Regularity results for nonlinear Young equations and applications

In this paper we provide sufficient conditions which ensure that the nonlinear equation dy(t)=Ay(t)dt+σ(y(t))dx(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathrm{d}y(t)=Ay(t)\mathrm{d}t+\sigma (y(t))\mathrm{d}x(t)$$\end{document}, t∈(0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in (0,T]$$\end{document}, with y(0)=ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(0)=\psi $$\end{document} and A being an unbounded operator, admits a unique mild solution such that y(t)∈D(A)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$y(t)\in D(A)$$\end{document} for any t∈(0,T]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\in (0,T]$$\end{document}, and we compute the blow-up rate of the norm of y(t) as t→0+\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t\rightarrow 0^+$$\end{document}. We stress that the regularity of y is independent of the smoothness of the initial datum ψ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\psi $$\end{document}, which in general does not belong to D(A). As a consequence we get an integral representation of the mild solution y which allows us to prove a chain rule formula for smooth functions of y.


Introduction
The Young integral has been introduced in [15], where the author defines extension of the Riemann-Stieltjes integral f dg when neither f nor g have finite total variation. In particular in [15] it is shown that, if f and g are continuous functions such that f has finite p-variation and g has finite q-variation, with p, q > 0 and p −1 +q −1 > 1, then the Stieltjes integral f dg is well-defined as a limit of Riemann sums. This was the starting point of the crucial extension to rough paths integration. Indeed, in [13] the author proves that it is possible to define the integral f dx also in the case when f has finite p-variation and x has finite q-variation with p, q > 0 and p −1 + q −1 < 1.
In this case, additional information on the function x is needed, which would play the role of iterated integrals for regular paths.
An alternative formulation of the integration over rough paths is provided in [6], where the author considers Hölder-like (semi)norms instead of p-variation norms. Namely, if f is α-Hölder continuous and g is η-Hölder continuous with α + η > 1 then the Young integral is well defined as the unique solution to an algebraic problem. Recently, a more general theory of rough integration, when α + η ≤ 1, has been introduced in [5].
Here, we consider only Young integrals and focus on the spatial regularity of solutions to infinite dimensional evolution equations leaving aside the enormous amount of results connected to the rough paths case culminating in the breakthrough on singular SPDEs (see, e.g., [7]). Namely, we consider the nonlinear evolution equation dy(t) = Ay(t)dt + σ (y(t))dx(t), t ∈ (0, T ], y(0) = ψ, (1.1) where A is the infinitesimal generator of a semigroup defined on a Banach space X with suitable regularizing properties and x is a η-Hölder continuous function with η > 1/2. Ordinary differential equations (in finite dimensional spaces) driven by an irregular path of Hölder regularity greater than 1/2 have been understood in full details since [16] (see also [10]). On the other hand, the infinite dimensional case was treated in [8] and then developed in [9] and [3], see also [14] for earlier results in the context of stochastic partial differential equations driven by an infinite dimensional fractional Brownian motion of Hurst parameter H > 1/2. In [3], problem (1.1) is formulated in a mild form where (S(t)) t≥0 is the analytic semigroup generated by the sectorial operator A, and the authors exploit the regularizing properties of S to show that, if the initial datum ψ is smooth enough (i.e., if it belongs to a suitable domain of the fractional powers (−A) α ), then Eq. (1.1) admits a unique mild solution with the same spatial regularity as the initial datum. The key technical point in [3] is to prove that the convolution This is one of the main difference with respect to the finite dimensional case, where the condition on the function f reads in terms of classical Hölder norms. Once that convolution (1.2) is well-defined, the smoothness of the initial datum ψ and suitable estimates on (1.2) allow the authors to solve the mild reformulation of Eq. (1.1) by a fixed point argument in the same Hölder-type function space introduced above. Our point in the present paper is that, if one looks a bit more closely to the tradeoff between Hölderianity in time and regularity in space of the convolution (1.2), one discovers that an extra regularity in space can be extracted by estimates, see Lemma 2.2. This allows us to show that the mild solution to Eq. (1.1), which in our situation is driven by a finite dimensional noise, is more regular than the initial datum (that nevertheless has to enjoy the same regularity assumptions as in [3]). Namely, y(t) belongs to D(A) for any t ∈ (0, T ] (see Theorem 3.1).
It is also worth mentioning that, when A is an unbounded operator, the mild formulation of Eq. (1.1) is the most suitable to prove existence and uniqueness of a solution since it allows to apply a fixed point argument in spaces of functions with a low degree of smoothness. On the other hand this formulation is too weak in several applications where an integral formulation of the equation helps a lot. Here, having proved that the mild solution y takes values in D(A), we are in a position to show that y admits an integral representation as well, i.e., it satisfies the equation: Moreover, starting from the above relation, we can also obtain a chain rule; in other words we show that we can differentiate with respect to time regular enough functions of the solution to Eq. (1.1). Finally, as an example of possible applications of the chain rule, we propose (in Hilbertian setting, see Proposition 5.1) a necessary conditions for the invariance of Hyperplanes under the action of solutions of equations driven by irregular paths. In the case of an ordinary differential equation with a rough path, this problem is addressed in [2], when the state space is finite dimensional and no unbounded operators are involved in the equation. The problem of the invariance of a convex set with respect to a general infinite dimensional evolution equation driven by a rough trajectory is still unexplored (see [1] and the references therein for corresponding results in the case of classical evolution equations). Summarizing, this paper can been described as a first step towards a systematic study, by the classical tools of semigroup theory, of smoothing properties of the mild solution to (1.1). We plan to go further in the analysis, first weakening the smoothness assumptions on ψ and, then, developing results analogous to those in this paper for equations driven by more irregular noises as in the case of rough paths.
The paper is structured as follows. In Sect. 2, we introduce the function spaces that we use and we recall some results taken from [6,9], slightly generalizing some of those results. In Sect. 3, we prove the existence and uniqueness of a mild solution to the nonlinear Young equation (1.1) when ψ belongs to a suitable space X α ⊂ X (which will be defined later), x is η-Hölder continuous for some η ∈ (1/2, 1) and α + η > 1. We show that this solution takes values in D(A) and estimate the blow-up rate of its X 1+μ -norm as t tends to 0 + , when μ ∈ [0, η + α − 1) (see Theorem 3.1). The smoothness of the mild solution strongly relies on the smoothing effect of the semigroup associated with operator A. In general, when A is the infinitesimal generator of a strongly continuous semigroup, such smoothing properties are not satisfied by the associated semigroup. Nevertheless, we still can prove the existence and uniqueness of the mild equation to Eq. (1.1) by a suitable choice of the spaces X α . Based on Theorem 3.1, in Sect. 4 we prove that the mild solution to (1.1) can be written in an integral form, which is used in Sect. 5 to prove the chain rule. By a simple example, we show how the availability both of a solution, which takes values in D(A), and of a chain rule, can be exploited to tackle the problem of the invariance of convex sets, when an unbounded operator A is involved. Finally, in Sect. 6 we provide two examples, one on the space of continuous functions and one on an L p state space, to illustrate our results. Notation. We denote by [a, b] 2 < the set {(s, t) ∈ R 2 : a ≤ s < t ≤ b}. Further, we denote by L (X α , X γ ) the space of linear bounded operators from X α into X γ , for each α, γ ≥ 0. For every A ⊆ R, C(A; X ) denotes the usual space of continuous functions from A into X endowed with the sup-norm. The subscript "b" stands for bounded. Finally, for every α ∈ (0, 1), C α (A; X ) denotes the subset of C b (A; X ) consisting of α-Hölder continuous functions. It is endowed with the norm

Function spaces and preliminary results
Throughout the paper, X denotes a Banach space and A : D(A) ⊆ X → X is a linear operator which generates a semigroup (S(t)) t≥0 . We further assume the following set of assumptions. Hypothesis 2.1. (i) For every α ∈ [0, 2), there exists a space X α (with the convention that X 0 = X and X 1 = D(A)) such that if β ≤ α then X α is continuously embedded into X β . We denote by K α,β a positive constant such that |x| β ≤ K α,β |x| α for every x ∈ X α ; (ii) for every ζ, α, γ ∈ [0, 2), ζ ≤ α, and μ, ν ∈ (0, 1] with μ > ν there exist positive constants M ζ,α,T , and C μ,ν,T , which depend on T , such that We refer the reader also to Sect. 3.1 for another choice of the spaces X α , which guarantees the validity of a part of Hypotheses 2.1.
We now introduce some operators which will be used extensively in this paper.
Remark 2.1. We stress that the continuity of the function a in [a, b] 2 < is implied by the strong continuity of the semigroup (S(t)) t≥0 in (0, +∞). No continuity assumption at t = 0 is required.
We recall some relevant results from [6] and [9]. In particular, we recall the definition of the Young integrals where f : [a, b] → X and x : [a, b] → R satisfy suitable assumptions. In particular, we assume the following condition on x.

Definition 2.3. For every
The above result reports for the construction of the "classical" Young integral. The following one, proved in [9,Sections 3 & 4], accounts the construction of Young type convolutions with the semigroup (S(t)) t≥0 .
Again, when x ∈ C 1 ([a, b]) the limit in (2.7) coincides with the Riemann-Stieltjes integral of the function S(t − ·) f with respect to the function x over the interval [s, t]. As above, this remark inspires the following definition (see again [9]).
For further use, we prove a slight extension of the estimate in [9, Theorem 4.1 (2)].

Lemma 2.1. Let f be a function in E
for every r ∈ [k, 1).

Proof. From Theorem 2.2 it follows that I S f is well-defined as Young convolution and
( Using condition (2.1)(ii)(a), we get Now, we prove that the Young convolution (2.9) can be split into the sum of two terms.
Proof. The proof is straightforward: it is enough to take into account the properties of Young convolution and the semigroup property of (S(t)) t≥0 .
Proof. From the definition of δ S and of I S f it follows that which combined with (2.14) yields the assertion.

Smoothness of mild solutions
We consider the following assumptions on the nonlinear term σ .
Hereafter, we assume that Hypothesis 2.2 with a = 0 and b = T > 0 and Hypothesis 3.1 hold true.

In particular, y(t) belongs to D(A) for every t ∈ (0, T ] and y
The proof follows the lines of [9, Theorem 4.3], but our assumptions are weaker. In particular, in [9] the authors assume that η > 2α, while we do not need this condition. Before proving Theorem 3.1, we state the following lemma, which is a straightforward consequence of Lemma 2.2. Lemma 3.1. Suppose that y is a mild solution to (3.2). Then, for every τ ∈ [0, T ] it holds that Proof of Theorem 3.1. We split the proof into some steps.

Remark 3.1. (i) Theorem 3.1 generalizes the results in [9, Theorem 4.3]. (ii) From the last part of
Step 3 in the proof of Theorem 3.1 it follows that y ∈ C((0, T ]; X μ ) for any μ ∈ [0, η + α). (iii) In Step 3 of the proof of Theorem 3.1 we have proved that for each r ∈ [α, 1) there exists a constant c such that for some constant c, independent of t. If ψ ∈ X γ for some γ ∈ [α, 1), then arguing as in estimate (3.27), we can easily show that we can replace α − r with (γ − r ) ∧ 0 in (3.36), with r ∈ [α, 1). Based on this estimate, (3.28) and (3.29), we conclude that for every β > 0 such that r + β < 1, every 0 < s < t ≤ T and some positive constants c * and c * * , independent of s and t. Since β < η + γ − r , from (3.37) we conclude that If γ − r − β ≥ 0 then the above estimate can be extended to s = 0. We will use these estimates in Sect. 5.

The case when the semigroup has no smoothing effects
The proof of Theorem 3.1 strongly relies on the smoothing effects on the semigroup (S(t)) t≥0 , i.e., on condition 2.1(a), which in general is not satisfied when the semigroup associated with operator A is merely strongly continuous. For instance, one may think to the semigroup of left-translations in the space of bounded and continuous functions over R d or in the usual L p (R d )-space related to the Lebesgue measure: the function S(t) f has the same degree of smoothness as the function f .
In the proof of Theorem 3.1, condition (2.1)(a) is heavily used to prove that the mild solution y to the nonlinear Young equation 3.

takes values to D(A).
In this subsection we show that partially removing condition (2.1)(a), i.e. assuming that it holds true only when α = ζ , and suitably choosing the intermediate spaces X α , the existence and uniqueness of a mild solution to Eq. (3.2) can still be guaranteed.
Proof. The proof follows the same lines as the first two steps of the proof of Theorem 3.1. The only difference is that, under these weaker assumptions, Lemma 2.1 can be applied only with r = k, so that estimate (3.11) now reads as follows: From this point on the proof of the theorem carries on as in the proof of the quoted theorem.
We now provide an example of intermediate spaces X α for which any strongly continuous semigroup satisfies Hypothesis 2.1(ii)(b) and 2.1(ii)(a), this latter at least with ζ = α.
Example 3.1. Let A be the generator of a strongly continuous semigroup (S(t)) t≥0 and for each α ∈ (0, 1) let us consider the Favard space endowed with the norm If α = k + β for some k ∈ N and β ∈ (0, 1), then endowed with the norm Each space F α is a Banach space when endowed with the norm · F α . Fix α ∈ R, x ∈ F α and t ∈ [0, +∞). For any s ∈ (0, 1], we can estimate Hence, S(t)x belongs to F α and S(t)x F α ≤ S(t) L(X ) x F α , so that Hypothesis 2.1(ii)(a), with ζ = α holds true if we take X α = F α .

The integral representation formula
Knowing that mild solutions take their values in D(A) we are in a position to prove that they solve equation (3.2) in a natural integral form.  1 − η, 1). We say that y solves equation (3.2) in the integral form if, for every t > 0, it satisfies the equation To prove that mild solutions verify (4.1), we first need to check that the integral is well defined as Young integral, when y is the unique mild solution to (3.2). But, if σ satisfies Hypothesis 3.1, then for every is well defined. Indeed, arguing as in the proof of (3.9) it can be easily checked that σ • f ∈ C α ([0, T ]; X ). Therefore, Theorem 2.1 guarantees that the integral in (4.2) is well-defined.
We can now prove that, under Hypotheses 2.1, 2.2 and 3.1, the mild solution y verifies (4.1) To prove this result, we first show that the mild solution to (3.2) can be approximated by mild solutions of classical problems. for some η > 1/2 and fix ψ ∈ X α for some α ∈ (0, 1/2) such that α + η > 1. For every n ∈ N, denote by y n the mild solution to (3.2) with x replaced by x n , and let y be the mild solution to (3.2). Then, the following properties are satisfied: (i) y n converges to y in E α ([0, T ]; X α ) as n tends to +∞; (ii) if we set J(t) = t 0 σ (y(u))dx(u) and J n (t) = t 0 σ (y n (u))dx n (u) for every t ∈ [0, T ] and n ∈ N, then J n converges to J in C η ([0, T ]; X ) as n tends to +∞.
Proof. (i) We split the proof into two steps. In the first one, we show the assertion when T is small enough and in the second step we remove this additional condition.
Step 2. If T * = T then we are done. Otherwise, let us fix T := (2T * ) ∧ T . For every t ∈ [T * , T ], from (3.5) we can write In Step 1 we have proved that y n (T * ) converges to y(T * ) in X α as n tends to +∞. Moreover, for every (s, t) ∈ [T * , T ] 2 < it holds that δ S S(· − T * )(y(T * ) − y n (T * ))(s, t) = 0. Hence, S(·−T * )(y(T * )− y n (T * )) E α ([T * , T ];X α ) vanishes as n tends to +∞. Repeating the same arguments as in Step 1, we conclude that and therefore y n converges to y in E α ([T * , T ]; X α ) as n tends to +∞. If T = T then the assertion follows. Otherwise by iterating this argument, we get the assertion in a finite number of steps.
(ii) As in the proof of property (i), we can write From (2.5), (3.7) and (3.10), we infer that for every t ∈ [0, T ], As far as the term J n 2 (0, t) is concerned, we argue similarly, taking advantage of the computations in (3.23) and estimate (3.24), and get To prove that J n converges to J in C η ([0, T ]; X ), now it suffices to note that (see and repeat the above computations to infer that for every n ∈ N. We are now ready to show that the mild solution y to (3.2) satisfies the integral representation formula (4.1).   . Then, the unique mild solution y to (3.38) with ψ ∈ X α , with α + η > 1, satisfies the equation Proof. The statement follows from Remark 3.2, and by repeating the computations in this section.

Chain rule for nonlinear Young equations
In this subsection we use the integral representation formula (4.1) of the unique mild solution y to problem (3.2) to prove a chain rule for F(·, y(·)), where F is a smooth function.
Theorem 5.1. Let F ∈ C 1 ([0, T ] × X ) be such that and F x is α-Hölder continuous with respect to t, locally uniformly with respect to x, and is locally γ -Hölder continuous with respect to x, uniformly with respect to t, for some α, γ ∈ (0, 1) such that η+αγ > 1. Further, let y be the unique mild solution to (3.2). Then, where Δy j = y(s n j ) − y(s n j−1 ), Δs n j = s n j − s n j−1 ,s n j = s n j−1 + θ n j (s n j − s n j−1 ), y j = y(s n j−1 ) + η n j (y(s n j ) − y(s n j−1 )) and θ n j , η n j ∈ (0, 1) are obtained from the mean-value theorem, for every j = 1, . . . , m n .
Analysis of the terms I 1,n and I 2,n . Since the function s → F t (s, y(s)) is continuous in [0, T ], I 1,n converges to As a byproduct, it follows that, if |Π(s, t)| ≤ δ, then |I 2,n | ≤ ε n j=1 Δs n j = ε(t − s) and this shows that I 2,n converges to 0 as n tends to +∞.
Analysis of the term I 3,n . Using (4. Choosing μ = 1 + ρ for some ρ < η + α − 1 and using (2.1)(b) we get Therefore, Ay is ρ-Hölder continuous in [ε, T ] for any ε ∈ (0, T and the right-hand side of the previous inequality vanishes as n tends to +∞. Let us consider the third term in the right-hand side of (5.1). From Theorem 2.1 and recalling that α + η > 1, we infer that (σ (y(u)) − σ (y(s n j−1 )))dx(u) Letting n tend to +∞ gives To conclude the study of I 3,n it remains to consider the term F x (s n j−1 , y(s n j−1 )), σ (y(s n j−1 )) (x(s n j ) − x(s n j−1 )).
For this purpose, we introduce the function g : [s, t] → R, defined by g(τ ) = F x (τ, y(τ )), σ (y(τ )) for every τ ∈ [s, t]. Let us prove that g ∈ C αγ ([s, t]). To this aim, we recall that Hence, we can estimate where the integral is well-defined as Young integral. From (5.2)-(5.5) we conclude that To complete the proof, we observe that I 4,n converges to 0 as n tends to +∞. This property can be checked arguing as we did for the term I 2,n , noting that F x is uniformly continuous in [0, T ] × y([0, T ]).
Summing up, we have proved that F(t, y(t)) − F(s, y(s)) = t s F t (u, y(u))du + t s F x (u, y(u)), Ay(u) du + t s F x (u, y(u)), σ (y(u)) dx(u), (5.6) for every 0 < s < t ≤ T . As s tends to 0 + , the left-hand side of (5.6) converges to F(t, y(t)) − F(0, y(0)). As far as the right-hand side is concerned, the first and the third term converge to the corresponding integrals over [0, t] since the functions u → F t (u, y(u)) and u → F x (u, y(u)) are continuous in [0, T ]. As far as the second term in the right-hand side of (5.6) is concerned, thanks to (3.4) with μ = 0 we can apply the dominated convergence theorem which yields the convergence to the integral over (0, t). The assertion in its full generality follows.
The same arguments as in the proof of Theorem 5.1 and Corollary 4.1 give the following result. for every (s, t) ∈ [0, T ].
As an immediate application of the chain rule, we provide necessary conditions, in the contexts of Hilbert spaces, for the invariance of the set K = {x ∈ X : x, ϕ ≤ 0} for the mild solution y to (3.38), where invariance means that, if ψ ∈ X α ∩ K , then y(t) belongs to K for any t ∈ [0, T ]. For this purpose, we assume that A : D(A) ⊂ X → X is a self-adjoint nonpositive closed operator which generates an analytic semigroup of bounded linear operators (S(t)) t≥0 on H and that the results so far proved hold true with X ζ = D((−A) ζ ) for any ζ ≥ 0. Proposition 5.1. Let Hypotheses 2.1, 2.2, 3.1 be fulfilled with η + α > 1. Let ϕ ∈ X ε for some ε ∈ [0, 1), ψ ∈ X ζ for some ζ ∈ [α, 1) and let K := {x ∈ X : x, ϕ ≤ 0} be invariant for y. The following properties are satisfied.

Examples
In this section, we provide two examples to which our results apply. We consider the second-order elliptic operator A, defined by Example 6.1. Let us assume that the coefficients of the operator A are bounded and β-Hölder continuous on R d , for some β ∈ (0, 1), and d i, j=1 q i j (x)ξ i ξ j ≥ μ|ξ | 2 for every x, ξ ∈ R d and some positive constant μ.