Abstract
We investigate existence, uniqueness and regularity for solutions of rough parabolic equations of the form \(\partial _{t}u-A_{t}u-f=(\dot X_{t}(x) \cdot \nabla + \dot Y_{t}(x))u\) on \([0,T]\times \mathbb {R}^{d}.\) To do so, we introduce a concept of “differential rough driver”, which comes with a counterpart of the usual controlled paths spaces in rough paths theory, built on the Sobolev spaces Wk,p. We also define a natural notion of geometricity in this context, and show how it relates to a product formula for controlled paths. In the case of transport noise (i.e. when Y = 0), we use this framework to prove an Itô Formula (in the sense of a chain rule) for Nemytskii operations of the form u↦F(u), where F is C2 and vanishes at the origin. Our method is based on energy estimates, and a generalization of the Moser Iteration argument to prove boundedness of a dense class of solutions of parabolic problems as above. In particular, we avoid the use of flow transformations and work directly at the level of the original equation. We also show the corresponding chain rule for F(u) = |u|p with p ≥ 2, but also when Y ≠ 0 and p ≥ 4. As an application of these results, we prove existence and uniqueness of a suitable class of Lp-solutions of parabolic equations with multiplicative noise. Another related development is the homogeneous Dirichlet boundary problem on a smooth domain, for which a weak maximum principle is shown under appropriate assumptions on the coefficients.
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Open Access funding provided by Projekt DEAL. The authors would like to thank the anonymous referee who significantly helped to improve the quality of this manuscript.
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Appendices
Appendix A: some technical proofs
1.1 A.1 Proof of Lemma 2.1
It is well-known that a multiplication operator Mf of the form Mfh := x↦f(x)h(x) for h ∈ L2, is bounded if and only if \(|f|_{L^{\infty }}<\infty ,\) and that the map \(f\in L^{\infty }\mapsto M_{f}\in {\mathcal{L}}(L^{2},L^{2})\) is an isometry (see for instance [57]). By an immediate generalization, for i = 1, 2, we see that the couple (j1,j2) defined as
is a continuous isomorphism, where \({\mathbb {D}}_{i},i=1,2,\) are equipped with the operator-norm topologies as in Definition 2.1.
Let \(t\mapsto B_{t}=X_{t}\cdot \nabla + {X^{0}_{t}}\) be in \(C^{1}(0,T;{\mathbb {D}}_{1})\) and, as in (??), define the canonical lift
By definition of \(B^{2}_{st} \), we have for 0 ≤ s ≤ t ≤ T:
where we recall the notation Xst := Xt − Xs, and where we introduce
The above integrals are understood in the sense of Bochner, in \(W^{3,\infty },\) \(W^{2,\infty }, W^{1,\infty }.\) As seen through immediate algebraic computations, the generalized Chen’s relations (??) hold in this case, since
Next, for almost every \(x\in {\mathbb R}^{d}\), an integration by parts in the time variable yields the identity
Denoting by \(\mathbb A_{st}^{ij}:={{\mathbb X}}_{st}^{ij}-\mathbb S_{st}^{ij},\) we further observe that Schwarz Theorem implies
since \(\mathbb A_{st}\) is antisymmetric. Hence, only the symmetric part of \({{\mathbb X}}\) contributes to the second order part of \(B^{2}_{st}\) in (A.2). This yields the desired expression, namely
To show (2.1), note that
This yields, by definition of [B]:
which is the claimed equality.
Now, pick any geometric differential rough driver B, and let \( {{\textbf B}} (n)\in C^{1}(0,T;{\mathbb {D}}_{1}),n\in {\mathbb N_0},\) be such that \({{\textbf B}}(n)\equiv S_{2}(B (n))\to _{\rho _{\alpha } } {{\textbf B}}.\) Making use of the isomorphisms (j1,j2) we see that the coefficients
converge to \((j_{1}^{-1}B^{1};j_{2}^{-1}B^{2}),\) in the space
In particular, one can take the limits in the identities (A.4), (A.6), (A.7), proving the corresponding relations for the limit B.
1.2 A.2 Renormalization property for geometric differential rough drivers
In what follows, we fix \(D\subset U\subset {\mathbb R}^{d}\) as in Section 4 and, recalling Notation 4.15, we will further denote by Ω := ΩD while \({\Omega }_{\epsilon } :={\Omega }^{D}_{\epsilon }. \)
Given Φ(⋅,⋅), we have for (x,y) ∈Ω, by definition of T𝜖:
where we introduce the new coordinates
Note that the Jacobian determinant of the map \(\chi \colon {\Omega } \to {\mathbb R}^{d}\times B_{1},\) (x,y)↦(x+,x−) is equal to 2−d (in fact \(\sqrt 2 \chi \) is a rotation). By a common abuse of notation, we will denote by ∇± the gradient with respect to the new coodinates x+(x,y) and x−(x,y). Formally, we have the relation ∇± = ∇x ±∇y.
The proof of Theorem 4.1 is based on the following result, whose proof is implicitly contained in [19], and therefore omitted.
Lemma A.1
Let V = σi(⋅)∂i be in \({\mathbb {D}}_{1}.\) For a generic function \(\psi :{\mathbb R}^{d}\to {\mathbb R},\) denote by Ψ(x,y) := ψ((x − y)/2), and let Vx (resp. Vy) be a shorthand for V ⊗id, (resp. id ⊗ V). For each k = 1, 2, 3 and \(\psi \in W^{3,\infty }\) with compact support in the unit ball \(B_{1}\subset {\mathbb R}^{d},\) it holds uniformly in 𝜖 ∈ (0, 1]:
for a.e. HCode \((x,y)\in {\mathbb R}^{d}\times {\mathbb R}^{d}\).
Proof Proof of the Theorem
-
Step 1:
the key estimate. We first show that for \({\Phi } \in W^{k,\infty }_{0}({\Omega } ),\) and with V as in Lemma A.1:
$$ |(\nabla_{\pm})^{k-1}(T_{\epsilon} )^{-1}(V_{x} + V_{y})T_{\epsilon} {\Phi}|_{L^{\infty}({\Omega} )} \leq C|\sigma |_{W^{k,\infty}}|{\Phi} |_{W^{k,\infty}_{0}({\Omega} )} . $$(A.9)By density, it will be enough to show (A.9) on functions of the form \({\Phi } (x,y)=\phi (\frac {x+y}{2})\psi (\frac {x-y}{2}),\) with ψ compactly supported in B1. For such Φ, we have
$$ \begin{array}{@{}rcl@{}} &&T_{\epsilon}^{-1}(V_{x} + V_{y})T_{\epsilon} {\Phi}(x,y) =T_{\epsilon}^{-1}(V_{x}+V_{y})\left[\phi (\frac{x+y}{2})\right]\psi (\frac{x-y}{2\epsilon })\\ &&+\phi (\frac{x+y}{2})T^{-1}_{\epsilon} (V_{x}+V_{y})\left[\psi (\frac{x-y}{2\epsilon })\right] =I_{\epsilon} +II_{\epsilon} . \end{array} $$Using the new coordinates, we have the following expression for the first term:
$$ I_{\epsilon}= \frac{1}{2}(\sigma (x_{+}+\epsilon x_{-})+\sigma (x_{+}-\epsilon x_{-}))\cdot \nabla \phi (x_{+})\psi (x_{-}) . $$By the commutation relations
$$ \nabla_{+}T_{\epsilon} =T_{\epsilon} \nabla_{+},\quad \text{and}\quad \nabla_{-} T_{\epsilon} =\epsilon^{-1} T_{\epsilon} \nabla_{-}. $$(A.10)it is then easily seen (see [38, Proposition 6.1] for details) that for k = 1, 2, 3 :
$$ \mathrm{ess sup}_{x_{+},x_{-}}|(\nabla_{\pm})^{k-1}I_{\epsilon} |\leq |\sigma|_{W^{k,\infty}}|{\Phi} |_{W^{k,\infty}} \leq C\omega_{B}(s,t)^{i\alpha } |{\Phi} |_{W^{k,\infty}} $$For the second term, we can use Lemma A.1, since by assumption ψ is supported on the unit ball of \({\mathbb R}^{d}.\) We have
$$ \mathrm{ess sup}_{x_{+},x_{-}}|(\nabla_{\pm})^{k-1}[II_{\epsilon} ]| \leq |\sigma |_{W^{k}}|{\Phi} |_{W^{k,\infty}} . $$(A.11) -
Step 2:
uniform estimates on the first component. For \(V\in {\mathbb {D}}_{1}\) define
and further let
(A.12)Particularizing (A.9) with \(V=B^{1}_{st}\in {\mathbb {D}}_{1}\) for fixed s,t, we see by definition of \({\Gamma }^{1,\epsilon }_{st}(B)\) that
for any k ∈{1, 2, 3}. This yields the first part of the claimed estimate.
Note that, since the bracket [B]st has order one (B is geometric), we can let V = [B]st in the previous computations in order to obtain
(A.13) -
Step 3:
uniform estimates on the second component. Recalling that [B] := B2 − B1 ∘ B1/2, we have by definition of \({\Gamma }^{2}_{st}(B)\):
$$ \begin{array}{@{}rcl@{}} {\Gamma}_{st}^{2,\epsilon }({{\textbf B}}) &=&T_{\epsilon}^{-1}\left( {B^{2}_{x}} +{B^{1}_{x}}{B^{1}_{y}} + {B^{2}_{y}}\right)_{st}T_{\epsilon} \\ &\equiv& T_{\epsilon}^{-1}\left( \frac12{B^{1}_{x}}{B^{1}_{x}}+[{{\textbf B}}]_{x} +{B^{1}_{x}}{B^{1}_{y}} + \frac{1}{2}{B^{1}_{y}}{B^{1}_{y}} +[{{\textbf B}}]_{y}\right)_{st}T_{\epsilon} \\ &=&T_{\epsilon}^{-1}\left( \frac12({B^{1}_{x}} + {B^{1}_{y}})^{2} + [{{\textbf B}}]_{x}+[{{\textbf B}}]_{y}\right)_{st}T_{\epsilon} . \end{array} $$Otherwise said, we have the algebraic identity
$$ {\Gamma}^{2,\epsilon}_{st}({{\textbf B}}) = \frac12{\Gamma}_{st}^{1,\epsilon}({{\textbf B}})\circ{\Gamma}_{st}^{1,\epsilon}({{\textbf B}}) + \mathcal B_{st}^{\epsilon} , \quad \text{where}\enskip \mathcal B_{st}^{\epsilon} :=T_{\epsilon}^{-1}([{{\textbf B}}]_{x}+[{{\textbf B}}]_{y})_{st}T_{\epsilon} . $$(A.14)But if k ∈{− 1, 0}, the estimate (A.13) shows that
$$ |\mathcal B^{\epsilon}_{st}|_{\mathcal L(W^{k,1},W^{k-1,1})}\leq C\omega_{B}(s,t)^{2\alpha } . $$(A.15)We can now conclude thanks to (A.15) and Step 2, since for k = 0,− 1 :
$$ \begin{array}{@{}rcl@{}} |{\Gamma}^{2,\epsilon }_{st}({{\textbf B}})|_{\mathcal L(W^{k,1},W^{k-2,1})}&\leq& \frac12|{\Gamma}^{1,\epsilon }_{st}({{\textbf B}})|_{\mathcal L(W^{k,1},W^{k-1,1})}|{\Gamma}_{st}^{1,\epsilon }({{\textbf B}})|_{\mathcal L(W^{k-1,1},W^{k-2,1})}\\ &&+|\mathcal B_{st}^{\epsilon} |_{\mathcal L(W^{k,1},W^{k-2,1})} \leq C\omega_{B}(s,t)^{2\alpha } , \end{array} $$which finishes the proof of Theorem 4.1.
□
Appendix B: Further remarks and comments
1.1 B.1 Uniqueness of the Gubinelli derivative
Let u be such that
where f ∈ L(0,T; W− 1,p) while \((g,g^{\prime })\in \mathcal D^{\alpha ,p}_{B},\) and write
It is natural to ask under which condition one can have uniqueness of the triple \((f;g,g^{\prime })\) such that \(u\simeq (f;g,g^{\prime }),\) a question that relates the Doob-Meyer decomposition for semi-martingales. Such uniqueness is certainly not true in general because our definition of a differential rough driver could accomodate that of \(\dot B:=\dot Z\partial _{x},\) where \(Z\in C^{\infty }(0,T;{\mathbb R}).\) Indeed, in this case one can arbitrarily choose \(g^{\prime }=0\) for any u and alternatively represent the element \(u\simeq (f;g,0)\) by writing instead \(u\simeq (f+\dot Z\partial _{x}g;0,0).\)
In the finite-dimensional case however (for instance replacing B by a path Z of \(\frac 1\alpha \)-finite variation with values in \({\mathbb R}\)), the decomposition (??) is indeed unique in the case where Z is truly rough [25], i.e. when there exists a dense set of times t ∈ [0,T] such that
The situation here is different in the sense that assuming B = Zσ ⋅∇ with Z as in (B.1) does not guarantee uniqueness of the couple (f,g) in (??). To wit, assume that d = 2, and let B as above with σ = (0, 1). If (f,g) satisfy (??), then it is immediately seen that any path of the form \(t\mapsto g_{t}(x,y)+\tilde g_{t}(x)\) where \(\tilde g \in {\mathcal V}_{1}^{\alpha } (0,T;L^{2}({\mathbb R}))\) is a function of the first variable only, will also satisfy (??). In this counterexample, one sees that the space variable plays an important role in the discussion, and that if one aims at the uniqueness of the above decomposition, then some “non-degeneracy” assumptions on the differential operator σ ⋅∇ are in order. Let us now formulate a natural sufficient condition under which uniqueness of the Gubinelli derivative holds.
Assume that we are given a family Bt of (non-necessarily differential) operators such that the mapping \([0,T]\to \cap _{-2\leq k\leq 0} \mathcal L(H^{k},H^{k-1}),\) t↦Bt is α-Hölder continuous, where as before α > 1/3. For notational simplicity, we denote in the sequel Bst := δBst.
Theorem B.1
Assume the existence of \(\gamma \in [\alpha ,\frac 32\alpha ),\) such that the following ellipticity condition is satisfied: there is a constant Λ > 0, such that for every φ in H− 1, and for each (s,t) ∈Δ∩ D2,
where we are given some dense subset D of [0,T].
Let \(u\in L^{\infty }(0,T;L^{2})\cap C^{\alpha } (0,T;H^{-1})\) and suppose that \(g,\tilde g\in C^{\alpha } (0,T;H^{-1})\) are both Gubinelli derivatives for u in the Hölder sense, by which we mean that
and similar for \(\tilde g.\) Then, \(g=\tilde g.\)
Proof
Fix (s,t) ∈Δ∩ D2. The assumption (B.2) implies that the bilinear form
is H− 1-coercive. Therefore, if \(F:H^{-1}\to {\mathbb R}\) is linear and continuous, the variational problem
admits a unique solution
Moreover, it is easily seen that the Riesz isomorphism between H− 2 and its dual identifies the dual of H− 1 with H− 3, hence the operator norm of \(\textbf {T}_{st}:(H^{-1})^{*}\simeq H^{-3}\to H^{-1}\) is estimated above as
Furthermore, if \(B^{\dagger }_{st}\) denotes the adjoint of Bst with respect to the H− 2-inner product, observe thanks to (B.3) that Tst is the inverse transform of
Let g be a Gubinelli derivative for u. From the above discussion, one infers the relation
By assumption on \(R^{u}_{st}:=\delta u_{st}-B_{st}g_{s},\) it holds
Hence, letting tn ↘ s, tn ∈ D, one sees that
This implies that gt is uniquely determined by the relation
thus proving our claim. □
Example B.1
Let d = 1, and consider a 1-dimensional, α-Hölder rough path \((Z^{1},Z^{2})\in \mathcal C^{\alpha } (0,T;{\mathbb R})\) such that for some D as above it holds
where we are given some constant γ ∈ [α, 2α) (this implies in particular true roughness for Z, in the sense of (B.1)). Moreover, let \(\sigma \in W^{3,\infty }\) be bounded below, namely such that there exist constants \(\underline {\sigma } >0\) with the property that \(\sigma (x)\geq \underline {\sigma }\), for almost every \(x\in {\mathbb R}^{d}.\)
Then, it is easily seen that (B.2) holds with the differential rough driver B given by Example 2.1 with ρ = 0, where \({\Lambda } ={\Lambda } (c,\underline {\sigma })>0.\)
1.2 B.2 Brackets
For a geometric rough path (Z1,μ,Z2,μν)1≤μ,ν≤m it is well-known that the symmetric part of Z2 is expressed in terms of Z1, as follows
and every (s,t) ∈Δ (see [50]). Alternatively, this means that the bracket \([Z]_{st}:={\text {sym}} Z^{2}_{st}-\frac 12(Z^{1}_{st})^{2}\) vanishes for geometric rough paths. By analogy, in the case of an differential rough driver B, we introduced the bracket as the following family of differential operators:
(see Lemma 2.1). In contrast with what is encountered in the classical theory, note that the bracket does not vanish in general for B geometric, which is a side effect of the non-commutativity of the algebra of differential operators. Nevertheless, we saw in Lemma 2.1 that, as a consequence of geometricity, Ł takes values in the space of \({\mathbb {D}}_{1}\). In particular, unless \(B^{1}_{st}\in {\mathbb {D}}_{0},\) we see that a a cancellation occurs, since in that case Łst has stricly lower order than \(B_{st}^{2}.\) This can be seen as a non-commutative counterpart of the fact that the bracket of geometric rough paths is zero.
Remark B.1
If B denotes a differential rough driver, then by definition of the bracket Ł in (B.5), we have
where \(\mathfrak {l}_{st}\) denotes the (generally unbounded) bilinear operator
To give a concrete example, consider a filtered probability space \(({\Omega } ,\mathcal A,\mathbb {P},\{\mathcal F_{t}\}_{t\in [0,T]}),\) let \(W:{\Omega } \times [0,T]\to {\mathbb R}\) be a Brownian motion, and fix \(V\in {\mathbb {D}}_{1}\setminus {\mathbb {D}}_{0}.\) Define the (random) differential rough driver \({{\textbf B}}^{\text {It\^{o}}}(\omega )\) by \(B^{\text {It\^{o}},1}_{st}:=(W_{t}-W_{s})V\) and, observing that \(\mathbb {P}\)-a.s., \({{\int \limits }_{s}^{t}}(W_{r}-W_{s})\mathrm {d} W_{r}=\frac 12[(W_{t}-W_{s})^{2}-(t-s)]\) (Itô sense), let
With this definition, we have
showing that \(\L \in {\mathbb {D}}_{2}\setminus {\mathbb {D}}_{1},\) almost surely.
Remark B.2
As seen in the above remark, if B is not geometric, its bracket Ł (see (B.5)) is generally not first order. In the stochastic context, this has to do with the violation of stochastic parabolicity assumption, as can be seen as follows. Using the notations of Remark (B.1), we see that in the proof of the product formula, the (??) must be changed to
If we let furthermore u = v where u is an L2-energy solution of (??), \({{\textbf B}}={{\textbf B}}^{\text {It\^{o}}},\) and ϕ = 1, we have
The latter competes with the term \(-2\lambda \iint _{[s,t]\times U}|\nabla u|^{2}\mathrm {d} x\mathrm {d} r,\) which is brought by the elliptic part of the equation. In particular, the usual technique to obtain the energy estimate on u fails, unless the coefficients of V are taken small with respect to λ. This illustrates the importance of the geometricity assumption in our results.
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Hocquet, A., Nilssen, T. An Itô Formula for rough partial differential equations and some applications. Potential Anal 54, 331–386 (2021). https://doi.org/10.1007/s11118-020-09830-y
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DOI: https://doi.org/10.1007/s11118-020-09830-y