Caracterización de la UCLV y del proceso de Atención Personalizada al

5.2.1 Set-up

As discussed in the previous section, the key issue which motivates our work is the dif- ference in behaviour between minimisers of the Freidlin-Wentzell action and the Onsager- Machlup functional. This difference is manifest whenT 1and is most cleanly described by considering the time scaleT = ε−1. The Γ-limit of the Onsager-Machlup functional (5.9) is studied, asε→0, under this time-rescaling, in [189]; the limit exhibits the undesir- able effects described in the preceding section. Our objective is to characterise theΓ-limit for the variational problems arising from best Gaussian approximation with respect to the Kullback-Leibler divergence.

Applying the time scalingt7→ε−1tto the equation (5.1) and noticing the boundary conditions (5.2), yields

dx(t) =−ε−1∇V(x(t))dt+√2dW(t), x(0) =x−, x(1) =x+.

(5.10)

The transformed SDE has an order one noise but a strong drift; it will be our object of study throughout the remainder of this chapter. For technical reasons, we make the following assumptions on the potentialV.

Assumptions 5.2.1. The potentialV appearing in(5.10)satisfies:

(A-1) V ∈C5(Rd);

(A-2) The set of critical points

E :={x∈Rd,∇V(x) = 0} (5.11) is finite and the HessianD2V(x)is non-degenerate for anyx∈E.

(A-3) Coercivity condition: ∃R >0such that inf

|x|>R

(A-4) Growth condition:

∃C1, C2>0andα ∈[0,2)such that for allx∈Rdand1≤i, j, k ≤d, lim sup ε→0 max ∂3 ∂xi∂xj∂xk Ψε(x) ,|Ψ_ε(x)| ≤C1eC2|x| α ; (5.13)

(A-5) V(x)→ ∞when|x| → ∞and there exitsR >0such that

2∆V(x)≤ |∇V(x)|2for|x| ≥R; (5.14)

(A-6) Monotonicity condition:

∃R >0such that|∇V(x1)| ≥ |∇V(x2)|if|x1| ≥ |x2| ≥R. (5.15)

Remark 5.2.2. (i) Conditions (A-2)-(A-3) are typical assumptions for provingΓ-convergence results for Ginzburg-Landau and related functionals [96,149]. The smoothness con- dition (A-1) is needed because our analysis involves a Taylor expansion of order three for Ψε. Furthermore, we will use conditions (A-4)-(A-6) to analyse the Γ-

convergence problem in this Chapter. These assumptions will be employed to simplify the expectation term in the Kullback-Leibler divergence (see the expression(5.41)).

(ii) The condition (A-5) is a Lyapunov type condition which guarantees that at small temperature (ε≤1) the solution to the SDE in(5.10)does not explode in finite time. The probability measure determined by this process is absolutely continuous with respect to the reference measure of the Brownian bridge. See [201, Chapter 2] for more discussions about the absence of explosion. Moreover, by the definition ofΨε,

(A-5) implies that for anyδ ∈Rthere exists a constantC >0depending onlyRand

δsuch that

|∇V(x)|2−εδ∆V(x)≥ −Cεfor anyx∈Rd. (5.16) Such lower bound will be used to prove the compactness of the functionals of interest (see Proposition5.4.4).

(iii) These conditions are not independent. For instance, the coercivity condition (A-3) can be deduced from the monotonicity condition (A-6) whenV(x) is non-constant for large|x|. Hence particularly (A-5) and (A-6) imply (A-3).

(iv) The set of functions satisfying conditions (A-1)-(A-7) is not empty: they are fulfilled by all polynomials. Therefore many classical potentials, such as the Ginzburg-Landau

double-well potentialV(x) = 1₄x2(1−x)2are included.

Forε > 0we denote byµεthe law of the above bridge processxdefined in (5.10)

andµ0 the law of the corresponding bridge for vanishing drift (V = 0) in (5.10). Then, by identical arguments to those yielding (5.7),µεis absolutely continuous with respect toµ0 and the Radon-Nikodym density is given by

dµε dµ0 (x) = 1 Zµ,ε exp − 1 2ε2 Z 1 0 Ψε(x(t))dt (5.17) whereΨεis given by (5.8) andZµ,εis the normalisation constant. Note that the extra factor

1

ε with respect to (5.7) is due to the time rescaling.

5.2.2 Notation

Throughout the chapter, we useC(or occasionallyC1andC2) to denote a generic positive constant which may change from one expression to the next and is independent of the temperature and any quantity of interest. We writeA . BifA ≤CB. Given an interval

I ⊂ R, letLp(I)andWm,p(I)withm ∈N,1 ≤p ≤ ∞be the standard Lebesgue and Sobolev spaces of scalar functions respectively. LetHm(I) = Wm,2(I). Fors ∈ [0,1], we setH₀s(I)to be the closure ofC₀∞(I)inHs(I)and equip it with the topology induced byHs(I). Define its dual spaceH−s(I) := (H₀s(I))0. Fors > 1/2, a function ofH₀s(I) has zero boundary conditions. Thanks to the Poincar´e inequality, theH1-semi-norm is an equivalent norm onH₀1(I). In the case thatI = (0,1), we simplify the notations by setting

H₀s=H₀s(0,1)andH−s=H−s(0,1).

We write scalar and vector variables in regular face whereas matrix-valued vari- ables, function spaces for vectors and matrices are written in boldface. Denote byS(d,R) the set of all real symmetricd×dmatrices and by Id the identity matrix of sized. Let Lp(0,1;Rd)andLp(0,1;S(d,R))be the spaces of vector-valued and symmetric matrix- valued functions with entries inLp(0,1)respectively. Similarly one can defineH1(0,1;Rd),

Hs

0(0,1;Rd)andH1(0,1;S(d,R)). For simplicity, we use the same notationLp(0,1)(re- spectivelyH1(0,1)) to denoteLp(0,1;S(d,R))andLp(0,1;Rd)(respectivelyH1(0,1;S(d,R)) andH1(0,1;Rd)). For anyA= (Aij)∈Lp(0,1;S(d,R))with1≤p≤ ∞, we define its

norm kAk_Lp_(0,1):=   d X i=1 d X j=1 kAijk2Lp_(0,1)   1 2 .

ForA= (Aij)∈H1(0,1;S(d,R)), the norm is defined by kAk_H1_(0,1):=   d X i=1 d X j=1 kAijk2_H1_(0,1)   1 2 . We also defineH1±(0,1) := H±1(0,1;Rd) := {x ∈ H1(0,1;Rd) : x(0) = x−, x(1) =

x+}. Denote byBV(I)the set ofRd-valued functions of bounded variations on an interval

I ⊂R.

For matricesA,B ∈ S(d,R) we write A ≥ B when A−B is positive semi- definite. The trace of a matrixAis denoted byTr(A). Denote byAT _{the transpose ofA}

and by|A|_F the Frobenius norm ofA. GivenA ∈ S(d,R) with the diagonalised form

A= PTΛP, we define the matrix matrix|A|:= PT|Λ|P. For matricesA = (Aij)and

B= (Bij), we write A:B= Tr(ABT) = d X i=1 d X j=1 AijBij.

Define the matrix-valued operator∂2_t :=∂_t2·Id. Fora >0, we define

L1_a(0,1) :=L1_a(0,1;S(d,R)) =A∈L1(0,1;S(d,R)) :A(t)≥a·Ida.e.on(0,1)

and

H1_a(0,1) :=H_a1(0,1;S(d,R)) =A∈H1(0,1;S(d,R)) :A(t)≥a·Ida.e.on(0,1) .

We writeAn*AinL1(0,1)whenAnconverges toAweakly inL1(0,1). LetH10(0,1) =

H₀1(0,1;Rd). DefineHs₀ =

d

z }| {

H₀s× · · · ×H₀s and let H−s be the dual. In addition, we define product spaces H := H1±(0,1)×H1(0,1),Ha := H1±(0,1)×H1a(0,1),X :=

L1(0,1)×L1(0,1)andX_a:=L1(0,1)×L1_a(0,1).

For a vector fieldv = (v1, v2,· · ·, vd), let∇v = (∂ivj)i,j=1,2,···,dbe its gradient,

which is a second order tensor (or matrix). Given a potentialV :Rd→R, denote byD2V

the Hessian ofV. Given a second order tensorT = (Tij)i,j=1,2,···,d, we denote by∇T

its gradient, which is a rank 3 tensor with(∇T)ijk = ∂Tij

∂xk . In particular, we useD 3_{V to} denote the gradient of the HessianD2V.

Finally we writeν µwhen the measureνis absolutely continuous with respect to

µand writeν ⊥µwhen they are singular. Throughout the chapter, we denote byN(m,Σ) the Gaussian measure onL2(0,1)with meanmand covariance operatorΣ. Moreover, the Gaussian measures considered in this chapter will always have the property that, almost

surely, draws from the measure are continuous functions on[0,1]and thus that point-wise evaluation is well-defined. Givenh ∈ L2(0,1), define the translation map Th by setting

T_hx=x+hfor anyx∈L2_{(0,1). Denote byT}∗

hµthe push-forward measure of a measure µonL2(0,1)under the mapT_h.