1210.3851, 2012.

[27] A. Doucet, N. de Freitas, and N. Gordon., editors. Sequential Monte Carlo Methods in Practice. Statistics for Engineering and Information Science. Springer, 2001.

[28] A. Doucet and A. M. Johansen. A tutorial on particle ﬁltering and smoothing: 15 years later. In Oxford Handbook of Nonlinear Filtering. Oxford University Press, 2011.

[29] A. Eberle and C. Marinelli. Lp estimates for feynman-kac propagators with time-dependent reference measures. Journal of mathematical analysis and applications, 365(1):120–134, 2010.

[30] A. Eberle and C. Marinelli. Quantitative approximations of evolving probability measures and sequential MCMC methods. IAM Bonn University, 2011.

[31] A. Eberle and C. Marinelli. Stability of nonlinear ﬂows of probability measures related to sequential MCMC methods. IAM Bonn University, 2011.

[32] N. J. Gordon, D. Salmond, and A. F. M. Smith. Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEE Proc. F, 1993.

[33] J. N. Hugonnier. The Feynman-Kac formula and pricing occupation time derivatives. Int. J. Theor. Appl. Finan., 2(2):153–178, 1999.

41 [34] B. Jeﬀeries. Evolution processes and the Feynman-Kac formula. Mathematics and its Applications. Kluwer Academic Publishers, 1996.

[35] N. Kantas, A. Doucet, S. S. Singh, and J. M. Maciejowski. An overview of sequential Monte Carlo methods for parameter estimation in general state-space models. In Proceedings IFAC System Identiﬁcation (SySid) Meeting, 2009.

[36] I. Karatzas and S. E. Shreve. Brownian motion and stochastic calculus, second ed. Graduate Texts in Mathematics. Springer-Verlag, New York, 1991.

[37] G. Kitagawa. Monte Carlo ﬁlter and smoother for non-Gaussian nonlinear state-space models. JCGS, 5(1):1–25, 1996. S.-Japan Joint Seminar on Statistical Time Series Analysis, pp.(1993).

[38] G. Kitagawa and W. Gersch. Smoothness Priors Analysis of Time Series. Lecture Notes in Statistics. Springer, 1996.

[39] I. Kosztin, B. Faber, and K. Schulten. Introduction to the Diﬀusion Monte Carlo method. Amer- ican Journal of Physics, 64:633–644, May 1996.

[40] T. Lelièvre, M. Rousset, and G. Stoltz. Computation of free energy diﬀerences through nonequi- librium stochastic dynamics: the reaction coordinate case. J. Comp. Phys., 222(2):624–643, 2007.

[41] T. Lelièvre, M. Rousset, and G. Stoltz. Free energy computation: a mathematical perspective. Imperial College Press, 2010.

[42] G.W. Peters, M. Briers, P. Shevchenko, and A. Doucet. Calibration and ﬁltering for multi factor commodity models with seasonality: incorporating panel data from futures contracts. Methodology and Computing in Applied Probability, pages 1–34, 2011.

[43] GW Peters, Y. Fan, and SA Sisson. On sequential monte carlo, partial rejection control and approximate bayesian computation. Statistics and Computing, pages 1–14, 2009.

[44] G.W. Peters, A.M. Johansen, and A. Doucet. Simulation of the annual loss distribution in operational risk via panjer recursions and volterra integral equations for value at risk and expected shortfall estimation. The Journal of Operational Risk, 2(3):29–58, 2007.

[45] Ph. Protter. Stochastic Integration and Diﬀerential Equations, 2nd ed. 2004.

[46] M. Rousset. On the control of an interacting particle approximation of Schrödinger ground states. SIAM J. Math. Anal., 38(3):824–844, 2006.