Диссертация (1137347), страница 12

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For a given Hamilton function of the form H(x, y) = V (y) + |x|2 , where2V is a potential and |x|2 the kinetic energy of a particle with unit mass, the associated stochasticHamiltonian system would correspond to b(Xs , Ys ) = −(∂y V (Ys ) + F (Xs , Ys )Xs ) in (4.7), where F isa friction term. When F > 0 natural questions arise concerning the asymptotic behaviour of (Xt , Yt ),48for instance, the geometric convergence to equilibrium for the Langevin equation is discussed inMattingly and Stuart [MSH02], numerical approximations of the invariant measures in Talay [Tal02],the case of high degree potential V is investigated in Hérau and Nier [HN04].4.2Parametrix in the degenerate caseIn this section we would like to introduce the parametrix technique which is possible to perform evenunder Hölder continuity assumptions on the coefficients.The unboundedness of the first order term imposes a more subtle strategy than in non-degeneratecase for the choice of the frozen Gaussian density.

We have to take into consideration the "geometry"of the deterministic differential equation associated to the first order terms of the operator. In otherwords, the corresponding flow must appear in the frozen density.We would like to keep considering the model similar to (4.7) but with Hölder continuity assumptions for coefficients instead of Lipschitz, namely, we consider Rd × Rd −valued processes that followthe dynamics:(dXt = b(Xt , Yt )dt + σ(Xt , Yt )dWt ,(4.9)dYt = Xt dt, t ∈ [0, T ],where b : R2d → Rd , σ : R2d → Rd ⊗ Rd are bounded coefficients that are Hölder continuous in space(this condition will be possibly relaxed for the drift term b) and W is a Brownian motion on somefiltered probability space (Ω, F, (Ft )t≥0 , P). In (4.9), T > 0 is a fixed deterministic final time.

Also,a(x, y) := σσ ∗ (x, y) is assumed to be uniformly elliptic.We point out that those assumptions (specified below) are actually sufficient to guarantee weakuniqueness for the solution of equation (4.9), see Remark 4.2.1.4.2.1AssumptionsFor better readability we now repeat the main assumptions of this Chapter, which we have introducedin Chapter 1.(AD1) (Boundedness of the coefficients).The components of the vector-valued function b(x, y) and the matrix-valued function σ(x, y) arebounded measurable.

Specifically, there exists a constant K s.t.sup|b(x, y)| +(x,y)∈R2dsup|σ(x, y)| ≤ K.(x,y)∈R2d(AD2) (Uniform Ellipticity).The matrix a := σσ ∗ is uniformly elliptic, i.e. there exists Λ ≥ 1, such that ∀(x, y, ξ) ∈ (Rd )3 ,Λ−1 |ξ|2 ≤ ha(x, y)ξ, ξi ≤ Λ|ξ|2 .(AD3) (Hölder continuity in space).For some γ ∈ (0, 1] , 0 < κ < ∞ it holds,|b(x, y) − b(x0 , y 0 )| + |σ(x, y) − σ(x0 , y 0 )|γγ/3≤ κ |x − x0 | + |y − y 0 |.Observe that the last condition also readily gives, thanks to the boundedness of σ that the diffusionmatrix a is also uniformly γ and γ/3 -Hölder continuous with respect to the variables x and yrespectively.We say that assumption (AD) holds when conditions (AD1)-(AD3) are in force.49Remark 4.2.1.

We point out that (AD) actually guarantees the well posedness of the martingaleproblem for the generator associated with the SDE (4.9) which in turns imply weak well-posednessfor (4.9). If b = 0 this readily follows from [Men11]. The weak well-posedness would in fact holdfor any γ1 , γ2 ∈ (0, 1], meant to be the respective Hölder continuity indexes for the variables x, y,in (A3) (see e.g.

Theorem 2.1 therein). Similarly, the well posedness would still hold for (4.9) forany bounded measurable b. The key point in the approach of [Men11] is indeed to have a so-calledsmoothing effect in time of an underlying “parametrix" kernel (introduced in the current work indefinition (4.18) below), which precisely holds for bounded drifts in the non-degenerate component(see again Theorem 2.1 in [Men11] and controls in Lemma 4.2.1 below).We will denote, from now on, by C a constant depending on the parameters appearing in (AD)and T . We reserve the notation c for constants that only depend on (AD) but not on T .

The valuesof C, c may change from line to line.4.2.2Parametrix expansion. DiffusionAlthough we have a lot in common with the parametrix technique presented in Chapter 2, there arealso some special properties which we need due to the structure of (4.9).Namely, the first step of the parametrix for degenerate diffusions also consists in approximating0 0the transition density p(T, (·, ·), (x0 , y 0 )) by a known Gaussian density p̃T,(x ,y ) (T, (·, ·), (x0 , y 0 )). The0 0choice obeys the following idea: in short time, p(T, (·, ·), (x0 , y 0 )) and p̃T,(x ,y ) (T, (·, ·), (x0 , y 0 )) are tobe close. We would like to emphasize that in our case, due to the measurability and boundedness ofthe function b(·, ·) it is just enough to use such a specific form of the Gaussian proxy, where the initialsystem does not depend on the trend function.

Similarly to the non-degenerate case in Chapter 2 wehave not incorporated the dependency on b(·, ·) into the frozen process directly.For non-smooth coefficients in (4.9) but satisfying (AD), it is then possible to use a mollificationprocedure, taking bη (x, y) := b ? ρη (x, y), ση (x, y) := σ ? ρη (x, y), x, y ∈ Rd where ρη is a smoothmollifying kernel and ? stands for the usual convolution operation and η ∈ [0, 1], the case η = 0 bydefinition will correspond to the initial process in (4.9).For mollified coefficients, the existence and smoothness of the density pη for the associated process(Xsη , Ysη ) follows from the Hörmander theorem (see e.g. [Hör67]). Thus, we can apply the parametrixtechnique directly for pη .Fixing the terminal point (x0 , y 0 ) at time T , we finally introduce the Gaussian system of the form:(T,(x0 ,y 0 )dX̃η,t= x + ση (x0 , y 0 − x0 (T − t))dWt ,(4.10)T,(x0 ,y 0 )T,(x0 ,y 0 )dỸη,t= y + X̃η,tdt.T,(x0 ,y 0 )Since the model (4.10) defines a Gaussian process, the transition density of (4.10) (p̃ηexists.Observe that in our settings the SDE (4.10) itself integrates as! Z t 00,(x,y),T,(x0 ,y 0 )X̃η,txx= Rt+Ru Bση (RT −u)dWu ,0,(x,y),T,(x0 ,y 0 )yy0Ỹη,t0(t, (x, y), (x̂, ŷ)))0<t≤T ;(x,y),(x̂,ŷ)∈(4.11)Id×d 0d×dwhere Rt =− the resolvent matrix associated with the linear system and B =tId×d Id×dId×d− the embedding matrix from Rd to R2d .0d×d50In particular, for a fixed t > 0 and a given starting point (x, y) in (4.11), we can write now theexact form of the transition density at time t for the frozen process:00,y )p̃t,(x(t, (x, y), (x0 , y 0 ))η=1d(2π) det(Ctη )1/2 0 0 xxxxexp − 12 h(Ctη )−1 (Rt−),R−i ,tyy0yy0Rt∗where Ctη = 0 Rt−u Bση ση∗ (x0 , y 0 − x0 (t − u))B ∗ Rt−udu.We have already introduced in (4.8) the generator for (4.9) and now it comes to the definition oft,(x0 ,y 0 ),ηthe frozen process (4.11) generator (L̃s)0≤s<t≤T :0 010 002,y ),ηTra(x,y−x(t−s))Dφ(x,y)+ hx, ∇y φ(x, y)i.(4.12)L̃t,(xφ(x,y)=ηxs2The density p̃η then readily satisfies the Kolmogorov Backward equation:t,(x0 ,y 0 ),η0 0p̃η (t − u, (x, y), (x0 , y 0 )) = 0,∂u p̃η (t − u, (x, y), (x , y )) + L̃u0 < u < t, (x, y), (x0 , y 0 ) ∈ R2d ,p̃η (t − u, (·, ·), (x0 , y 0 )) → δ(x0 ,y0 ) (.).(4.13)t−u↓0On the other hand, since the density of (Xsη , Ysη ) is smooth, it must satisfy the Kolmogorovforward equation (see e.g.

Dynkin [Dyn65]). For a given starting point (x, y) ∈ R2d at time 0,∂u pη (u, (x, y), (x0 , y 0 )) − L∗ pη (u, (x, y), (x0 , y 0 )) = 0, 0 < u ≤ t, (x, y) ∈ R2d ,pη (u, (x, y), .) → δ(x,y) (.),u↓0(4.14)where L∗ stands for the adjoint (which is well defined since the coefficients are smooth) of thegenerator L in (4.8).Let us remind for a given c > 0 and for all (x, y), (x0 , y 0 ) ∈ R2d the Kolmogorov-type density,introduced in Chapter 1: 0cd 3d/2|x − x|2|y 0 − y − (x + x0 )t/2|20 0pc,K (t, (x, y), (x , y )) :=exp −c+3,(2πt2 )d4tt3(4.15)which also enjoys the semigroup property, i.e. for any 0 ≤ s < t ≤ T,Zpc,K (s, (x, y), (w, z))pc,K (t − s, (w, z), (x0 , y 0 ))dwdz = pc,K (t, (x, y), (x0 , y 0 )).(4.16)R2dThe subscript K in the notation pc,K stands for Kolmogorov-like equations, and pc,K (t, (x, y), (·, ·))denotes the transition density of!√Z t2Wtc,Kc,Kc,KXt , Yt:= x + 1/2 , y +Xs ds .c0Observe carefully that the density in (4.15) exhibits a multiscale behaviour.

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