Non-stationary phase of the MALA algorithm

Transcription

1 Stoch PDE: Anal Comp 018) 6: on-tationary phae of the MALA algorithm Juan Kuntz 1 Michela Ottobre Andrew M. Stuart 3 Received: 3 Augut 017 / Publihed online: 17 April 018 The Author) 018 Abtract The Metropoli-Aduted Langevin Algorithm MALA) i a Markov Chain Monte Carlo method which create a Markov chain reverible with repect to a given target ditribution, π, with Lebegue denity on R ; it can hence be ued to approximately ample the target ditribution. When the dimenion i large a key quetion i to determine the computational cot of the algorithm a a function of. The meaure of efficiency that we conider in thi paper i the expected quared umping ditance ESJD), introduced in Robert et al. Ann Appl Probab 71):110 10, 1997). To determine how the cot of the algorithm in term of ESJD) increae with dimenion, we adopt the widely ued approach of deriving a diffuion limit for the Markov chain produced by the MALA algorithm. We tudy thi problem for a cla of target meaure which i not in product form and we addre the ituation of practical relevance in which the algorithm i tarted out of tationarity. We thereby ignificantly extend previou work which conider either meaure of product form, when the Markov chain i tarted out of tationarity, or non-product meaure defined via a denity with repect to a Gauian), when the Markov chain i tarted in tationarity. In order to work in thi non-tationary and non-product etting, ignificant new analyi i B Michela Ottobre michelaottobre@gmail.com Juan Kuntz uankuntz@protonmail.com Andrew M. Stuart atuart@caltech.edu 1 Imperial College London, London SW7 AZ, UK Mathematic Department, Heriot Watt Univerity, Edinburgh EH14 4AS, UK 3 Department of Computing and Mathematical Science, California Intitute of Technology, Paadena, CA 9115, USA

2 Stoch PDE: Anal Comp 018) 6: required. In particular, our diffuion limit comprie a tochatic PDE coupled to a calar ordinary differential equation which give a meaure of how far from tationarity the proce i. The family of non-product target meaure that we conider in thi paper are found from dicretization of a meaure on an infinite dimenional Hilbert pace; the dicretied meaure i defined by it denity with repect to a Gauian random field. The reult of thi paper demontrate that, in the non-tationary regime, the cot of the algorithm i of O 1/ ) in contrat to the tationary regime, where it i of O 1/3 ). Keyword Markov Chain Monte Carlo Metropoli-Aduted Langevin Algorithm Diffuion limit Optimal caling Mathematic Subect Claification Primary 60J; Secondary 60J0 60H10 1 Introduction 1.1 Context Metropoli Hating algorithm are Markov Chain Monte Carlo MCMC) method ued to ample from a given probability meaure, referred to a the target meaure. The baic mechanim conit of employing a propoal tranition denity qx, y) in order to produce a reverible Markov chain {x k } k=0 for which the target meaure π i invariant [11]. At tep k of the chain, a propoal move y k i generated by uing qx, y), i.e. y k qx k, ). Then uch a move i accepted with probability αx k, y k ): α x k, y k) = min {1, π y k) q y k, x k) } π x k) q x k, y k). 1.1) The computational cot of thi algorithm when the tate pace ha high dimenion i of practical interet in many application. The meaure of computational cot conidered in thi paper i the expected quared umping ditance, introduced in [19] and related work. Roughly peaking [we will be more precie about thi in the next Sect. 1., ee comment before 1.8)], if the ize of the propoal move i too large, i.e. if we propoe move which are too far away from the current poition, then uch move tend to be frequently reected; on the other hand, if the algorithm propoe move which are too cloe to the current poition, then uch move will be mot likely accepted, however the chain will have not moved very far away. In either extreme cae, the chain tend to get tuck and will exhibit low mixing, and thi i more and more true a the dimenion of the tate pace increae. It i therefore clear that one need to trike a balance between thee two oppoite cenario; in particular, the optimal ize of the propoed move i.e., the propoal variance) will depend on. If the propoal variance cale with like ζ,foromeζ>0, then we will ay that the cot of the algorithm, in term of ESJD, i of the order ζ. A widely ued approach to tackle thi problem i to tudy diffuion limit for the algorithm. Indeed the caling ued to obtain a well defined diffuion limit correpond

3 448 Stoch PDE: Anal Comp 018) 6: to the optimal caling of the propoal variance ee Remark 1.1). Thi problem wa firt tudied in [19], for the Random Walk Metropoli algorithm RWM); in thi work it i aumed that the algorithm i tarted in tationarity and that the target meaure i in product form. In the cae of the MALA algorithm, the ame problem wa conidered in [0,1], again in the tationary regime and for product meaure. In thi etting, the cot of RWM ha been hown to be O), while the cot of MALA i O 1 3 ). The ame O 1 3 ) caling for MALA, in the tationary regime, wa later obtained in the etting of non-product meaure defined via denity with repect to a Gauian random field [17]. In the paper [6] extenion of thee reult to non-tationary initialization were conidered, however only for the Gauian target. For Gauian target, RWM wa hown to cale the ame in and out of tationarity, whilt MALA cale like O 1 ) out of tationarity. In [1,13] the RWM and MALA algorithm were tudied out of tationarity for quite general product meaure and the RWM method hown again to cale the ame in and out of tationarity. For MALA the appropriate caling wa hown to differ in and out of tationarity and, crucially, the caling out of tationarity wa hown to depend on a certain moment of the potential defining the product meaure. In thi paper we contribute further undertanding of the MALA algorithm when initialized out of tationarity by conidering non-product meaure defined via denity with repect to a Gauian random field. Conidering uch a cla of meaure ha proved fruitful, ee e.g. [15,17]. Relevant to thi trand of literature, i alo the work [5]. In thi paper our primary contribution i the tudy of diffuion limit for the the MALA algorithm, out of tationarity, in the etting of general non-product meaure, defined via denity with repect to a Gauian random field. Significant new analyi i needed for thi problem becaue the work of [17] relie heavily on tationarity in analyzing the acceptance probability, whilt the work of [13] ue propagation of chao technique, unuitable for non-product etting. The challenging diffuion limit obtained in thi paper i relevant both to the picture ut decribed and, in general, due to the widepread practical ue of the MALA algorithm. The undertanding we obtain about the MALA algorithm when applied to realitic non-product target i one of the main motivation for the analyi that we undertake in thi paper. The diffuion limit we find i given by an SPDE coupled to a one-dimenional ODE. The evolution of uch an ODE can be taken a an indicator of how cloe the chain i to tationarity ee Remark 1.1 for more detail on thi). The caling adopted to obtain uch a diffuion limit how that the cot of the algorithm i of order 1/ in the non-tationary regime, a oppoed to what happen in the tationary phae, where the cot i of order 1/3. It i important to recognize that, for meaure abolutely continuou with repect to a Gauian random field, algorithm exit which require O1) tep in and out of tationarity; ee [7] for a review. Such method were uggeted by Radford eal in [16], and developed by Alex Beko for conditioned tochatic differential equation in [4], building on the general formulation of Metropoli Hating method in [3]; thee method are analyzed from the point of view of diffuion limit in [18]. It thu remain open and intereting to tudy the MALA algorithm out of tationarity for non-product meaure which are not defined via denity with repect to a Gauian random field; however the reult in

4 Stoch PDE: Anal Comp 018) 6: [1] demontrate the ubtantial technical barrier that will exit in trying to do o. An intereting tarting point of uch work might be the tudy of non i.i.d. product meaure a pioneered by Bédard [,3]. 1. Setting and the main reult Let H,,, ) be an infinite dimenional eparable Hilbert pace and conider the meaure π on H, defined a follow: dπ dπ 0 exp Ψ), π 0 := 0, C). 1.) That i, π i abolutely continuou with repect to a Gauian meaure π 0 with mean zero and covariance operator C. Ψ i ome real valued functional with domain H H, Ψ : H R. Meaure of the form 1.) naturally arie in Bayeian nonparametric tatitic and in the tudy of conditioned diffuion [10,]. In Sect. we will give the precie definition of the pace H and identify it with an appropriate Sobolev-like ubpace of H denoted by H in Sect. ).The covariance operator C i a poitive, elf-adoint, trace cla operator on H, with eigenbai {λ,φ }: Cφ = λ φ,, 1.3) and we aume that the et {φ } i an orthonormal bai for H. We will analye the MALA algorithm deigned to ample from the finite dimenional proection π of the meaure 1.) on the pace X := pan{φ } H 1.4) panned by the firt eigenvector of the covariance operator. otice that the pace X i iomorphic to R. To clarify thi further, we need to introduce ome notation. Given a point x H, P x) := n φ, x φ i the proection of x onto the pace X and we define the approximation of functional Ψ and covariance operator C: Ψ := Ψ P and C := P C P. 1.5) With thi notation in place, our target meaure i the meaure π on X = R ) defined a dπ x) = M Ψ e Ψ x), π0 := 0, C ), 1.6) dπ 0 where M Ψ i a normalization contant. otice that the equence of meaure {π } approximate the meaure π in particular, the equence {π } converge to π in the Hellinger metric, ee [, Section 4] and reference therein). In

5 450 Stoch PDE: Anal Comp 018) 6: order to ample from the meaure π in 1.6), we will conider the MALA algorithm with propoal where y k, = x k, + δc log π x k, ) + δ C 1/ ξ k,, 1.7) ξ k, = ξ i φ i, i=1 ξ i D 0, 1) i.i.d., and δ>0 i a poitive parameter. We rewrite y k, a y k, = x k, δ x k, + C Ψ x k, )) + δ C 1/ ξ k,. The propoal define the kernel q and enter in the accept-reect criterion α, which i added to preerve detailed balance with repect to π more detail on the algorithm will be given in Sect..). The propoal i a dicretization of a π -invariant diffuion proce with time tep δ; in the MCMC literature δ i often referred to a the propoal variance. The accept-reect criterion compenate for the dicretization, which detroy the π -reveribility. A crucial parameter to be appropriately choen in order to optimize the performance of the algorithm i δ; uch a choice will depend on the dimenion of the tate pace. To be more precie, et δ = l ζ, where l, ζ are two poitive parameter, the latter being, for the time, the mot relevant to thi dicuion. A explained when outlining the context of thi paper, if ζ i too large o that δ i too mall) then the algorithm will tend to move very lowly; if ζ i too big, then the propoed move will be very large and the algorithm will tend to reect them very often. In thi paper we how that, if the algorithm i tarted our of tationarity then, in the non-tationary regime, the optimal choice of ζ i ζ = 1/. In particular, if δ = l/ 1.8) then the acceptance probability i O1). Furthermore, tarting from the Metropoli Hating chain {x k, } k, we define the continuou interpolant x ) t) = 1/ t k)x k+1, + k + 1 1/ t)x k,, t k t < t k+1, where t k = k. 1.9) 1/ Thi proce converge weakly to a diffuion proce. The precie tatement of uch a reult i given in Theorem 4. and Sect. 4 contain heuritic argument which explain how uch a reult i obtained). In proving the reult we will ue the fact that W t) i a H -valued Brownian motion with covariance C, with H a Hilbert) ubpace of H and C the covariance in thi pace. Detail of thee pace are given in Sect., ee in particular.4) and.5). Below C[0, T ]; H ) denote the pace of H -valued continuou function on [0, T ], endowed with the uniform topology; α l, h l and b l

6 Stoch PDE: Anal Comp 018) 6: are real valued function, which we will define immediately after the tatement, and x k, denote the th component of the vector x k, X with repect to the bai {φ 1,...,φ } more detail on thi notation are given in Sect..1.) Main Reult Let {x k, } k be the Metropoli Hating Markov chain to ample from π and contructed uing the MALA propoal 1.7) i.e. the chain.14)) with δ choen to atify 1.8). Then, for any determinitic initial datum x 0, = P x 0 ), where x 0 i any point in H, the continuou interpolant x ) defined in 1.9) converge weakly in C[0, T ]; H ) to the olution of the SDE dxt) = h l St)) xt) + C Ψxt)) ) dt + h l St)) dwt), x0) = x 0, 1.10) where St) R + := { R : 0} olve the ODE dst) = b l St)) dt, 1 S0) := lim x 0, λ. 1.11) In the above the initial datum S0) i aumed to be finite and W t) i a H -valued Brownian motion with covariance C. The function α l, h l, b l : R R in the previou tatement are defined a follow: α l ) = 1 e l 1)/ 1.1) h l ) = lα l ) 1.13) ) b l ) = l1 ) 1 e l 1)/ = 1 )h l ). 1.14) Remark 1.1 We make everal remark concerning the main reult. Since the effective time-tep implied by the interpolation 1.9)i 1/,themain reult implie that the number of tep required by the Markov chain in it nontationary regime i O 1/ ). A more detailed dicuion on thi fact can be found in Sect. 4. otice that Eq. 1.11) evolve independently of Eq. 1.10). Once the MALA algorithm.14) i introduced and an initial tate x 0 H i given uch that S0) i finite, the real valued double) equence S k,, i=1 x 0, i S k, := 1 i=1 x k, i λ i 1.15) tarted at S0 := 1 i well defined. For fixed, {S k, } λi k i not, in general, a Markov proce however it i Markov if e.g. Ψ = 0). Conider the continuou interpolant S ) t) of the equence S k,, namely

7 45 Stoch PDE: Anal Comp 018) 6: S ) t)= 1/ t k)s k+1, + k + 1 1/ t)s k,, t k t <t k+1, t k = k ) In Theorem 4.1 we prove that S ) t) converge in probability in C[0, T ]; R) to the olution of the ODE 1.11) with initial condition S 0 := lim S0. Once uch a reult i obtained, we can prove that x ) t) converge to xt). We want to tre that the convergence of S ) t) to St) can be obtained independently of the convergence of x ) t) to xt). LetSt): R R be the olution of the ODE 1.11). We will prove ee Theorem 3.1) that St) 1at ; thi i alo conitent with the fact that, in tationarity, S k, converge to 1 a for every k > 0), ee Remark 4.1.In view of thi and the above comment, St) or S k, ) can be taken a an indication of how cloe the chain i to tationarity. Moreover, notice that h l 1) = l; heuritically one can then argue that the aymptotic behaviour of the law of xt), the olution of 1.10), i decribed by the law of the following infinite dimenional SDE: dzt) = lzt) + C Ψzt)))dt + ldwt). 1.17) It wa proved in [9,10] that 1.17) i ergodic with unique invariant meaure given by 1.). Our deduction concerning computational cot i made on the aumption that the law of 1.10) doe indeed tend to the law of 1.17), although we will not prove thi here a it would take u away from the main goal of the paper which i to etablih the diffuion limit of the MALA algorithm. In[1,13] the diffuion limit for the MALA algorithm tarted out of tationarity and applied to i.i.d. target product meaure i given by a non-linear equation of McKean-Vlaov type. Thi i in contrat with our diffuion limit, which i an infinite-dimenional SDE. The reaon why thi i the cae i dicued in detail in [14, Section 1.]. The dicuion in the latter paper i in the context of the Random Walk Metropoli algorithm, but it i conceptually analogou to what hold for the MALA algorithm and for thi reaon we do not pell it out here. In thi paper we make tronger aumption on Ψ than are required to prove a diffuion limit in the tationary regime [17]. In particular we aume that the firt derivative of Ψ i bounded, wherea [17] require only boundedne of the econd derivative. Removing thi aumption on the firt derivative, or howing that it i neceary, would be of interet but would require different technique to thoe employed in thi paper and we do not addre the iue here. Remark 1. The propoal we employ in thi paper i the tandard MALA propoal. It can be een a a particular cae of the more general propoal introduced in [4, equation 4.)] ee alo [1]; in our notation thi propoal can be written a y k+1, = x k, + δ { 1 θ)x k, θy k+1, C Ψ x k, )} + δξ k,. 1.18) In the above, θ [0, 1] i a parameter. The choice θ = 0 correpond to our propoal. When θ = 1/, the reulting algorithm i well poed in infinite dimenion; a a

8 Stoch PDE: Anal Comp 018) 6: conequence a diffuion limit i obtained, in and out of tationarity, without caling δ with repect to ; eeremark4.3. When θ = 1/ the algorithm all uffer from the cure of dimenionality: it i neceary to cale δ inverely with a power of to obtain an acceptable acceptance probability. In thi paper we tudy how the efficiency decreae with when θ = 0; reult analogou to the one we prove here will hold for any θ = 1/, but proving them at thi level of generality would lengthen the article without adding inight. Furthermore, for non-gauian prior practitioner might ue the algorithm with θ = 0 and o our reult hed light on that cae; if the prior i actually Gauian practitioner hould ue the algorith with θ = 1. There i no reaon to ue any other value of θ in practice, a far a we are aware. 1.3 Structure of the paper The paper i organized a follow. In Sect. we introduce the notation and the aumption that we ue throughout thi paper. In particular, Sect..1 introduce the infinite dimenional etting in which we work, Sect.. dicue the MALA algorithm and the aumption we make on the functional Ψ and on the covariance operator C. Section 3 contain the proof of exitence and uniquene of olution for the limiting Eq. 1.10) and 1.11). With thee preliminarie in place, we give, in Sect. 4, the formal tatement of the main reult of thi paper, Theorem 4.1 and 4.. In thi ection we alo provide heuritic argument outlining how the main reult are obtained. The complete proof of thee reult build on a continuou mapping argument preented in Sect. 5. The heuritic of Sect. 4 are made rigorou in Sect In particular, Sect. 6 contain ome etimate of the ize of the chain ump and the growth of it moment, a well a the tudy of the acceptance probability. In Sect. 7 and 8 we ue thee etimate and approximation to prove Theorem 4.1 and 4., repectively. Reader intereted in the tructure of the proof of Theorem 4.1 and 4. but not in the technical detail may wih to kip the enuing two ection Sect. and 3) and proceed directly to the tatement of thee reult and the relevant heuritic dicued in Sect. 4. otation, algorithm, and aumption In thi ection we detail the notation and the aumption Sect..1 and.3, repectively) that we will ue in the ret of the paper..1 otation Let H,,, ) denote a real eparable infinite dimenional Hilbert pace, with the canonical norm induced by the inner-product. Let π 0 be a zero-mean Gauian meaure on H with covariance operator C. By the general theory of Gauian meaure [8],C i a poitive, trace cla operator. Let {φ,λ } 1 be the eigenfunction and eigenvalue of C, repectively, o that 1.3) hold. We aume a normalization under which {φ } 1

9 454 Stoch PDE: Anal Comp 018) 6: form a complete orthonormal bai of H. Recalling 1.4), we pecify the notation that will be ued throughout thi paper: x and y are element of the Hilbert pace H; the letter i reerved to denote the dimenionality of the pace X where the target meaure π i upported; x i an element of X = R imilarly for y and the noie ξ ); for any fixed, x k, i the kth tep of the chain {x k, } k X contructed to ample from π ; x k, i i the ith component of the vector x k,, that i x k, i := x k,,φ i with abue of notation). For every x H, we have the repreentation x = 1 x φ, where x := x,φ. Uing thi expanion, we define Sobolev-like pace H, R, with the innerproduct and norm defined by x, y = x y and x = x. The pace H,, ) i alo a Hilbert pace. otice that H 0 = H. Furthermore H H H for any > 0. The Hilbert Schmidt norm C aociated with the covariance operator C i defined a x C := λ x = x,φ, x H, and it i the Cameron Martin norm aociated with the Gauian meaure 0, C). Such a norm i induced by the calar product λ x, y C := C 1/ x, C 1/ y, x, y H. Similarly, C define a Hilbert Schmidt norm on X, x := C x,φ, x X,.1) λ which i induced by the calar product x, y C := C 1/ x, C 1/ y, x, y X. For R,letL : H H denote the operator which i diagonal in the bai {φ } 1 with diagonal entrie, L φ = φ,

10 Stoch PDE: Anal Comp 018) 6: o that L 1 φ = φ. The operator L let u alternate between the Hilbert pace H and the interpolation pace H via the identitie: x, y = L 1 x, L 1 y and x = L 1 x. Since L 1/ φ k = φ k = 1, we deduce that { ˆφ k := L 1/ φ k } k 1 form an orthonormal bai of H. An element y 0, C) can be expreed a y = D λ ρ φ with ρ 0, 1) i.i.d..) If λ <, then y can be equivalently written a y = λ )ρ L 1/ D φ ) with ρ 0, 1) i.i.d..3) For a poitive, elf-adoint operator D : H H, it trace in H i defined a Trace H D) := φ, Dφ. We tre that in the above {φ } i an orthonormal bai for H,, ). Therefore, if D : H H, it trace in H i Trace H D) = L 1 φ, DL 1 φ. Since Trace H D) doe not depend on the orthonormal bai, the operator D i aid to be trace cla in H if Trace H D) < for ome, and hence any, orthonormal bai of H. Becaue C i defined on H, the covariance operator 1 C = L 1/ CL 1/.4) i defined on H. Thu, for all the value of r uch that Trace H C ) = λ <, we can think of y a a mean zero Gauian random variable with covariance operator 1 In thi paper, we commit a light abue of our notation by writing C to mean the covariance operator on the Sobolev-like ubpace H and C to mean that on the finite dimenional ubpace X a defined in 1.5). We ditinguih thee two by alway employing a the ubcript for the latter, and lower cae letter uch a or r for the former.

11 456 Stoch PDE: Anal Comp 018) 6: C in H and C in H [ee.) and.3)]. In the ame way, if Trace H C )<, then W t) = λ w t)φ = λ r w t) ˆφ,.5) where {w t)} 1 a collection of i.i.d. tandard Brownian motion on R, can be equivalently undertood a an H-valued C-Brownian motion or a an H -valued C -Brownian motion. We will make ue of the following elementary inequality, x, y = x ) y ) x y, x H, y H..6) Throughout thi paper we tudy equence of real number, random variable and function, indexed by either or both) the dimenion of the pace on which the target meaure i defined or the chain tep number k. In doing o, we find the following notation convenient. Two double) equence of real number {A k, } and {B k, } atify A k, B k, if there exit a contant K > 0 independent of and k) uch that A k, KB k,, for all and k uch that {A k, } and {B k, } are defined. IftheA k, and B k, are random variable, the above inequality mut hold almot urely for ome determinitic contant K ). IftheA k, and B k, are real-valued function on H or H, A k, = A k, x) and B k, = B k, x), the ame inequality mut hold with K independent of x,for all x where the A k, and B k, are defined. A i cutomary, R + := { R : 0} and for all b R + we let [b] =n if n b < n + 1 for ome integer n. Finally, for time dependent function we will ue both the notation St) and S t interchangeably.. The algorithm A natural variant of the MALA algorithm tem from the obervation that π i the unique tationary meaure of the SDE dy t = C log π Y t )dt + dw t,.7) where W i an X -valued Brownian motion with covariance operator C.The algorithm conit of dicretiing.7) uing the Euler-Maruyama cheme and adding

12 Stoch PDE: Anal Comp 018) 6: a Metropoli accept-reect tep o that the invariance of π i preerved. The variant on MALA which we tudy i therefore a Metropoli Hating algorithm with propoal y k, = x k, δ x k, + C Ψ x k, )) + δc 1/ ξ k,,.8) where ξ k, := ξ k, φ, ξ k, 0, 1) i.i.d. We tre that the Gauian random variable ξ k, i are independent of each other and of the current poition x k,. Motivated by the conideration made in the introduction and that will be made more explicit in Sect. 4.1), in thi paper we fix the choice δ := l..9) 1/ If at tep k the chain i at x k,, the algorithm propoe a move to y k, defined by Eq..8). The move i then accepted with probability where, for any x, y R X, α x k,, y k, ) := π y k, ) q y k,, x k, ) π x k, ) q x k, ),.10), yk, q x, y ) e 1 4δ y x ) δ log π x ) C..11) If the move to y k, i accepted then x k+1, = y k,, if it i reected the chain remain where it wa, i.e. x k+1, = x k,. In hort, the MALA chain i defined a follow: x k+1, := γ k, y k, + 1 γ k, )x k,, x 0, := P x 0),.1) where in the above γ k, D Bernoulli α x k,, y k, )) ;.13) that i, conditioned on x k,, y k, ), γ k, ha Bernoulli law with mean α x k,, y k, ). Equivalently, we can write γ k, = 1 { U k, α x k,,y k,)}, with U k, D Uniform [0, 1], independent of x k, and ξ k,. For fixed, the chain {x k, } k 1 live in X = R and ample from π. However, in view of the fact that we want to tudy the caling limit of uch a chain a,

13 458 Stoch PDE: Anal Comp 018) 6: the analyi i cleaner if it i carried out in H; therefore, the chain that we analye i the chain {x k } k H defined a follow: the firt component of the vector x k H coincide with x k, a defined above; the remaining component are not updated and remain equal to their initial value. More preciely, uing.8) and.1), the chain x k can be written in a component-wie notation a follow: x k+1 i = x k+1, i [ = x k, l i γ k, 1/ ] l + 1/ λ i ξ k, x k, i + [ C Ψ x k, )] i i.14) ) and x k+1 i = x k i = 0 i ) For the ake of clarity, we pecify that [C Ψ x k, )] i denote the ith component of the vector C Ψ x k, ) H. From the above it i clear that the update rule.14) only update the firt coordinate with repect to the eigenbai of C) of the vector x k. Therefore the algorithm evolve in the finite-dimenional ubpace X. From now on we will avoid uing the notation {x k } k for the extended chain defined in H, a it can be confued with the notation x, which intead i ued throughout to denote a generic element of the pace X. We conclude thi ection by remarking that, if x k, i given, the propoal y k, only depend on the Gauian noie ξ k,. Therefore the acceptance probability will be interchangeably denoted by α x, y ) or α x,ξ )..3 Aumption In thi ection, we decribe the aumption on the covariance operator C of the Gauian meaure π 0 0, C) and thoe on the functional Ψ. We fix a ditinguihed D exponent 0 and aume that Ψ : H R and Trace H C )<. In other word, H i the pace that we were denoting with H in the introduction. Since Trace H C ) = λ,.16) the condition Trace H C )< implie that λ 0a. Therefore the equence {λ } i bounded: λ C,.17) for ome contant C > 0 independent of. For each x H the derivative Ψx) i an element of the dual LH, R) of H, compriing the linear functional on H. However, we may identify LH, R) = H

14 Stoch PDE: Anal Comp 018) 6: and view Ψx) a an element of H for each x H. With thi identification, the following identity hold Ψx) LH,R) = Ψx)..18) To avoid technical complication we aume that the gradient of Ψx) i bounded and globally Lipchitz. More preciely, throughout thi paper we make the following aumption. Aumption.1 The functional Ψ and covariance operator C atify the following: 1. Decay of Eigenvalue λ of C: there exit a contant κ> + 1 uch that κ λ κ.. Domain of Ψ : the functional Ψ i defined everywhere on H. 3. Derivative of Ψ : The derivative of Ψ i bounded and globally Lipchitz: Ψx) 1, Ψx) Ψy) x y..19) Remark.1 The condition κ> + 1 enure that Trace H C )<. Conequently, π 0 ha upport in H π 0 H ) = 1). Example.1 The functional Ψx) = 1 + x atifie all of the above. Remark. Our aumption on the change of meaure that i, on Ψ ) are le general than thoe adopted in [14,17] and related literature ee reference therein). Thi i for purely technical reaon. In thi paper we aume that Ψ grow linearly. If Ψ wa aumed to grow quadratically, which i the cae in the mentioned work, finding bound on the moment of the chain {x k, } k 1 much needed in all of the analyi) would become more involved than it already i, ee Remark C.1. However, under our aumption, the meaure π or π ) i till, generically, of non-product form. We now explore the conequence of Aumption.1. The proof of the following lemma can be found in Appendix A. Lemma.1 Suppoe that Aumption.1 hold. Then 1. The function C Ψx) i bounded and globally Lipchitz on H, that i C Ψx) 1 and C Ψx) C Ψy) x y..0) Therefore, the function Fz) := z C Ψz) atifie Fx) Fy) x y and Fx) 1 + x..1)

15 460 Stoch PDE: Anal Comp 018) 6: The function Ψx) i globally Lipchitz and therefore alo Ψ x) := ΨP x)) i globally Lipchitz: Ψ y) Ψ x) y x..) Before tating the next lemma, we oberve that by definition of the proection operator P we have that Ψ = P Ψ P..3) Lemma. Suppoe that Aumption.1 hold. Then the following hold for the function Ψ and for it the gradient: 1. If the bound.19) hold for Ψ, then they hold for Ψ a well: Ψ x) 1, Ψ x) Ψ y) x y..4). Moreover, and C Ψ x) 1,.5) C Ψ x) 1. C.6) We tre that in.4).6) the contant implied by the ue of the notation ee end of Sect..1) i independent of. Latly, in what follow we will need the fact that, due aumption on the covariance operator, E C 1/ ξ 1, uniformly in,.7) where ξ := ξ φ and ξ i D 0, 1) i.d.d., ee [15,.3)] or [14, firt proof of Appendix A] 3 Exitence and uniquene for the limiting diffuion proce The main reult of thi ection are Theorem 3.1, 3. and 3.3. Theorem 3.1 and 3. are concerned with etablihing exitence and uniquene for Eq. 1.10) and 1.11), repectively. Theorem 3.3 tate the continuity of the Itô map aociated with Eq. 1.10) and 1.11). The proof of the main reult of thi paper Theorem 4.1 and 4.) rely heavily on the continuity of uch map, a we illutrate in Sect. 5. Once Lemma 3.1 below i etablihed, the proof of the theorem in thi ection are completely analogou to the proof of thoe in [14, Section 4]. For thi reaon, we omit them and refer the reader to [14]. In what follow, recall that the definition of the function α l, h l and b l ha been given in 1.1), 1.13) and 1.14), repectively.

16 Stoch PDE: Anal Comp 018) 6: Lemma 3.1 The function α l ), h l ) and h l ) are poitive, globally Lipchitz continuou and bounded. The function b l ) i globally Lipchitz and it i bounded above but not below. Moreover, for any l>0, b l ) i trictly poitive for [0, 1), trictly negative for > 1 and b l 1) = 0. Proof of Lemma 3.1 When > 1, α l ) = 1 while for 1 α l ) ha bounded derivative; therefore α l ) i globally Liphitz. A imilar reaoning give the Liphitzianity of the other function. The further propertie of b l are traightforward from the definition. In the cae of 1.11) we have the following. Theorem 3.1 For any initial datum S0) >0, there exit a unique olution St) to the ODE 1.11). The olution i trictly poitive for any t > 0, it i bounded and ha continuou firt derivative for all t 0. In particular lim t St) = 1 and 0 min{s0), 1} St) max{s0), 1}. 3.1) For 1.10) we have that: Theorem 3. Let Aumption.1 hold and conider Eq. 1.10), where W t) i any H -valued C -Brownian motion and St) i the olution of 1.11). Then for any initial condition x 0 H and any T > 0 there exit a unique olution of Eq. 1.10) in the pace C[0, T ]; H ). Conider the determinitic equation dzt) =[ zt) C Ψzt))]h l St)) dt + dζt), z0) = z 0 3.) and dst) = b l St)) dt + dwt), S0) = S 0, 3.3) where S i the olution of 1.11), z 0 H, S 0 R, and ζ and w are function in C[0, T ]; H ) and C[0, T ]; R), repectively. Throughout the paper, we endow the pace C[0, T ]; H ) and C[0, T ]; R) with the uniform topology. The following i the tarting point of the continuou mapping argument preented in Sect. 5. Theorem 3.3 Suppoe that Aumption.1 i atified. Both 3.) and 3.3) have unique olution in C[0, T ]; H ) and C[0, T ]; R), repectively. The Itô map J 1 : H C[0, T ]; H ) C[0, T ]; H ) z 0,ζ) z

17 46 Stoch PDE: Anal Comp 018) 6: and J : R + C[0, T ]; R) C[0, T ]; R) S 0,w) S are continuou. 4 Main theorem and heuritic of proof In order to tate the main reult, we firt et H {x := H 1 : lim x i i=1 λ i } <, 4.1) where we recall that in the above x i := x,φ i. Theorem 4.1 Let Aumption.1 hold and let δ = l/ 1. Let x 0 H and T > 0. Then, a, the continuou interpolant S ) t) of the equence {S k, } k R + defined in 1.16)) and tarted at S 0, = 1 i=1 x 0 i /λ i, converge in probability in C[0, T ]; R) to the olution St) of the ODE 1.11) with initial datum S 0 := lim S 0,. For the following theorem recall that the olution of 1.10) i interpreted preciely through Theorem 3. a a proce driven by an H valued Brownian motion with covariance C, and olution in C[0, T ]; H ). Theorem 4. Let Aumption.1 hold let δ = l/ 1. Let x 0 H and T > 0. Then, a, the continuou interpolant x ) t) of the chain {x k, } k H defined in 1.9) and.14), repectively) with initial tate x 0, := P x 0 ), converge weakly in C[0, T ]; H ) to the olution xt) of Eq. 1.10) with initial datum x 0. We recall that the time-dependent function St) appearing in 1.10) i the olution of the ODE ), tarted at S0) := lim i=1 x 0 i /λ i. Both Theorem 4.1 and 4. aume that the initial datum of the chain x k, i aigned determinitically. From our proof it will be clear that the ame tatement alo hold for random initial data, a long a i) x 0, i not drawn at random from the target meaure π or from any other meaure which i a change of meaure from π i.e. we need to be tarting out of tationarity) and ii) S 0, and x 0, have bounded moment bounded uniformly in ) of ufficiently high order and are independent of all the other ource of noie preent in the algorithm. otice moreover that the convergence in probability of Theorem 4.1 i equivalent to weak convergence, a the limit i determinitic. The rigorou proof of the above reult i contained in Sect In the remainder of thi ection we give heuritic argument to utify our choice of caling δ 1/ and we explain how one can formally obtain the fluid) ODE limit 1.11) forthe

18 Stoch PDE: Anal Comp 018) 6: double equence S k, and the diffuion limit 1.10) for the chain x k,. We tre that the argument of thi ection are only formal; therefore, we often ue the notation, to mean approximately equal. That i, we write A B when A = B+ term that are negligible a tend to infinity; we then utify thee approximation, and the reulting limit theorem, in the following Sect Heuritic analyi of the acceptance probability A oberved in [17, equation.1)], the acceptance probability.10) can be expreed a α x,ξ ) = 1 e Q x,ξ ), 4.) where, uing the notation.1), the function Q x,ξ)can be written a Q x,ξ ) := δ y x 4 C C [ δ x = C 1/ C ξ ) + r x,ξ ) 4.3) )] δ3 x C 4 C ) δ 3/ δ5/ x, C 1/ ξ C + rψ x,ξ ). 4.4) We do not give here a complete expreion for the term r x,ξ ) and r Ψ x,ξ ). For the time being it i ufficient to point out that r x,ξ ) := I + I 3 rψ x,ξ ) := r x,ξ ) δ δ 3) + x, C Ψ x ) C δ3 C Ψ x ) 4 C + δ5/ C Ψ x ), C 1/ ξ C 4.5) where I and I 3 will be defined in 6.10) and 6.11), repectively. Becaue I and I3 depend on Ψ, r Ψ contain all the term where the functional Ψ appear; moreover rψ vanihe when Ψ = 0. The analyi of Sect. 6 ee Lemma 6.4) will how that with our choice of caling, δ = l/ 1/,thetermr and rψ are negligible for large). Let u now illutrate the reaon behind our choice of caling. To thi end, et δ = l/ ζ and oberve the following two imple fact: S k, = 1 x k, λ = 1 x k, 4.6) C

19 464 Stoch PDE: Anal Comp 018) 6: and C 1/ ξ = C ξ i, 4.7) the latter fact being true by the Law of Large umber. eglecting the term containing Ψ,attepk of the chain we have, formally, Q x k,,ξ k+1, ) l ) 1 ζ S k, 1 4.8) l ζ S k, l3/ 1 3ζ)/ xk,, C 1/ ξ k, C + l5/ i=1 4.9) 1 5ζ)/ xk,, C 1/ ξ k, C. 4.10) The above approximation which, we tre again, i only formal and will be made rigorou in ubequent ection) ha been obtained from 4.4) by etting δ = l/ ζ and uing 4.6) and 4.7), a follow: δ [ x ] C 1/ C ξ 4.8), 4.11) C δ 3 x C 4 δ3/ x, C 1/ ξ C 4.9), + δ5/ x, C 1/ ξ C = 4.10). Looking at the decompoition 4.8) 4.10) of the function Q, we can now heuritically explain the reaon why we are lead to chooe ζ = 1/ when we tart the chain out of tationarity, a oppoed to the caling ζ = 1/3 when the chain i tarted in tationarity. Thi i explained in the following remark. Remark 4.1 Firt notice that the expreion 4.4) and the approximation 4.8) 4.10) for Q are valid both in and out of tationarity, a the firt i only a conequence of the definition of the Metropoli Hating algorithm and the latter i implied ut by the propertie of Ψ and by our definition. If we tart the chain in tationarity, i.e. x0 π where π ha been defined in 1.6)), then x k, π for every k 0. A we have already oberved, π i abolutely continuou with repect to the Gauian meaure π0 0, C ); becaue all the almot ure propertie are preerved under thi change of meaure, in the tationary regime mot of the etimate of interet need to be hown only for x π0. In particular if x π0 then x can be repreented a x = i=1 λ i ρ i φ i, where ρ i are i.i.d. 0, 1). Therefore we can ue the law of large number and oberve that x = C i=1 ρ i.

20 Stoch PDE: Anal Comp 018) 6: Suppoe we want to tudy the algorithm in tationarity and we therefore make the choice ζ = 1/3. With the above point in mind, notice that if we tart in tationarity then by the Law of Large number 1 i=1 ρ i = S k, 1a, with peed of convergence 1/ ). Moreover, if x π0,bythe Central Limit Theorem the term x, C 1/ ξ C / i O1) and converge to a tandard Gauian. With thee two obervation in place we can then heuritically ee that, with the choice ζ = 1/3 the term in 4.10) are negligible a while the term in 4.9) areo1). The term in 4.8) can be better undertood by looking at the LHS of 4.11) which, with ζ = 1/3 and x π0, can be rewritten a l /3 ρ i ξ i ). 4.1) i=1 The expected value of the above expreion i zero. If we apply the Central Limit Theorem to the i.i.d. equence { ρ i ξ i } i,4.1) how that 4.8) i O 1/ /3 ) and therefore negligible a. In concluion, in the tationary cae the only O1) term are thoe in 4.9); therefore one ha the heuritic approximation Q x,ξ) ) l3 4, l3. For more detail on the tationary cae ee [17]. If intead we tart out of tationarity the choice ζ = 1/3 i problematic. Indeed in [6, Lemma 3] the author tudy the MALA algorithm to ample from an - dimenional iotropic Gauian and how that if the algorithm i tarted at a point x 0 uch that S0) <1, then the acceptance probability degenerate to zero. Therefore, the algorithm tay tuck in it initial tate and never proceed to the next move, ee [6, Figure ] to be more precie, a increae the algorithm will take longer and longer to get untuck from it initial tate; in the limit, it will never move with probability 1). Therefore the choice ζ = 1/3 cannot be the optimal one at leat not irrepective of the initial tate of the chain) if we tart out of tationarity. Thi i till the cae in our context and one can heuritically ee that the root of the problem lie in the term 4.8). Indeed if out of tationarity we till chooe ζ = 1/3 then, like before, 4.9) i till order one and 4.10) i till negligible. However, looking at 4.8), if x 0 i uch that S0) <1 then, when k = 0, 4.8) tend to minu infinity; recalling 4.), thi implie that the acceptance probability of the firt move tend to zero. To overcome thi iue and make Q of order one irrepective of the initial datum) o that the acceptance probability i of order one and doe not degenerate to 0 or 1 when,wetakeζ = 1/; in thi way the term in 4.8) are O1), all the other are mall. Therefore, the intuition leading the analyi of the non-tationary regime hinge on the fact that, with our caling, Q x k,,ξ k, ) l Sk, 1); 4.13)

21 466 Stoch PDE: Anal Comp 018) 6: hence α x k,,ξ k, ) = 1 e Q x k,,ξ k, ) ) α l S k, ), 4.14) where the function α l on the RHS of 4.14) i the one defined in 1.1). The approximation 4.13) i made rigorou in Lemma 6.4, while 4.14) i formalized in Sect. 6.1 ee in particular Propoition 6.1). Finally, we mention for completene that, by arguing imilarly to what we have done o far, if ζ<1/ then the acceptance probability of the firt move tend to zero when S0) <1. If ζ > 1/ then Q 0, o the acceptance probability tend to one; however the ize of the move i mall and the algorithm explore the phae pace lowly. Remark 4. otice that in tationarity the function Q i, to leading order, independent of ξ; that i, Q and ξ are aymptotically independent ee [17, Lemma 4.5]). Thi can be intuitively explained becaue in tationarity the leading order term in the expreion for Q i the term with δ 3 x. We will how that alo out of tationarity Q and ξ are aymptotically independent. In thi cae uch an aymptotic independence can, roughly peaking, be motivated by the approximation 4.13), a the interpolation of the chain S k, converge to a determinitic limit). The aymptotic correlation of Q and the noie ξ i analyed in Lemma 6.5. Remark 4.3 When one employ the more general propoal 1.18), auming Ψ 0, the expreion for Q become Q x k,, y k, ) = δ 4 1 θ) x k, C y k, C ). So, if θ = 1/, the acceptance probability would be exactly one for every ), i.e. the algorithm would be ampling exactly from the prior hence there i no need of recaling δ with. 4. Heuritic derivation of the weak limit of S k, Let Y be any function of the random variable ξ k, and U k, introduced in Sect..), for example the chain x k, itelf. Here and throughout the paper we ue E x 0 [Y ] to denote the expected value of Y with repect to the law of the variable ξ k, and U k,, with the initial tate x 0 of the chain given determinitically; in other word, E x 0Y ) denote expectation with repect to all the ource of randomne preent in Y. We will ue the [ notation E k [Y ] for the conditional expectation of Y given x k,, E k [Y ] := E x 0 Y x k, ] we hould really be writing Ek in place of E k,butto improve readability we will omit the further index ). Let u now decompoe the chain S k, into it drift and martingale part: S k+1, = S k, + 1 b k, l + 1 1/4 Dk,, 4.15)

22 Stoch PDE: Anal Comp 018) 6: where b k, l := E k [S k+1, S k, ] 4.16) and [ D k, := 1/4 S k+1, S k, 1 b k, x k, )]. 4.17) l In thi ubection we give the heuritic which underly the proof, given in ubequent ection, that the approximate drift b k, l = b k, l x k, ) converge to b l S k, ), where b l i the drift of 1.11), while the approximate diffuion D k, tend to zero. Thi formally give the reult of Theorem 4.1. Let u formally argue uch a convergence reult. By 4.6) and.1), S k+1, = 1 x k+1, λ Therefore, again by 4.6), = 1 γ k, y k, + 1 γ k, ) ) x k,. C C 4.18) b k, l = [ E k S k+1, S k, ] = 1 [ E k γ k, y k, C x k, C = 1 E k [ 1 e Q x k,,y k,) ) y k, x k, C )] C )], 4.19) where the econd equality i a conequence of the definition of γ k, with a reaoning, completely analogou to the one in [14, lat proof of Appendix A], ee alo 4.4). Uing 4.3) with δ = l/ ), the fact that r i negligible and the approximation 4.13), the above give b k, l = E k [S k+1, S k, ] 4 1 e l ) ) S k, 1 / l S k, 1 ) =b l S k, ). l The above approximation i made rigorou in Lemma 7.5. A for the diffuion coefficient, it i eay to check ee proof of Lemma 7.) that E k [S k+1, S k, ] <. Hence the approximate diffuion tend to zero and one can formally deduce that the interpolant of) S k, converge to the ODE limit 1.11). otice that S k, i only a function of x k,.

23 468 Stoch PDE: Anal Comp 018) 6: Heuritic analyi of the limit of the chain x k,. The drift-martingale decompoition of the chain x k, i a follow: x k+1, = x k, + 1 1/ Θk, + 1 1/4 Lk, 4.0) where Θ k, = Θ k, x k, ) i the approximate drift Θ k, := E k [x k+1, x k, ] 4.1) and L k, := 1/4 [ x k+1, x k, 1 Θ k, x k, )] 4.) i the approximate diffuion. In what follow we will ue the notation Θx, S) for the drift of Eq. 1.10), i.e. Θx, S) = Fx)h l S), x, S) H R, 4.3) with Fx) defined in Lemma.1. Again, we want to formally argue that the approximate drift Θ k, x k, ) tend to Θx k,, S k, ) 3 and the approximate diffuion L k, tend to the diffuion coefficient of Eq. 1.10) Approximate drift A a preliminary conideration, oberve that E k γ k, C 1/ ξ k, ) = E k 1 e Q x k,,ξ k, ) ) C 1/ ξ k, ), 4.4) ee [14, equation 5.14)]. Thi fact will be ued throughout the paper, often without mention. Coming to the chain x k,, a direct calculation baed on.8) and on.1) give x k+1, x k, = γ k, δ x k, + C Ψ x k, )) + γ k, δc 1/ ξ k,. 4.5) Therefore, with the choice δ = l/,wehave Θ k, = [ E k x k+1, x k, ] [ 1 = le k e Q x k,,ξ k, ) ) x k, + C Ψ ))] k, x + 1/4 le k [ 1 e Q x k,,ξ k, ) ) C 1/ ξ k, ] 4.6) 3 ote that in the limit the dependence of the drift on S k, become explicit.

24 Stoch PDE: Anal Comp 018) 6: The addend in 4.6) i aymptotically mall ee Lemma 6.5 and notice that thi addend would ut be zero if Q and ξ k, were uncorrelated); hence, uing the heuritic approximation 4.13) and 4.14), Θ k, = E k [x k+1, x k, ] lα l S k, ) x k, + C Ψ )) k, x 1.13) = h l S k, ) x k, + C Ψ x k, )) ; 4.7) the right hand ide of the above i preciely the limiting drift Θx k,, S k, ) Approximate diffuion We now look at the approximate diffuion of the chain x k, : By definition, L k, := 1/4 x k+1, x k, E k x k+1, x k, )). L k, E k = x E k+1, k x k, ) E k x k+1, x k,. 4.8) By 4.7) the econd addend in the above i aymptotically mall. Therefore L k, E k x E k+1, k x k,.1),4.5) γ le k, k C 1/ ξ k, ) = le k λ 1 e Q x k,,ξ k, ) k, ξ. The above quantity i carefully tudied in Lemma 6.6. However, intuitively, the heuritic approximation 4.14) and the aymptotic independence of Q and ξ that 4.14) i a manifetation of) uffice to formally derive the limiting diffuion coefficient [i.e. the diffuion coefficient of 1.10)]: L k, E k l λ E k l [ 1 e Q x k,,y k,) ) ξ k, ] [ 1 λ E k e l S 1) k, / ) k, ξ ]

25 470 Stoch PDE: Anal Comp 018) 6: l λ 1 e l ) S k, 1 / ) l TraceC )α l S k, ) 1.13) = TraceC ) h l S k, ). 5 Continuou mapping argument In thi ection we outline the argument which underlie the proof of our main reult. In particular, the proof of Theorem 4.1 and 4. hinge on the continuou mapping argument that we illutrate in the following Sect. 5.1 and 5., repectively. The detail of the proof are deferred to the next three ection: Sect. 6 contain ome preliminary reult that we employ in both proof, in Sect. 7 contain the the proof of Theorem 4.1 and Sect. 8 that of Theorem Continuou mapping argument for 3.3) Let u recall the definition of the chain {S k, } k and of it continuou interpolant S ), introduced in 1.15) and 1.16), repectively. From the definition 1.16) of the interpolated proce and the drift-martingale decompoition 4.15) of the chain {S k, } k we have that for any t [t k, t k+1 ), [ S ) t) = 1/ t k) S k, + 1 b k, l + 1 1/4 Dk, = S k, + t t k )b k, l + 1/4 t t k )D k,. Iterating the above we obtain S ) t) = S 0, + t t k )b k, l + 1 k 1 =0 ] + k + 1 t 1/ )S k, b, l + w t), where w t) := 1 k 1 1/4 D, + 1/4 t t k )D k, t k t < t k ) =0 The expreion for S ) t) can then be rewritten a t S ) t) = S 0, + b l S ) v))dv +ŵ t), 5.) 0 having et ŵ t) := e t) + w t), 5.3)

26 Stoch PDE: Anal Comp 018) 6: with e t) := t t k )b k, l + 1 Equation 5.) how that k 1 =0 b, l S ) = J S 0,, ŵ ), t b l S ) v))dv. 5.4) 0 where J i the Itô map defined in the tatement of Theorem 3.3. By the continuity of the map J, if we how that ŵ converge in probability in C[0, T ]; R) to zero, then S ) t) converge in probability to the olution of the ODE 1.11). We prove convergence of ŵ to zero in Sect. 7.Inviewof5.3), we how the convergence in probability of ŵ to zero by proving that both e Lemma 7.1) and w Lemma 7.) converge in L Ω; C[0, T ]; R)) to zero. Becaue {S 0, } i a determinitic equence that converge to S 0, we then have that S 0,, ŵ ) converge in probability to S 0, 0). 5. Continuou mapping argument for 3.) We now conider the chain {x k, } k H, defined in.14). We act analogouly to what we have done for the chain {S k, } k. So we tart by recalling the definition of the continuou interpolant x ),Eq.1.9) and the notation introduced at the beginning of Sect An argument analogou to the one ued to derive 5.) how that for any t [t k, t k+1 ) where and x ) t) = x 0, + t t k )Θ k, + 1 k Θ, + η t) t = x 0, + Θx ) v), Sv))dv +ˆη t), 5.5) 0 ˆη t) := d t) + υ t) + η t), 5.6) η t) := 1/4 t t k )L k, + 1 k 1 1/4 L,, 5.7) d t) := t t k )Θ k, + 1 k 1 t Θ, Θx ) v), S ) v))dv, 5.8) =0 0 t [ ] υ t) := Θx ) v), S ) v)) Θx ) v), Sv)) dv. 5.9) 0 =0

27 47 Stoch PDE: Anal Comp 018) 6: Equation 5.5) implie that x ) = J 1 x 0,, ˆη ), 5.10) where J 1 i Itô map defined in the tatement of Theorem 3.3. In Sect. 8 we prove that ˆη converge weakly in C[0, T ]; H ) to the proce η, where the proce η i the diffuion part of Eq. 1.10), i.e. ηt) := t 0 hl Sv))dW v, 5.11) with W v a H -valued C -Brownian motion. Looking at 5.6), we prove the weak convergence of ˆη to η by the following tep: 1. We prove that d converge in L Ω; C[0, T ]; H )) to zero Lemma 8.1);. uing the convergence in probability in C[0, T ]; R)) ofs ) to S, wehow convergence in probability in C[0, T ]; H ))ofυ to zero Lemma 8.); 3. we how that η converge in weakly in C[0, T ]; H ) to the proce η, defined in 5.11) Lemma 8.3). Becaue {x 0, } i a determinitic equence that converge to x 0, the above three tep and Slutky Theorem) imply that x 0,, ˆη ) converge weakly to x 0,η). ow oberve that xt) = J 1 x 0,ηt)), where xt) i the olution of the SDE 1.10). The continuity of the map J 1 Theorem 3.3), 5.10) and the Continuou Mapping Theorem then imply that the equence {x ) } converge weakly to the olution of the SDE 1.10), thu etablihing Theorem Preliminary etimate and analyi of the acceptance probability Thi ection gather everal technical reult. In Lemma 6.1 we tudy the ize of the ump of the chain. Lemma 6. contain uniform bound on the moment of the chain {x k, } k and {S k, } k, much needed in Sect. 7 and 8. In Section 6.1 we detail the analyi of the acceptance probability. Thi allow u to quantify the correlation between γ k, and the noie ξ k,, Sect. 6.. Throughout the paper, when referring to the function Q defined in 4.3), we ue interchangeably the notation Q x k,, y k, ) and Q x k,,ξ k, ) a we have already remarked, given x k,, the propoal y k, i only a function of ξ k, ). Lemma 6.1 Let q 1/ be a real number. Under Aumption.1 the following hold: y E k, k x k, q 1 ) q/ 1 + x k, q 6.1) and E k y k, x k, q C S k, ) q + q/. 6.)