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SOME REMARK ON THE ASYMPTOTIC
VARIANCE IN A DRIFT ACCELERATED
DIFFUSION
A. Said
To cite this version:
DRIFT ACCELERATED DIFFUSION
A. OULED SAID
Introduction
The main purpose of this paper is to give some general formula for the asymptotic variance of a drifted diffusion as the drift is increased by a large constant. Some formula for the asymptotic variance when the amplitude of the drift grows to infinity was given by Hwang et al (2015) and also by Duncan et al (2016). Our intention is to give some unifying formula for the asymptotic variance which contains as special cases the expression of Hwang et al (2015) and the one of Duncan et al (2016). Our result shows an interesting identity between projections on some particular subspace in different Sobolov-spaces.
1. Main results
Let U : Rn −→ R be a given energy function and π be the probability measure with density proportional to e−U (x). If one is interested in sampling from π, then a widely used diffusion is the time reversible Langevin equation
(1) dXt= −∇U (Xt) dt +
√ 2 dBt,
where Bt is a Brownian motion. The diffusion Xt has equilibrium
distribu-tion π. Perturbing the reversible diffusion (1) by adding a drift term, which preserves the mesure π, results in a stochastic equation of the form
(2) dXtb = −∇U (Xtb) dt + √2 dBt + cb(Xtb)dt.
If the vector field b is supposed to be weighted divergence-free with respect to π, i.e., div(be−U ) = 0, then the resulting diffusion has also equilibrium distribution π.
The generator of the diffusion defined in (1) has the form Lf = ∆f − ∇U · ∇f.
Note that L is a self-adjoint operator on L2(Rd, π). We will suppose that L has only discrete spectrum. This is the case if some suitable growth conditions are satisfied by U (see A1, A2 et A3 in the next section).
Date: December 2, 2019.
2010 Mathematics Subject Classification. Primary 60J60; Secondary 60J35, 47D0, 34K08.
Key words and phrases. non-reversible diffusion, spectral measure, fast incompressible drift, asymptotic variance .
2 A. OULED SAID
We denote by ϕ1, ϕ1, ϕ2, ... the eigenfunctions of the operator L
correspond-ing to the eigenvalues λ0, λ1, λ2, .... Since L is a self-adjoint operator on
L2(Rd, π), it is known that the eigenfunctions form an orthonormal basis of
the Hilbert space H = f ∈ L2(Rd, π) : Z Rd f dπ = 0 .
For some q ∈ R, we introduce the normed space H1 which contains all the functions f =P∞ i=1aiϕi satisfying : kf kHq = ∞ X i=1 |ai|2λqi < ∞.
It then follows that the operator L is a symmetric bijection from H2 to H. The generator of the modified diffusion introduced in (2) is given by
Lcbf = Lf + cb · ∇f.
The operator Lcb is also a bijection from H2 to H. Suppose that for f ∈ H
there exists a solution g ∈ H2 to the Poisson equation Lcbg = −f.
Under additional assumptions on the distribution π and the vector field b, the following central limit theorem was proved in Bhattacharya (1982) and also in Kipnis and Varadhan (1986)(see [1], [6]) :
t−12 1 t Z t 0 f (Xsb)ds − Z Rd f dπ d −→ N (0, σ2cb(f )) as t → 0. The asymptotic variance in this result is given by the following expression :
(3) σcb2(f ) = 2hf, gi.
We can ask the question, how this asymptotic variance behaves as the drift is accelerated to infinity by multiplying the vector field b by a large constant c. It was proved in Hwang et al (2015) that
lim |c|→∞σ 2 cb(f ) = 2kP L −1/2 f k2,
where P is the orthogonal projection in H to N = ker(L−1/2b · ∇L−1/2). An-other representation for the limit of the asymptotic variance was published in Duncan et al (2016). Their proof is based on a method from Bhattacharya et al (1989).They showed that
lim |c|→∞σ 2 cb(f ) = 2kP L −1 f k2H1,
where P is the orthogonal projection in H1 to N = ker(L−1b · ∇). One might ask whether, there is a way to relate both expressions.
We will prove in Theorem 2.3 under suitable conditions the following result for any p ∈0,12
lim
|c|→∞σ 2
where P is the orthogonal projection in H1−2pto N = ker(Lp−1b · ∇L−p). It is clear that for p = 12 we obtain the expression of Hwang et al (2015) and for p = 0 the expression of Duncan et al (2016).
2. An Expression for the Asymptotic Variance
Our goal is to obtain formula for σ2cb(f ) which shows the explicit dependency from the parameter c. The operator L is symmetric and negative definite, it follows that L−1 is a well-defined bounded operator from H2 to H which is symmetric and negative definite. In addition we have the following norm identity kL−pf k2H1 = ∞ X k=1 |ak|2kL−pϕkk2H1 = ∞ X k=1 |ak|2kλ−pϕkk2H1 = ∞ X k=1 |ak|2λ−2pkϕ kk2H1 = ∞ X k=1 |ak|2λ1−2p = kf k2 H1−2p.
Therefore the operator L−p is a bounded self-adjoint bijection from H1−2p to H1. Similarly, we have kLp−1f k2H = kf k2H2−2p and it follows that the
operator Lp−1 is a bounded self-adjoint bijection from H2−2p to H.
Generalizing the approach of Hwang et al (2015) and Duncan et al (2016), we can define the operator B = icLp−1b · ∇L−p from H1−2p to H1−2p. The operator B has a dense domain of definition dom(B) in H1−2p. We can see
this as follows
H1−2p −→ HL−p 1 −→ Hb·∇ L−→ Hp−1 2−2p⊂ H1−2p.
Note that in our general situation b · ∇ is a densely defined operator from H1 to H. Since L−p and Lp−1 are symmetric and b · ∇ is anti-symmtric, we
get for all f and g ∈ H1−2p hLp−1b · ∇L−pf, gi H1−2p = hL1−2pLp−1b · ∇L−pf, gi = hL−pC · ∇L−pf, gi = −hf, L−pb · ∇L−pgi = −hf, Lp−1b · ∇L−pgi H1−2p.
Also note that if the vector field b is bounded, then the operator B is a compact operator.
It can easily be seen that B is symmetric. As in Hwang et al (2015), we shall make the following assumptions
(A1) For all ε > 0, there is a cε> 0 such that
|b · ∇U | + |D2U | ≤ ε|∇U |2+ c ε,
where D2U denotes the Hessian matrix of U .
4 A. OULED SAID
Those assumptions guarantees that the invariant measure π exists and that the resulting generator L is self-adjoint on H. Moreover it follows that the spectrum of L is discrete.
The following two lemma are generalizations of some results from Hwang et al (2015) to our situation.
Lemma 2.1. The operator B is essentially self-adjoint.
Proof. Note that B is symmetric with respect to the scalar product on H1−2p. By a corollary to Theorem VIII.3 in [5], we need to check that the range Ran(B ± i) of B ± i is dense in H1−2p. We have Ran(B ± i) ⊂ H2−2p. For any g ∈ H2−2p then LL−p(−ig) provides a well-defined element of H. Since Lcbis onto H there exists a h ∈ H2 such that Lcbh = LL−p(−ig). Now
it follows that
cLp−1b · ∇L−pf = Lp−1(Lcb− L)L−pf = Lp−1LcbL−pLp−1Lh − f
= Lp−1Lcbh − f = Lp−1LL−p(−ig) − f = −ig − f.
We conclude that (B + i)f = i(−ig − f + f ) = g.
Similarly, we can check that Ran(B − i) ⊃ H1−2p by considering h ∈ H2 such that L−cbh = LL−p(ig) and using the fact that L−cb is onto L2. A
similar calculation shows that Lp−1b · ∇L−pf = f − ig. We conclude then that, (B − i)f = g.
The result follows.
Lemma 2.2. We have
(4) L−1cb = L−p(I − iB)−1Lp−1 Proof. First we note that we can rewrite Lcb as follows:
(L + cb · ∇)−1 = (L−p+1(I + cLp−1b · ∇L−p)Lp)−1 = L−p(I + cLp−1b · ∇L−p)−1Lp−1 = L−p(I − iB)−1Lp−1.
This implies that L−p(I − iB)−1Lp−1 is the inverse of Lcb.
Using the fact that Lcb is invertible on H, we can obtain an explicit
charac-terisation of the asymptotic variance introduced in (3).
Theorem 2.3. Assume (A1), (A2) and (A3). For all f ∈ H1, one has lim
c→∞σ 2
cb(f ) = 2kP Lp−1f k2H1−2p,
associated with the essentially self adjoint operator B on H1−2p (see [5] page 224 for a definition). We have
hL−1cb f, f i = hL−p(I − iB)−1Lp−1f, f i = hL1−2p(I − iB)−1Lp−1f, Lp−1f i = hL1−2p(I − iB)−1Lp−1f, Lp−1f i = h(I − iB)−1Lp−1f, Lp−1f iH1−2p = Z σ(B)\{0} 1 1 + c2y2µLp−1f(dy),
where the last equality is justified by the fact that the spectrum for the operator B is located on the real line.
Consequently, we obtain the following formula for the asymptotic variance
σcb2(f ) = hP Lp−1f, P Lp−1f iH1−2p+
Z
σ(B)\{0}
1
1 + c2y2µLp−1f(dy),
where P is the orthogonal projection in H1−2p to N = ker(iLp−1b · ∇L−p). Therefore, we have
lim
c→∞σ 2
cb(f ) = 2kP Lp−1f k2H1−2p.
This proves the theorem.
Remark : A similar proof can be used to show similar results on a compact Riemannian manifold.
Acknowledgment
I would like to thank my supervisors Brice Franke and Mondher Damak for his help and advice during this work and his availability for answering all my questions.
This work was supported by the Tunisian-French cooperation (PHC-UTIQUE) project CMCU2016 Number 16G1505.
References
[1] R. N. Bhattacharya : On the functional central limit theorem and the law of the iterated logarithm for Markov processes. Z. Wahrsch. Verw. Gebiete, 60(2):185-201,1982.
[2] R. N. Bhattacharya, V. N. Gupta and H. K Walker. : Asymptotics of solute dispersion in periodic porous media. SIAM Journal on Applied Mathematics, Vol. 49, Feb, pp. 86-98(1989)
[3] A. Duncan, T. Lelievre and G. Pavliotis : Variance reduction using nonre-versible langevin samplers. Journal of Statistical Physics, Vol. 163, pp 457-49(May 2016).
[4] C. Hwang, R. Normand. and S-J. Wu : Variance reduction for diffusions. Stochastic Processes and their Applications, vol 125(9) : 3522-3540 (2015). [5] M.Reed and B.Simon : Methods of Modern Mathematical Physics I : Functional
6 A. OULED SAID
[6] C.Kipnis and S.Varadhan : Central limit theorem for additive functionals of reversible Markov processes and applications to simple exclusions. Commun. Math. Phys. 104(1), 1-19 (1986).
Universit´e de Brest, UMR 6205 Laboratoire de Math´ematiques de Bretagne Atlantique, 6 Avenue Le Gorgeu, CS 93837, 29238 Brest, cedex 3, France.
