Диссертация (1137347), страница 10
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Again, we have to consider these two terms separately.39Term ∆21 H h,ε . We first write from (3.54):∆1 ϕh,ελ (tk , z, y, w)noϕhλ (T h (tk , z, w), T h (tk , z, w)) − ϕhλ (T h (tk , z, w), T0h (tk , y, w)) −="#nohϕhλ (T h (tk , z, w), Tεh (tk , z, w)) − ϕhλ (T h (tk , z, w), T0,ε(tk , y, w))"noh+ ϕhλ (T h (tk , z, w), Tεh (tk , z, w)) − ϕhλ (T h (tk , z, w), T0,ε(tk , y, w))#nohhhhhh− ϕλ (Tε (tk , z, w), Tε (tk , z, w)) − ϕλ (Tε (tk , z, w), T0,ε (tk , y, w))"−nohϕhλ,ε (Tεh (tk , z, w), Tεh (tk , z, w)) − ϕhλ,ε (Tεh (tk , z, w), T0,ε(tk , y, w))#noh− ϕhλ (Tεh (tk , z, w), Tεh (tk , z, w)) − ϕhλ (Tεh (tk , z, w), T0,ε(tk , y, w))=:3X∆1i ϕh,ελ (tk , z, y, w).(3.56)i=1We now state some useful controls for the analysis. Namely, setting:D(tk , z, y, w) := T h (tk , z, w)T h (tk , z, w)∗ − T0h (tk , y, w)T0h (tk , y, w)∗ ,hhDε (tk , z, y, w) := Tεh (tk , z, w)Tεh (tk , z, w)∗ − T0,ε(tk , y, w)T0,ε(tk , y, w)∗ ,we have from (A3) and equation (3.4) for q = +∞ :(|D| + |Dε |)(tk , z, y, w) ≤ c̄(h2 + h3/2 |w| + h(1 ∧ |z − y|)γ |w|2 ),|D − Dε |(tk , z, y, w) ≤ c̄∆ε,γ,∞ (h2 + h3/2 |w| + h(1 ∧ |z − y|)γ |w|2 ).(3.57)From the definition of ϕhλ in (3.51), equation (3.56), the control (3.57) and Lemma 3.1.8, we get:|∆11 ϕh,ελ |(tk , z, y, w)≤ c̄∆ε,γ,∞ψc,c1 (tj − tk , y − (z + λT h (tk , z, w))) 2(h + h3/2 |w| + h(1 ∧ |z − y|)γ |w|2 ).(tj − tk )40(3.58)We would similarly get from Lemma 3.1.8 and (3.57):|∆13 ϕh,ελ |(tk , z, y, w)≤ c̄∆ε,γ,∞ψc,c1 (tj − tk , y − (z + λTεh (tk , z, w))) 2(h + h3/2 |w| + h(1 ∧ |z − y|)γ |w|2 ),(tj − tk )|∆12 ϕh,ελ |(tk , z, y, w)≤ψc,c1 (tj − tk , y − (z + θλT h (tk , z, w) + (1 − θ)λTεh (tk , z, w)))(tj − tk )3/2|(T h − Tεh )(tk , z, w)||Dε |ψc,c1 (tj − tk , y − (z + θλT h (tk , z, w) + (1 − θ)λTεh (tk , z, w)))≤ c̄∆ε,γ,∞(tj − tk )3/2×× (h2 + h3/2 |w| + h(1 ∧ |z − y|)γ |w|2 )(h + h1/2 |w|),(3.59)for some θ ∈ (0, 1), using as well (3.49) and (3.4) for the last inequality.
The point is now to get ridof the transitions appearing in the function ψc,c1 . We separate here the two assumptions at hand.- Under (IG), it suffices to remark that by the convexity inequality |z − y − Θ|2 ≥ 21 |z − y|2 − |Θ|2 ,for all Θ ∈ Rd :c |Θ|2cd/2c |z − y|2ψc,c1 (tj − tk , y − z − Θ) ≤ c1exp−exp.4 tj − tk2 tj − tk(2π(tj − tk ))d/2Now, if Θ is one of the above transitions or linear combination of transitions, we get from (3.49):c |z − y|2(c/2)d/2cψc,c1 (tj − tk , y − z − Θ) ≤ c1exp −exp( K22 |w|2 ),d/24 tj − tk2(2π(tj − tk ))(3.60)up to a modification of c1 observing that h/(tj − tk ) ≤ 1 and with K2 as in (A1).
Since c canbe chosen small enough in the previous controls, up to deteriorating the concentration propertiesin Lemma 3.1.8, the last term can be integrated by the standard Gaussian density fξ appearing in(3.55). We thus derive, from (3.60), (3.58), (3.59) and the definition in (3.56), up to modifications ofc, c1 :|∆1 ϕh,ελ |(tk , z, y, w) ≤|w||z − y|γ |w|2∆ε,γ,∞ hc̄ψc,c1 (tj − tk , z − y) exp(c|w|2 ) 1 ++,tj − tk(tj − tk )1/2which plugged into (3.55) yields up to modifications of c̄, c, c1 :|∆21 H h,ε (tk , tj , z, y)| ≤ c̄∆ε,γ,∞ (1 ∨ T (1−γ)/2 )ψc,c1 (tj − tk , z − y).(tj − tk )1−γ/2(3.61)- Under(IP), we only detail the computations for the off diagonal regime |z − y| ≥ c(tj − tk )1/2 whichis the most delicate to handle.
In this case, we have to discuss according to the position of w w.r.t.y − z. With the notations of (A2), introduce D := {w̄ ∈ Rd : {Λh}1/2 |w̄| ≤ |z − y|/2}. If w ∈ D,41then, still from (3.58), (3.59),h,ε(|∆11 ϕh,ελ | + |∆13 ϕλ |)(tk , z, y, w)≤ c̄∆ε,,γ,∞ψc,c1 (tj − tk , y − z) 2(h + h3/2 |w| + h(1 ∧ |z − y|)γ |w|2 ),(tj − tk )|∆12 ϕh,ελ |(tk , z, y, w)≤ c∆ε,γ,∞ψc,c1 (tj − tk , y − z) 2(h + h3/2 |w| + h(1 ∧ |z − y|)γ |w|2 )(h + h1/2 |w|).(tj − tk )3/2On the other hand, when w 6∈ D we use fξ to make the off-diagonal bound of ψc,c1 (tj − tk , y − z)appear. Namely, we can write:fξ (w) ≤ c≤ c1≤c(1 + |w|)M(1 +1(1 +1|z−y| M −(d+4))h1/2|z−y|)M −(d+4)(tj −tk )1/21(1 + |w|)d+41,(1 + |w|)d+4(3.62)where the last splitting is performed in order to integrate the contribution in |w|3 coming from theupper bound for |∆12 ϕh,ελ | in (3.59).
Plugging the above controls in (3.55) yields:h,ε|∆H21(tk , tj , z, y)| ≤∆ε,γ,∞ Φc,c1 (tj − tk , z − y).(tj − tk )1−γ/2(3.63)We emphasize that in the case of innovations with polynomial decays, the control on the difference ofthe kernels again induces a loss of concentration of order γ in order to equilibrate the time singularity.Term ∆22 H h,ε . This term can be handled with the same arguments as ∆21 H h,ε . For the sake ofcompleteness we anyhow specify how the different contributions appear.
Namely, with the notationsof (3.54) and (3.55):∆2 ϕh,ελ (tk , z, y, w) =1Zdµ0nDDT1 ϕhλ (T0h (tk , y, w) + µ(T h (tk , z, w) − T0h (tk , y, w)), T0h (tk , y, w)),ET h (tk , z, w) − T0h (tk , y, w)Dhhh− DT1 ϕhλ,ε (T0,ε(tk , y, w) + µ(Tεh (tk , z, w) − T0,ε(tk , y, w)), T0,ε(tk , y, w)),EohTεh (tk , z, w) − T0,ε(tk , y, w)=nZ0−nZ011nDdµ DT1 ϕhλ (T0h (tk , y, w) + µ(T h (tk , z, w) − T0h (tk , y, w)), T0h (tk , y, w)), hEoh(T (tk , z, w) − T0h (tk , y, w)) − (Tεh (tk , z, w) − T0,ε(tk , y, w))hDhhhdµ DT1 ϕhλ,ε (T0,ε(tk , y, w) + µ(Tεh (tk , z, w) − T0,ε(tk , y, w)), T0,ε(tk , y, w))i−DT1 ϕhλ (T0h (tk , y, w) + µ(T h (tk , z, w) − T0h (tk , y, w)), T0h (tk , y, w)) ,EohTεh (tk , z, w) − T0,ε(tk , y, w)h,ε=: (∆21 ϕh,ελ + ∆22 ϕλ )(tk , z, y, w).42−3/2In ∆21 ϕh,ε,λ we have sensitivities of order 3 for the density, giving time singularities in (tj − tk )which are again equilibrated by the the multiplicative factor:|T0h (tk , y, w)[T0h (tk , y, w)]∗ |h×|(T h (tk , z, w) − T0h (tk , y, w)) − (Tεh (tk , z, w) − T0,ε(tk , y, w))|≤ c̄(h2 + h3/2 |w| + h|w|2 )∆ε,γ,∞ (h + h1/2 (1 ∧ |z − y|)γ |w|),where the last inequality is obtained similarly to (3.57) using as well (3.4).
The same kind of controlscan be established for ∆22 ϕh,ελ . Anyhow, the analysis of this term leads to investigate the differenceof third order derivatives, which finally yields contributions involving derivatives of order four. Thisis what induces the final concentration loss under (IP), i.e. we need to integrate a term in |w|4 (seealso equation (3.62) in which we performed the splitting of fξ on the off-diagonal region to integratea contribution in |w|3 ).We can thus claim thath,ε(tk , tj , z, y)| ≤|∆H22∆ε,γ,∞ Φc,c1 (tj − tk , z − y).(tj − tk )1−γ/2Plugging the above control and (3.63) (or (3.61) under (IG) into (3.55) we derive:|∆H2h,ε (tk , tj , z, y)| ≤∆ε,γ,∞ Φc,c1 (tj − tk , z − y),(tj − tk )1−γ/2which together with (3.53) and the decomposition (3.52) completes the proof.From Lemmas 3.1.8 and 3.1.9 the proof of Theorem 3.1.2 is achieved, under (IG), following thesteps of Lemmas 3.1.6 and 3.1.7, using the Hölder inequalities for the differences of the drift termsfor q ∈ (d, +∞).The point is that we want to justify the following inequality under (IP) and q = +∞:|(p̃h ⊗h H h,(r)− p̃hε ⊗h Hεh,(r) )(ti , tj , x, y)|×(3.64)r+1 γ rΓ( 2 ))c1 }Γ(1 + r γ2 )rγcd/2y−xQ(tj − ti ) 2 .M −(d+5+γ)d/21/2(tj − ti )(tj − ti ) /c≤ (r + 1)∆ε,γ,∞{(1 ∨ T(1−γ)/2The only delicate point, w.r.t.
the analysis performed for diffusions, consists in controlling theconvolutions of the densities with polynomial decay. To this end, we can adapt a technique used byKolokoltsov [Kol00] to investigate convolutions of "stable like" densities. Set m := M − (d + 5 + γ)dand denote for all 0 ≤ i < j ≤ N, x ∈ Rd by qm (tj − ti , x) := (tj −tc i )d/2 QM −(d+5+γ) (tj −tix)1/2 /cthe density with polynomial decay appearing in Lemmas 3.1.8 and 3.1.9. Let us consider for fixedi < k < j, (x, y) ∈ (Rd )2 the convolution:ZIt1k (ti , tj , x, y) :=dzqm (tk − ti , z − x)qm (tj − tk , y − z).(3.65)Rd- If |x−y| ≤ c(tj −ti )1/2 (diagonal regime for the parabolic scaling), it is easily seen that one of the twodensities in the integral (3.65) is homogeneous to qm (tj − ti , y − x).
Namely, if (tk − ti ) ≥ (tj − ti )/2,qm (tk − ti , z − x) ≤cd/2 cm(tk −ti )d/2≤(2c)d/2 cm(tj −ti )d/2≤ c̃qm (tj − ti , y − x). Thus,Z1Itk (ti , tj , x, y) ≤ c̃qm (tj − ti , y − x)dzqm (tj − tk , y − z) = c̃qm (tj − ti , y − x).Rd43If (tk − ti ) < (tj − ti )/2, the same operation can be performed taking qm (tj − tk , y − z) out of theintegral, observing again that in that case qm (tj − tk , y − z) ≤ c̃qm (tj − ti , y − x).- If |x − y| > c(tj − ti )1/2 (off-diagonal regime), we introduce A1 := {z ∈ Rd : |x − z| ≥ 21 |x − y|},A2 := {z ∈ Rd : |z − y| ≥ 21 |x − y|}. Every z ∈ Rd belongs at least to one of the {Ai }i∈{1,2} .
Letus assume w.l.o.g. that z ∈ A2 . Then |z − y| ≥ 2c (tj − ti )1/2 ≥ 2c (tj − tk )1/2 so that the densityqm (tj − tk , y − z) is itself in the off-diagonal regime. Write:ZZdzqm (tk − ti , z − x)qm (tj − tk , y − z) ≤dzqm (tk − ti , z − x)A2A2m≤(m−d)/2cm 2 (tj − ti )|x − y|mcm (tk − ti )(m−d)/2|z − y|mZdzqm (tk − ti , z − x) ≤ c̄qm (tj − ti , y − x),A2recalling that, under (IP), m > d for the last but one inequality.
The same operation could beperformed on A1 .We have thus established that, there exist c̄ > 1 s.t. for all 0 ≤ i < k < j, (x, y) ∈ (Rd )2 :It1k (ti , tj , x, y) ≤ c̄qm (tj − ti , y − x).From the controls of Lemma 3.1.9 and following the strategy of Lemma 3.1.7, we will be led to considerconvolutions of the previous type involving Γ functions. The above strategy thus yields (3.64) byinduction.44Chapter 4Degenerate diffusions4.14.1.1IntroductionHypoellipticityWe would like to study the development and applications of the parametrix technique for a certainclass of degenerate diffusions.We will specifically focus on the Kolmogorov like diffusions (named after the seminal work ofKolmogorov [Kol34] which later on inspired Hörmander’s general theory of hypoellipticity [Hör67]).Discussing the hypoellipticity concepts, we would like to introduce the class of hypoelliptic differentialoperators.A partial differential operator L with C ∞ coefficients in an open set Ω ⊂ Rd is called hypoelliptic(on Ω) in case for every distribution u in Ω we have that u is a C ∞ function in every open setwhenever Lu is a C ∞ function.Although necessary and sufficient conditions for constant coefficients for L to be hypoelliptichave been known for quite some time before [Hör67] , see e.g.
[Hör63], the technique obviously wasnot adapted for the general case. For instance, Kolmogorov [Kol34] constructed an example of thefundamental solution of the equation:∂2u∂u ∂u+x−= f,2∂x∂y∂t(4.1)which is a C ∞ function outside the diagonal. This means, the corresponding operator is hypoellipticwhich was not in the framework of the existing sufficient conditions, derived before [Hör67].Inspired by [Kol34], Hörmander in his paper [Hör67] studied a characterization of hypoellipticsecond order differential operators L with real C ∞ coefficients. Namely, if A0 , .
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