Abstract
We establish strong uniqueness for a class of degenerate SDEs of weak Hörmander type under suitable Hölder regularity conditions for the associated drift term. Our approach relies on the Zvonkin transform which requires to exhibit good smoothing properties of the underlying parabolic PDE with rough, here Hölder, drift coefficients and source term. Such regularizing effects are established through a perturbation technique (forward parametrix approach) which also heavily relies on appropriate duality properties on Besov spaces. For the method employed, we exhibit some sharp thresholds on the Hölder exponents for the strong uniqueness to hold.
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Notes
We will use throughout the paper the notation \([\![\cdot ,\cdot ]\!]\) for integer intervals.
For simplicity reasons, we restrain our considerations to rotationally invariant stable processes with generator 1/2 the usual fractional Laplacian.
By “small enough” we mean that there exists a time \({\mathcal {T}}>0\) depending only on known parameters in (A) s.t. for any \(T\le {\mathcal {T}} \) the statement of the theorem holds.
Pay attention that this is not the case for L whose coefficients do not have the required smoothness in (\(\mathbf{T}_{\beta }\)) to compute the corresponding Lie brackets.
Meaning that the freezing curve \({\varvec{\theta }}\) solves the corresponding ODE associated with (1.1) in a forward form.
Observe that for this contribution, from (2.25) the bound would hold for any freezing parameter \({\varvec{\xi }}\). We choose here to take \({\varvec{\xi }}={\mathbf {x}}\) for the compatibility with the other terms \((D_{{\mathbf {x}}_1}H_{l,{k}}^{{{\varvec{\xi }}}} (s,{\mathbf {x}}))_{{k}\in \{2,3\}}\) for which this specific choice is indeed needed.
Since \(I_{1,3}^{\varvec{\xi }}(s,{\mathbf {x}})=0 \) for all \({\mathbf {x}}\in {\mathbb {R}}^{nd} \), the contribution is clearly 0 for \(l=1\).
Note that in this case we have that for any s in (t, T] the off-diagonal regime holds.
Recall indeed that what we are able to control is precisely the Hölder moduli of the derivatives \(D_{{\mathbf {y}}_m}u(s,\cdot ) \) w.r.t. the variables \(\ell \le m\).
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Acknowledgements
We would like to thank the anonymous referees for their careful reading and comments which improved a lot the presentation of the manuscript. For the first author, this work has been partially supported by the ANR project ANR-15-IDEX-02. For the third author, the article was prepared within the framework of a subsidy granted to the HSE by the Government of the Russian Federation for the implementation of the Global Competitiveness Program.
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Appendices
A Sensitivity results for the mean: Proof of Lemma 16
In order to prove Lemma 16, we first need to establish some controls on the sensitivity of the flows, see Lemma 17 below. Those results are obtained under the sole assumption (A) and remain valid for the mollification procedure of the coefficients considered in (AM). We will then proceed to the final proof of Lemma 16 in Section A.2.
For our analysis, we now introduce the spatial homogeneous distance, which basically reflects the various scales of the system already seen e.g. in Proposition 5. Namely, for \(({\mathbf {x}},{\mathbf {x}}')\in {\mathbb {R}}^{nd} \), we define:
The distance is homogeneous in the sense that, for any \(\lambda \!>\!0,\ {\mathbf {d}}(\lambda ^{- 1/2}{\mathbb {T}}_\lambda {\mathbf {x}},\lambda ^{-1/2}{\mathbb {T}}_\lambda {\mathbf {x}}')=\lambda ^{1/2}{\mathbf {d}}({\mathbf {x}},{\mathbf {x}}') \).
1.1 A.1 A first sensitivity result for the flow
Lemma 17
(Control of the flows). Under (A), there exists \(C:=C(\mathbf{(A)},T)\) s.t. for all spatial points \(({\mathbf {x}},{\mathbf {x}}')\in ({\mathbb {R}}^{nd})^{2},\ {{\mathbf {d}}({\mathbf {x}},{\mathbf {x}}')\le 1}, 0\le t<s\le T{\le 1}\) and \(i \in [\![1,n]\!]\):
The flow, \({\varvec{\theta }}_{s,t}\) is, somehow, locally “almost” Lipschitz continuous in space w.r.t. the homogeneous distance \({\mathbf {d}}\), up to a time additive term. This time contribution is a consequence of the non-Lipschitz continuity of the drift \({\mathbf {F}}\). The analysis of the sensitivity of the flow was already done for \({\mathbf {F}}\) Lipschitz continuous in Proposition 4.1 of [48], and Appendix A.1 in [12] with different Hölder regularity of \({\mathbf {F}}\). Actually, as we consider a smoother drift than in [12], the following lemma can be seen as a by-product of Lemma 12 therein.
For the sake of completeness, we provide the corresponding, and more direct, analysis below.
Proof
The analysis mainly relies on Grönwall type arguments coupled with suitable mollification techniques, because \({\mathbf {F}}\) is not Lipschitz continuous, and appropriate Young inequalities in order to make the intrinsic scales associated to the spatial variables appear.
Let \(\delta \in {\mathbb {R}}^n\) be the vector whose entries \(\delta _{i} >0\) correspond to the mollification parameter of the drift \({\mathbf {F}}_i \) in the \(i^{\mathrm{th}} \) variable for \(i\in [\![2,n]\!]\). Namely, for all \(\ v\in [0,T],\ {\mathbf {z}}\in {\mathbb {R}}^{nd} \), \(i\in [\![2,n]\!]\), we define
with \(\rho _{\delta _{i}}(w):=(1/\delta _{i}^{d})\rho \left( w/\delta _{i} \right) \) where \( \rho :{\mathbb {R}}^{d}\rightarrow {\mathbb {R}}_{+}\) is a usual mollifier, namely \(\rho \) has compact support and \(\int _{{\mathbb {R}}^{d}} \rho ({\mathbf {z}})d{\mathbf {z}}=1 \). Eventually, we write \({\mathbf {F}}^\delta (v,{\mathbf {z}}):=({\mathbf {F}}_1 (v,{\mathbf {z}}),{\mathbf {F}}_2^\delta (v,{\mathbf {z}}), {\ldots },{\mathbf {F}}_n^\delta (v,{\mathbf {z}})) \). With a slight abuse of notation in the previous definitions, since the first component \({\mathbf {F}}_1\) is not mollified. The sublinearity of \({\mathbf {F}}_1\) is actually enough to obtain the desired control.
To be at the good current time scale for the contributions associated with the mollification, we pick \(\delta _{i}\) in order to have \(C:=C(\mathbf{(A)},T)>0\) s.t. for all \({\mathbf {z}}\in {\mathbb {R}}^{nd}\), \(u \in [t,s]\):
By the previous definition of \({\mathbf {F}}^\delta \) in (A.2), identity (A.3) will be implied from \((\mathbf {T})_{\beta }\) if:
Hence, we choose from now on, for each \(i\in [\![2,n ]\!]\):
Next, let us control the last components of the flow. By the definition of \({\varvec{\theta }}_{s,t}\) in (2.4), we get:
observing for the last inequality that since \(\beta _n>(2n-2)/(2n-1)\) and \(\delta _n \) is meant to be small, \(\delta _{n}^{\beta _n}\le \delta _n^{(2n-2)/(2n-1)} \).
Hence by Grönwall’s lemma, we get:
using (A.5) for the last inequality. For the \((n-1)^\mathrm{{th}}\) component, the situation is quite different in the sense that we have to handle the non-Lipschitz continuity of \({\mathbf {F}}^\delta _{n-1}\) in its \(n^\mathrm{{th}}\) variable. Write:
from our choice of \(\delta _{n-1} \) in (A.5) for the second inequality. We also exploited for the last inequality that, since under (\(\mathbf{T}_{\beta }\)), \(\beta _n>(2n-2)/(2n-1) \), \({\beta _n(n-\frac{1}{2})}>n-1>n-3/2 \) and \(0\le t<s\le T \) where T is small, then \((v-t)^{\beta _n(n-1/2)}\le (v-t)^{n-3/2} \). Also, since \({\mathbf {d}}({\mathbf {x}},{\mathbf {x}}')\le 1 \), the same arguments yield \(|({\mathbf {x}}-{\mathbf {x}}')_n|^{\beta _n}\le |({\mathbf {x}}-{\mathbf {x}}')_n|^{(2n-2)/(2n-1)}\le |({\mathbf {x}}-{\mathbf {x}}')_n|^{(2n-3)/(2n-1)} \).
From (A.7), which still holds true replacing s by any \(\tilde{s}\in [t,s] \), we deduce that taking the supremum over \(\tilde{s}\in [t,s] \):
Taking then the supremum in \(w\in [t,s]\) in the above integral, we obtain:
From Young’s inequality we now derive:
recalling for the last inequality that \(s-t\le 1 \), and since \(\beta _n>(2n-2)/(2n-1), 1-\beta _n<1/(2n-1)\), we also have \((s-t)^{1/(1-\beta _n)}<(s-t)^{2n-1}<(s-t)^{n- 3/2} \). Plugging the above control into (A.8), we obtain up to a modification of C:
Plugging the above inequality into (A.6) we derive:
using again the Young inequalities \(|({\mathbf {x}}-{\mathbf {x}}')_n|^{(2n-3)/(2n-1)}(s-t) \le C(|({\mathbf {x}}-{\mathbf {x}}')_n|+(s-t)^{n- 1/2})\) and \(|({\mathbf {x}}-{\mathbf {x}}')_{n-1}|(s-t)\le C\big (|({\mathbf {x}}-{\mathbf {x}}')_{n-1}|^{(2n-1)/(2n-3)}+(s-t)^{n- 1/2}\big ) \) for the last inequality. Iterating the procedure, we get:
Anagolously, for \(i\in [\![2,n]\!]\), we obtain:
We explicitly see from the above equation that each entry of the difference of the starting points appears at its intrinsic scale for the homogeneous distance \({\mathbf {d}}\) introduced in (A.1).
Remark 6
Observe that equations (A.10) and (A.11) are available for any fixed time \(s\in [t,T] \).
The first term, i.e. for \(i=1 \) is controlled slightly differently. In other words, for any \(\tilde{s}\in [t,s] \), write:
which in turn implies from (A.11), Remark 6 and convexity inequalities:
Recalling \(\beta _j>(2j-2)/(2j-1) \) and \(0\le t<s\le T\le 1 \), \({\mathbf {d}}({\mathbf {x}},{\mathbf {x}}')\le 1 \), we get:
Write now from the Young inequality:
We eventually derive from (A.12) that:
which gives the statement for \(i=1\). Plugging now this inequality into (A.11), we get for each \(i\in [\![2,n ]\!]\):
using again for the last identity the Young inequality to derive that \((s-t)^{i-1}{\mathbf {d}}({\mathbf {x}},{\mathbf {x}}') \le C\big ((s-t)^{i-1/2}+{\mathbf {d}}^{2i-1} ({\mathbf {x}},{\mathbf {x}}') \big ) \). The proof is complete. \(\square \)
1.2 A.2 Sensitivity results for the mean: final proof of Lemma 16
Again through the analysis, we assume w.l.o.g. that \({\mathbf {d}}({\mathbf {x}},{\mathbf {x}}')\le 1\). We choose here to prove, for the sake of completeness, a slightly more general result than the one stated in the Lemma. Namely, we do not restrict to the case where \({\mathbf {x}},{\mathbf {x}}' \) only differ on their \(i^\mathrm{th}\) component but consider arbitrary given points \({\mathbf {x}},{\mathbf {x}}'\in {\mathbb {R}}^{nd} \) to emphasize how the specific structure of the drift leads to the control of the linearization error in terms of the homogeneous distance.
The control is done with a distinction of two contributions to handle.
By the proxy definition in (2.3), we deduce that the mean value of \(\tilde{{\mathbf {X}}}^{m, {\varvec{\xi }}}_v\), \( {\mathbf {m}}_{v,t}^{{\varvec{\xi }}}\) is s.t.
The sub-triangular structure of \(D{\mathbf {F}}\) yields that for each \(i \in [\![2,n]\!]\):
Also, since \({\mathbf {m}}_{v,t}^{{\mathbf {x}}}({\mathbf {x}}')_1= {\mathbf {x}}_1'+\int _t^s dv {\mathbf {F}}_1(v,{\varvec{\theta }}_{v,t}({\mathbf {x}}))\), so we obtain that \([{\mathbf {m}}_{v,t}^{\mathbf {x}}({\mathbf {x}}')_{1}-{\varvec{\theta }}_ {v,t}({\mathbf {x}})_{1}]={\mathbf {x}}'_1-{\mathbf {x}}_1 \), we then obtain by iteration that:
with the convention that for \(i=1\), \( \sum _{k=2}^i=0\). From the above control, equation (A.13) and the dynamics of the flow, and because the starting points are the same, the contributions involving differences of the spatial points (\({\mathbf {x}}'-{\mathbf {x}})\) or flows only appear in iterated time integrals, we obtain:
We derive from the previous Lemma 17 (control of the flows) recalling again that \(\beta _j>(2j-2)/(2j-1) \) and \({\mathbf {d}}({\mathbf {x}},{\mathbf {x}}')\le 1, 0\le t<s\le T\le 1 \):
In particular, for \(s=t_0=t+c_0{\mathbf {d}}^2({\mathbf {x}},{\mathbf {x}}')\) with \(c_0<1\), the previous equation yields:
using again \({\mathbf {d}}({\mathbf {x}},{\mathbf {x}}') \le 1\) for the middle term. After summing and by convexity inequalities, we eventually deduce:
This concludes the proof of Lemma 16. \(\square \)
B Parametrix expansion with different freezing points
In this section we show how the parametrix expansion (3.17) involving different freezing points can be derived. This can actually be done from the Duhamel formulation up to an additional discontinuity term. Restarting from (2.26) we can indeed rewrite from the Markov property that for given \((t,{\mathbf {x}}')\in [0,T]\times {\mathbb {R}}^{nd} \) and any \( r\in (t,T], {\varvec{\xi }}'\in {\mathbb {R}}^{nd}\):
Differentiating the above expression in \(r\in (t,T]\) yields for any \({\varvec{\xi }}'\in {\mathbb {R}}^{nd}\):
Denoting by \(t_0\in (t,T]\) the time at which we change the freezing point and integrating (B.2) on \([t,t_0]\) for a first given \({\varvec{\xi }}' \) and between \([t_0 ,T]\) with a possibly different \(\tilde{{\varvec{\xi }}}'\) yields:
Recalling that \(u(T,\cdot )=0 \) (terminal condition), the above equation rewrites:
We see that for \({\varvec{\xi }}' \ne \tilde{{\varvec{\xi }}} '\) we have an additional discontinuity term deriving from the change of freezing point along the time variable. The above equation is precisely (3.17).
C Auxiliary results concerning the multi-scale Gaussian densities, their derivatives and some related objects
In order to be self contained, we gather in this section the proof of some results related to the Gaussian dynamics in (2.3). Namely, we provide a complete proof of Proposition 4 and some auxiliary related results used throughout the previous proofs. We here freely use the notations of Sect. 2.1.
1.1 C.1 About the objects appearing in the multi-scale density
1.1.1 C.1.1 Good scaling properties of the covariance matrix: proof of Proposition 4
We recall the correspondence between the notations of [22] and those of the current article.
-
Notations and Assumptions from [22]. Consider the Gaussian process with dynamics
$$\begin{aligned} d{\mathbf {G}}_t ={\mathbf {L}}_t {\mathbf {G}}dt +B\Sigma _t dW_t \end{aligned}$$(C.1)where \( (\Sigma _t)_{t\in [ 0,T]}\) is a measurable deterministic \({\mathbb {R}}^d\otimes {\mathbb {R}}^d \) valued family s.t. \(\mathbf {A}_t:= \Sigma _t \Sigma _t^* \) has uniformly non-degenerate spectrum, i.e. there exists \(\Lambda \ge 1 \) s.t. for any \(t\in [0,T],\ \mathrm{Spec} (\mathbf {A}_t)\in [\Lambda ^{-1},\Lambda ] \), and the measurable deterministic \({\mathbb {R}}^{nd}\otimes {\mathbb {R}}^{nd}\) valued family \(({\mathbf {L}}_t)_{t\in [ 0,T]} \) is such that for any \(t\in [0,T] \):
$$\begin{aligned} {\mathbf {L}}_t=\left( \begin{array}{cccccc} \mathbf {0}_{d,d}&{} \cdots &{}\cdots &{} \cdots &{}\mathbf {0}_{d,d}\\ \alpha _t^{1} &{} \mathbf {0}_{d,d} &{}\cdots &{} \cdots &{} \mathbf {0}_{d,d}\\ \mathbf {0}_{d,d}&{}\alpha _t^2&{} \ddots &{}\mathbf {0}_{d,d}\\ \vdots &{} \ddots &{} \ddots &{}\ddots &{} \vdots \\ \mathbf {0}_{d,d}&{} \cdots &{} \mathbf {0}_{d,d}&{} \alpha _t^{n-1}&{} \mathbf {0}_{d,d} \end{array}\right) , \end{aligned}$$(C.2)where the \((\alpha _{t}^{i})_{i\in [\![1, n-1]\!]} \) are \({\mathbb {R}}^d\otimes {\mathbb {R}}^d \) valued.
This is special case of assumption \((\mathbf {A})^{\mathrm{linear}}\) in [22]. Proposition 3.4 of that reference states that, whenever for all \(i\in [\![1,n-1]\!]\) and \(t\in [0,T] \), \(\alpha _t^i \) belongs to \({\mathcal {E}}_i \) (closed convex subset of \(GL_d({\mathbb {R}}) \)), there exists a constant \(c\ge 1\) depending on \({\mathcal {E}}_i \), \(\Lambda , \kappa \) s.t. \(\max _{i\in [\![1,n-1]\!]}\sup _{t\in [0,T]}|\alpha _t^i|\le \kappa \), n, d such that the Gaussian process \(({\mathbf {G}}_t)_{t\in [0,T]} \) introduced in (C.1) satisfies a good scaling property with constant c in the sense of Definition 3.2 of [22]. Precisely, denoting by \(\big ({\mathbf {R}}(s,t)\big )_{0\le t,s\le T } \) the resolvent matrix associated with \(({\mathbf {L}}_t)_{t\ge 0} \), the covariance matrix
$$\begin{aligned} {\mathbf {K}}_t=\int _0^tds {\mathbf {R}}_{t,s}B \mathbf {A}_s B^*{\mathbf {R}}_{t,s}^* \end{aligned}$$of the random variable \({\mathbf {G}}_t \) satisfies that for any \({\mathbf {y}}\in {\mathbb {R}}^{nd} \),
$$\begin{aligned} c^{-1}t^{-1}|{\mathbb {T}}_t {\mathbf {y}}|^2\le \langle {\mathbf {K}}_t{\mathbf {y}},{\mathbf {y}}\rangle \le c t^{-1}|{\mathbb {T}}_t {\mathbf {y}}|^2, \end{aligned}$$or equivalently, for all \(t\in (0,T] \):
$$\begin{aligned} c^{-1}t|{\mathbb {T}}_t^{-1} {\mathbf {y}}|^2\le \langle {\mathbf {K}}_t^{-1} {\mathbf {y}},{\mathbf {y}}\rangle \le c t|{\mathbb {T}}_t^{-1} {\mathbf {y}}|^2. \end{aligned}$$ -
Derivation of the Proposition 4 from the previous results of [22]. From the dynamics of the process \((\tilde{{\mathbf {X}}}_v^{(\tau ,{\varvec{\xi }})})_{v\in [t,T]} \) given in (2.3) and starting from \({\mathbf {x}}\) at time t, see also the associated integrated expression in (2.5), it can be seen that the covariance matrix of the random variable \(\tilde{{\mathbf {X}}}_v^{(\tau ,{\varvec{\xi }})} \) writes, with the notations of Sect. 2.1:
$$\begin{aligned} \tilde{{\mathbf {K}}}_{v,t}^{(\tau ,{\varvec{\xi }})}:=\int _t^vdu {\tilde{{\mathbf {R}}}}^{(\tau ,{\varvec{\xi }})} (v,u) Ba(u,{\varvec{\theta }}_{u,\tau }({\varvec{\xi }}))B^*{\tilde{{\mathbf {R}}}}^{(\tau ,{\varvec{\xi }})}(v,u)^*, \end{aligned}$$as given in the statement of Proposition 4. This covariance matrix also corresponds to the one of a Gaussian process with dynamics (C.1) setting for fixed \(0\le t<s\le T \) and for any \(r\in [0,s-t]\)
$$\begin{aligned} {{\mathbf {L}}}_r=\left( \begin{array}{ccccc} {\mathbf {0}}_{d,d} &{} \cdots &{} \cdots &{}\cdots &{} {\mathbf {0}}_{d,d}\\ D_{{\mathbf {z}}_1}{\mathbf {F}}_{2}(t+r,{\varvec{\theta }}_{t+r,\tau }({\varvec{\xi }})) &{} {\mathbf {0}}_{d,d} &{}\cdots &{}\cdots &{}{\mathbf {0}}_{d,d}\\ {\mathbf {0}}_{d,d} &{} D_{{\mathbf {z}}_2} {\mathbf {F}}_{3}(t+r,{\varvec{\theta }}_{t+r,\tau }^{2:n}({\varvec{\xi }}))&{} {\mathbf {0}}_{d,d}&{} {\mathbf {0}}_{d,d} &{}\vdots \\ \vdots &{} {\mathbf {0}}_{d,d} &{} \ddots &{} \ddots &{}\vdots \\ {\mathbf {0}}_{d,d} &{}\cdots &{} {\mathbf {0}}_{d,d} &{} D_{{\mathbf {z}}_{n-1}}{\mathbf {F}}_{n}(t+r,{\varvec{\theta }}_{t+r,\tau }^{n-1:n}({\varvec{\xi }})) &{} {\mathbf {0}}_{d,d} \end{array}\right) , \end{aligned}$$(C.3)and
$$\begin{aligned} \Sigma _r=\sigma (t+r,{\varvec{\theta }}_{t+r,\tau }({\varvec{\xi }})). \end{aligned}$$Since the resolvent \(({\mathbf {R}}(r,v))_{v\in [0,s-t]}\) associated with \(({\mathbf {L}}_r)_{r\in [0,s-t]} \) writes \({\mathbf {R}}(r,v)= \tilde{{\mathbf {R}}}^{(\tau ,{\varvec{\xi }})}(t+r,t+v)\), one readily derives that the covariance matrix \( {\mathbf {K}}_{s-t}\) of \({\mathbf {G}}_{t-s}={\mathbf {R}}(t-s,0){\mathbf {x}}+\int _{0}^{t-s}{\mathbf {R}}(t-s,r)B\Sigma _r dW_r\) coincides with \(\tilde{{\mathbf {K}}}_{s,t}^{(\tau ,{\varvec{\xi }})} \). Since \({\mathbf {K}}_{s-t}=\tilde{{\mathbf {K}}}_{s,t}^{(\tau ,{\varvec{\xi }})} \) satisfies a good scaling property, this proves Proposition 4.
1.1.2 C.1.2 Scaling Properties of the resolvent
We here aim at proving the following control. There exists \(\hat{C}_1:=\hat{C}_1(\mathbf{(A)},T)\) s.t.
We will proceed following the arguments of Lemma 3.6 in [22] specifying how they can apply to the current setting. Let us restart from the previous linear Gaussian dynamics. Namely, for fixed \(t\in [0,T]\), let \(({\mathbf {G}}_s)_{s\in [0,t]}\) be as in (C.1) and introduce the scaled process \(\hat{{\mathbf {G}}}_s^t=t^{1/2}{\mathbb {T}}_t^{-1} {\mathbf {G}}_{s t} ,\ s\in [0,1]\). Intuitively, from the previously established good scaling property of \({\mathbf {G}}\), all the components of the process \(\hat{{\mathbf {G}}}^t \) actually evolve at a macro scale, i.e. its covariance matrix is of order one at time \(s=1\).
It is then easily checked that \(\hat{{\mathbf {G}}}_s^t \) satisfies (C.1), i.e. \(d\hat{{\mathbf {G}}}_s^t =\hat{{\mathbf {L}}}_s^t \hat{{\mathbf {G}}}_s^t ds+B\hat{\Sigma }_s^t d\hat{W}_s^t\) with:
and importantly that from the specific subdiagonal structure considered in (C.3) \(\hat{{\mathbf {L}}}_{s}^t= {\mathbf {L}}_{st}\). In any case, we obtain \(|\hat{{\mathbf {L}}}_{s}^t| \le (1\vee T^n)\kappa \) as soon as \(\sup _{s\in [0,1] }| {\mathbf {L}}_{st}| \le \kappa \).
It is then clear that denoting by \(\hat{{\mathbf {R}}}^t \) the resolvent associated with \( (\hat{{\mathbf {L}}}_{s}^t)_{{s\in [0,1]}}\) it holds that there exists \(\hat{C}_1:=\hat{C}_1(\mathbf{(A)},T) \) s.t. for all \(s_0,s_1\in [0,1] \), \(|\hat{{\mathbf {R}}}^t(s_1,s_0)|\le \hat{C}_1 \). On the other hand, direct computations also yield that
where \({\mathbf {R}}\) stands for the resolvent associated with \({\mathbf {L}}\). The final bound (C.4) on the rescaled resolvent associated with the frozen process \(((s-t)^{1/2}{\mathbb {T}}_{s-t}^{-1}\tilde{{\mathbf {X}}}_v^{(\tau ,{\varvec{\xi }}), (t,{\mathbf {x}})})_{v\in [t,s]}\) is eventually derived from the same previous correspondence exhibited to prove the good scaling property of Proposition 4 in the previous paragraph.
1.2 C.2 Proof of Lemma 14
This Section is dedicated to the proof of the Lemma 14 about the Hölder controls for the derivatives of the frozen densities in the diagonal-diagonal regime, i.e. when \({\mathbf {x}},{\mathbf {x}}' \in {\mathbb {R}}^{nd}\) only differ in their \(i^\mathrm{th} \), d-dimensional component and \(s\in {\mathcal {S}}_i^c \), i.e. \(c_0|({\mathbf {x}}-{\mathbf {x}})_i|^{\frac{2}{2i-1}}\le s-t\le T-s \).
Write:
From (2.15) in Proposition 5 we thus derive:
Now, from the definition of \(\hat{p}_{C^{-1}}\) in (2.15), recalling as well from (2.7) that \({\mathbf {x}}\mapsto {\mathbf {m}}_{s,t}^{{\varvec{\xi }}}({\mathbf {x}}):={\mathbf {m}}_{s,t}^{(t,{\varvec{\xi }})}({\mathbf {x}})\) is affine, we get:
Using the rescaling arguments of the proof of Proposition 5 on the resolvent (see equation (2.18)), we then get \((s-t)^{ 1/2}|{\mathbb {T}}_{s-t}^{-1} {\tilde{{\mathbf {R}}}^{(t,{\varvec{\xi }})}(s,t)}({\mathbf {x}}-{\mathbf {x}}')| \le C (s-t)^{ 1/2}|{\mathbb {T}}_{s-t}^{-1}({\mathbf {x}}-{\mathbf {x}}')|=C(s-t)^{-i+1/2} |({\mathbf {x}}'-{\mathbf {x}})_i|\le C\), from the very definition of \({\mathcal {S}}_i^c \). Hence,
Recalling that for given \({\mathbf {x}},{\mathbf {x}}' \) in the global diagonal regime and \(s\in {\mathcal {S}}_i^c, |{\mathbf {x}}_i-{\mathbf {x}}_i'|\le c_0^{-(i-\frac{1}{2})}(s-t)^{i-\frac{1}{2}}\), we derive
which from (C.6) and (C.7) precisely gives (3.27) recalling that \(\alpha _i=1/(2i-1) \) (see the definition in the statement of Lemma 10).
D Proof of Lemma 12 through reverse Taylor formula
Proof of Lemma 12
We assume here, for the sake of simplicity and without loss of generality, that \(d=1\) (scalar case). When \(d>1\), the proof below can be reproduced componentwise. Fix \(l\in [\![2,n]\!]\) and \(k\in ]\!]l+1,n]\!]\). For \({\mathbf {y}}\in {\mathbb {R}}^{nd}, {\mathbf {y}}'=({\mathbf {y}}_1,\cdots ,{\mathbf {y}}_{k-1},{\mathbf {y}}_k',{\mathbf {y}}_{k+1},\cdots , {\mathbf {y}}_n) \), write:

where
. The first two terms can be dealt directly.

For \(\Delta _l^{3}(s,{\mathbf {y}}',{\mathbf {y}})\), we use an explicit reverse Taylor expansion which yields:

Taking \(\delta _l\) s.t. \(\delta _l \alpha _l=1-\delta _l\), which implies that \(\delta _l= (1+\alpha _l)^{-1}\), gives in (D.9) and (D.10) a global bound of order
. Since we now recall from Lemma 11 that \({\alpha _l=1/(2l-1)} \), we get
We then write from (D.8) and the definition of
that:
which gives the result. \(\square \)
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Chaudru de Raynal, PÉ., Honoré, I. & Menozzi, S. Strong regularization by Brownian noise propagating through a weak Hörmander structure. Probab. Theory Relat. Fields 184, 1–83 (2022). https://doi.org/10.1007/s00440-022-01150-z
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DOI: https://doi.org/10.1007/s00440-022-01150-z
Mathematics Subject Classification
- Primary 60H30
- 60H10
- Secondary 34F05
- 35H10