Restoration of Well-Posedness of Infinite-dimensional Singular ODE's via Noise

In this paper we aim at generalizing the results of A. K. Zvonkin and A. Y. Veretennikov on the construction of unique strong solutions of stochastic differential equations with singular drift vector field and additive noise in the Euclidean space to the case of infinite-dimensional state spaces. The regularizing driving noise in our equation is chosen to be a locally non-H\"{o}lder continuous Hilbert space valued process of fractal nature, which does not allow for the use of classical construction techniques for strong solutions from PDE or semimartingale theory. Our approach, which does not resort to the Yamada-Watanabe principle for the verification of pathwise uniqueness of solutions, is based on Malliavin calculus.


Introduction
The main objective of this paper is the construction of (unique) strong solutions of infinite-dimensional stochastic differential equations (SDEs) with a singular drift and additive noise. In fact, we want to derive our results from the perspective of a rather recently established theory of stochastic regularization (see [19] and the references therein) with respect to a new general method based on Malliavin calculus and another variational technique which can be applied to different types of SDEs and stochastic partial differential equations (SPDEs).
In order to explain the concept of stochastic regularization, let us consider the first-order ordinary differential equation (ODE) for all t, x and y with constants C 1 , C 2 < ∞. However, well-posedness in the sense of existence and uniqueness of solutions may fail, if the vector field b lacks regularity, that is if e.g. b is not Lipschitz continuous. In this case, the ODE (1) may not even admit the existence of a solution in the case H = R d .
On the other hand, the situation changes, if one integrates on both sides of the ODE (1) and adds a "regularizing" noise to the right hand side of the resulting integral equation.
More precisely, if H = R d , well-posedness of the ODE (1) can be restored via regularization by a Brownian (additive) noise, that is by a perturbation of the ODE (1) given by the SDE where (B t ) t∈[0,T ] is a Brownian motion in R d and ε > 0. If the vector field b is merely bounded and measurable, it turns out that the SDE (2) -regardless how small ε is -possesses a unique (global) strong solution, that is a solution (X x t ) t∈[0,T ] , which as a process is a measurable functional of the driving noise (B t ) t∈ [0,T ] . This surprising and remarkable result was first obtained by A. K. Zvonkin [41] in the one-dimensional case, whose proof, using PDE techniques, is based on a transformation ("Zvonkin-transformation"), that converts the SDE (2) into a SDE without drift part. Subsequently, this result was generalized by A. Y. Veretennikov [39] to the multi-dimensional case. Much later, that is 35 years later, Zvonkin's and Veretennikov's results were extended by G. Da Prato, F. Flandoli, E. Priola and M. Röckner [13] to the infinite-dimensional setting by using estimates of solutions of Kolmogorov's equation on Hilbert spaces. In fact, the latter authors study mild solutions (X t ) t∈[0,T ] to the SDE where (W t ) t∈[0,T ] is a cylindrical Brownian motion on H, A : D(A) → H a negative self-adjoint operator with compact resolvent, Q : H → H a non-negative definite self-adjoint bounded operator and b : H → H. Here, the authors prove for b ∈ L ∞ (H; H) under certain conditions on A and Q the existence of a unique mild solution, which is adapted to a completed filtration generated by (W t ) t∈ [0,T ] . So restoration of well-posedness of the ODE (1) with a singular vector field is established via regularization by both the cylindrical Brownian noise (W t ) t∈[0,T ] and A, which cannot be chosen to be the zero operator.
Other works in this direction in the infinite-dimensional setting based on different methods are e.g. A. S. Sznitman [38], A. Y. Pilipenko, M. V. Tantsyura [36] in connection with systems of McKean-Vlasov equations and G. Ritter, G. Leha [25] in the case of discontinuous drift vector fields of a rather specific form. We also refer to the references therein.
In this article, we aim at restoring well-posedness of singular ODE's by using a certain non-Hölder continuous additive noise of fractal nature. More specifically, we want to analyze solutions to the following type of SDE: where the H−valued regularizing noise (B t ) t∈[0,T ] is a stationary Gaussian process with locally non-Hölder continuous paths given by Here {λ k } k≥1 ⊂ R, {e k } k≥1 is an orthonormal basis of H and {B H k · } k≥1 are independent one-dimensional fractional Brownian motions with Hurst parameters H k ∈ (0, 1 2 ), k ≥ 1, such that H k ց 0 for k → ∞. Under certain (rather mild) growth conditions on the Fourier components b k , k ≥ 1, of the singular vector field b : [0, T ] × H → H (see (22) and (23)), which do not necessarily require that all b k are equal (compare e.g. to [38]), we show in this paper the existence of a unique (global) strong solution to the SDE (3) driven by the non-Markovian process (B t ) t∈[0,T ] .
Our approach for the construction of strong solutions to (3) relies on Malliavin calculus (see e.g. D. Nualart [32]) and another variational technique, which involves the use of spatial regularity of local time of finite-dimensional approximations of B t . In contrast to the above mentioned works (and most of other related works in the literature), the method in this paper is not based on PDE, Markov or semimartingale techniques. Furthermore, our technique corresponds to a construction principle, which is diametrically opposed to the commonly used Yamada-Watanabe principle (see e.g. [40]): Using the Yamada-Watanabe principle, one combines the existence of a weak solution to a SDE with pathwise uniqueness to obtain strong uniqueness of solutions. So Weak existence + Pathwise uniqueness ⇒ Strong uniqueness .
This tool is in fact used by many authors in the literature. See e.g. the above mentioned authors or I. Gyöngy, T. Martinez [22], I. Gyöngy, N. V. Krylov [21], N. V. Krylov, M. Röckner [24] or S. Fang, T. S. Zhang [18], just to mention a few.
However, using our approach, verification of the existence of a strong solution, which is unique in law, provides strong uniqueness: Strong existence + Uniqueness in law ⇒ Strong uniqueness .
See also H. J. Engelbert [17] in the finite-dimensional Brownian case regarding the latter construction principle.
In order to briefly explain our method in the case of time-homogeneous vector fields, we mention that we apply an infinite-dimensional generalization of a compactness criterion for square integrable Brownian functionals in L 2 (Ω), which is originally due to G. Da Prato, P. Malliavin, and D. Nualart [32], to a doublesequence of strong solutions Here {b d,ε } d∈N,ε>0 is an approximating double-sequence of vector fields of the singular drift b, which are smooth and live on d−dimensional subspaces of H. The application of the above mentioned compactness criterion (for each fixed t), however, requires certain (uniform) estimates with respect to the Malliavin derivative D t of X d,ε t in the direction of a cylindrical Brownian motion. For this purpose, the Malliavin derivative D · : Malliavin differentiable random variables and L HS (H, H) is the space of Hilbert-Schmidt operators from H to H) in connection with a chain rule is applied to both sides of (4) and one obtains the following linear equation: where b d,ε ′ is the derivative of b d,ε , ·, · H the inner product and K H a certain kernel function defined for Hurst parameters H n ∈ (0, 1 2 ). We remark here that this type of linearization based on a stochastic derivative D t actually corresponds to the Nash-Moser principle, which is used for the construction of solutions of (non-linear) PDE's by means of linearization of equations via classical derivatives. See e.g. J. Moser [31].
In a next step we then can derive a representation of D s X d,ε t (under a Girsanov change of measure) in (5) which is not based on derivatives of b d,ε by using Picard iteration and the following variational argument:  ) and f : R dn −→ R is a smooth function with compact support. Here D α stands for a partial derivative of order |α| with respect a multi-index α. Further, L n κ (t, z) is a spatially differentiable local time of B d · on a simplex scaled by non-negative integrable function κ(s) = κ 1 (s)...κ n (s). Then, using the latter we can verify the required estimates for the Malliavin derivative of the approximating solutions in connection with the above mentioned compactness criterion and we finally obtain (under some additional arguments) that for each fixed t Finally, let us also mention a series of papers, from which our construction method gradually evolved: We refer to the works [27], [28], [29], [30] in the case of finite-dimensional Brownian noise. See [20] in the Hilbert space setting in connection with Hölder continuous drift vector fields. In the case of SDEs driven by Lévy processes we mention [23]. Other results can be found in [6], [1] with respect to SDEs driven by fractional Brownian motion and related noise. See also [7] in the case of "skew fractional Brownian motion", [5] with respect to singular delay equations and [8] in the case of Brownian motion driven mean-field equations.
We shall also point to the work of R. Catellier and M. Gubinelli [11], who prove existence and path by path uniqueness (in the sense of A. M. Davie [15]) of strong solutions of fractional Brownian motion driven SDEs with respect to (distributional) drift vector fields belonging to the Besov-Hölder space B α ∞,∞ , α ∈ R. The approach of the authors is based inter alia on the theorem of Arzela-Ascoli and a comparison principle based on an average translation operator. In the distributional case, that is α < 0, the drift part of the SDE is given by a generalized non-linear Young integral defined via the topology of B α ∞,∞ . See also D. Nualart, Y. Ouknine [33] in the one-dimensional case.
The structure of our article is as follows: In Section 2 we introduce the mathematical framework of this paper. Further, in Section 3 we discuss some properties of the process B · and weak solutions of the SDE (3). Section 4 is devoted to the construction of unique strong solutions to the SDE (3). Finally, in Section 5 examples of singular vector fields for which strong solutions exist are given.

Notation.
For the sake of readability we assume throughout the paper that 1 ≤ T < ∞ is a finite time horizon. We define H to be an infinite-dimensional separable real-valued Hilbert space with scalar product ·, · H and orthonormal basis {e k } k≥1 .
Denote by · H the induced norm on H defined by x H := x, x 1 2 H , x ∈ H. For every x ∈ H and k ≥ 1 we denote by x (k) := x, e k H the projection onto the subspace spanned by e k , k ≥ 1. Loosely speaking we are referring to the subspace spanned by e k , k ≥ 1, as the k-th dimension. In line with this notation we denote the projection of the SDE (3) on the subspace spanned by e k , k ≥ 1, by X (k) := X, e k H . Moreover we can write the SDE (3) as an infinite dimensional system of real-valued stochastic differential equations, namely where b k and B (k) are the projections on the subspace spanned by e k , k ≥ 1, of b and B, respectively. Note here that the function b k : [0, T ] × H → R has still domain [0, T ] × H. Furthermore, we define the truncation operator π d , d ≥ 1, which maps an element x ∈ H onto the first d dimensions, by The truncated space π d H is denoted by H d . We define the change of basis operator τ : H → ℓ 2 by where { e k } k≥1 is an orthonormal basis of ℓ 2 . It is easily seen that the operator τ is a bijection and we denote its inverse by τ −1 : ℓ 2 → H. Further frequently used notation: • Let (X , A, µ) denote a measurable space and (Y, · Y ) a normed space. Then L 2 (X ; Y) denotes the space of square integrable functions X over X taking values in Y and is endowed with the norm • The space L 2 (Ω, F ) denotes the space of square integrable random variables on the sample space Ω measurable with respect to the σ-algebra F . • We define B x := x + B.
• For any vector u we denote its transposed by u ⊤ . • We denote by Id the identity operator.
• The Jacobian of a differentiable function is denoted by ∇.
• For any multi-index α of length d and any d-dimensional vector u we define • Let A be some countable set. Then we denote by #A its cardinality.

Fractional Calculus.
In the following we give some basic definitions and properties on fractional calculus. For more insights on the general theory we refer the reader to [34] and [37]. Let a, b ∈ R with a < b, f, g ∈ L p ([a, b]) with p ≥ 1 and α > 0. We define the left-and right-sided Riemann-Liouville fractional integrals by Here Γ denotes the gamma function.
Furthermore, for any given integer p ≥ 1, let I α a + (L p ) and I α b − (L p ) denote the images of L p ([a, b]) by the operator I α a + and I α b − , respectively. If 0 < α < 1 as well as f ∈ I α a + (L p ) and g ∈ I α b − (L p ), we define the left-and right-sided Riemann-Liouville fractional derivatives by and respectively. The left-and right-sided derivatives of f and g defined in (10) and (11) admit moreover the representations Last, we get by construction that similar to the fundamental theorem of calculus for all f ∈ I α a + (L p ), and D α a with Hurst parameter H ∈ (0, 1 2 ) on a complete probability space (Ω, F , P) is defined as a centered Gaussian process with covariance function Subsequently we give a brief outline of how a fractional Brownian motion can be constructed from a standard Brownian motion. For more details we refer the reader to [32].
Recall the following result (see [32, Proposition 5.1.3]) which gives the kernel of a fractional Brownian motion and an integral representation of R H (t, s) in the case of H < 1 2 . (14) where and β is the beta function, satisfies Subsequently, we denote by W a standard Brownian motion on the complete filtered probability space (Ω, F , is the natural filtration of W augmented by all P-null sets. Using the kernel given in (14) it is well known that the fractional Brownian motion B H has a representation Note that due to representation (16) the natural filtration generated by B H is identical to F W . Furthermore, equivalent to the case of a standard Brownian motion, it exists a version of Girsanov's theorem for fractional Brownian motion which is due to [16,Theorem 4.9]. In the following we state the version given in [33, Theorem 3.1].
But first let us define the isomorphism From (17) and the properties of the Riemann-Liouville fractional integrals and derivatives (12) and (13), the inverse of K H is given by . It can be shown (see [33]) that if ϕ is absolutely continuous where ϕ ′ denotes the weak derivative of ϕ.

Theorem 2.2 (Girsanov's theorem for fBm) Let u = (u t ) t∈[0,T ] be a process with integrable trajectories and set
Then the shifted process B H is an F W -fractional Brownian motion with Hurst parameter H under the new probability measure P defined by d P dP = E T . Remark 2.3. Theorem 2.2 can be extended to the multi-and infinite-dimensional cases, which will be considered in this paper primarily. Indeed, note first that the measure change in Girsanov's theorem acts dimension-wise. In particular, consider the two dimensional shifted process where B H 1 and B H 2 are two fractional Brownian motions with Hurst parameters H 1 and H 2 generated by the independent standard Brownian motions W (1) and W (2) , respectively, and u (1) and u (2) are two shifts fulfilling the conditions of Theorem 2.2.
Then the measure change with respect to the stochastic exponential Here, B H 1 is a fractional Brownian motions with respect to the measure P defined T . Note that B H 2 is still a fractional Brownian motion under P, since W (1) and W (2) are independent. Applying Girsanov's theorem again with respect to the stochastic exponential yields the two dimensional process where B H 1 and B H 2 are independent fractional Brownian motions with respect to the measureP defined by Repeating iteratively yields the stochastic exponential -if well-defined - Finally, we give the property of strong local non-determinism of the fractional Brownian motion B H with Hurst parameter H ∈ (0, 1 2 ) which was proven in [35,Lemma 7.1]. This property will essentially help us to overcome the limitations of not having independent increments of the underlying noise.

Cylindrical fractional Brownian motion and weak solutions
We start this section by defining the driving noise be a sequence of independent one-dimensional standard Brownian motions on a joint complete probability space (Ω, F , P). We define the cylindrical Brownian motion W taking values in H by its natural filtration augmented by the P-null sets. Moreover, we define a sequence of Hurst parameters H : with the following properties: Using H we construct the sequence of fractional Brownian motions where the kernel K H k (·, ·) is defined as in (14). Note that the fractional Brownian motions {B H k } k≥1 are independent by construction. Consequently, we define the cylindrical fractional Brownian motion B H with associated sequence of Hurst parameters H by Nevertheless, the cylindrical fractional Brownian motion B H is not in the space L 2 (Ω; H). That is why we consider the operator Q : H → H defined by for a given sequence of non-negative real numbers λ : ∈ ℓ 1 . In particular, Q is a self-adjoint operator and we have that the weighted cylindrical fractional Brownian motion Proof. Note first that due to [10][Theorem 1] for any fractional Brownian motion Using monotone convergence and (21) we have that Before we come to the next result, let us recall the notion of a weak solution and uniqueness in law.

Definition 3.2
The sextuple (Ω, F , F, P, B, X) is called a weak solution of stochastic differential equation (3) satisfies the usual conditions of right-continuity and completeness, (20), and (iii) X = (X t ) t∈[0,T ] is a continuous, F-adapted, H-valued process satisfying P-a.s.
Remark 3.3. For notational simplicity we refer solely to the process X as a weak solution (or later on as a strong solution) in the case of an unambiguous stochastic basis (Ω, F , F, P, B). Definition 3. 4 We say a weak solution X 1 with respect to the stochastic basis (3) is weakly unique or unique in law, if for any other weak solution X 2 of (3) on a potential other stochastic basis (Ω 2 , F 2 , F 2 , P 2 , B 2 ) it holds that

Moreover, the solution is unique in law.
Proof. Let {W (k) } k≥1 be a sequence of independent standard Brownian motions on the filtered probability space (Ω, F , F, Q). Consider the cylindrical fractional Brownian motion B H generated by {W (k) } k≥1 as defined in (19) with associated sequence of Hurst parameters H. We define the stochastic exponential E by In order to show that the stochastic exponential E is well-defined we first have to verify that for every k ≥ 1 Consequently, under the probability measure P, defined by dP , is a cylindrical fractional Brownian motion due to Theorem 2.2 and Remark 2.3. Therefore, is a weak solution of SDE (3). Since the probability measures Q ≈ P are equivalent, the solution is unique in law.

Strong Solutions and Malliavin Derivative
After establishing the existence of a weak solution, we investigate under which conditions SDE (3) has a strong solution. Therefore, let us first recall the notion of a strong solution and moreover the notion of pathwise uniqueness.
The cause of this paper is to establish the existence of strong solutions of stochastic differential equation (3) where y = (y 1 , . . . , y d ) and K : H → H is the defined by  (22) which merely act on d dimensions and are sufficiently smooth 2) For every d ≥ 1 and ε > 0, we prove that the SDE has a unique strong solution which is Malliavin differentiable 3) We show that the double-sequence of strong solutions X d,ε t converges weakly to E X t |F W t , where X t is the unique weak solution of SDE (3) 4) Applying a compactness criterion based on Malliavin calculus, we prove that the double-sequence is relatively compact in L 2 (Ω, F W t ) 5) Last, we show that X t is adapted to the filtration F B and thus is a strong solution of SDE (3) 4.1. Approximating double-sequence. Recall the truncation operator π d , d ≥ 1, defined in (6) and the change of basis operator τ defined in (7). We define the operator π d : Let ϕ ε , ε > 0, be a mollifier on R d such that for any locally integrable function f : [0, T ] × R d → R d and for every t ∈ [0, T ] the convolution f (t, ·) * ϕ ε is smooth and almost everywhere with respect to the Lebesgue measure. Finally, we define for every d ≥ 1 and ε > 0 the double-sequence b d,ε : Analogously to (25), we define for t ∈ [0, T ] and Due to the definition of the mollifier ϕ ε we have that for every for almost every (t, z) ∈ [0, T ] × R d with respect to the Lebesgue measure. Thus, due to (28) and the canonical properties of the truncation operator we have that Due to the assumptions on b we further get for every p ≥ 2 using dominated convergence that Hence, we can speak of an approximating double-sequence {b d,ε } d≥1,ε>0 of the drift coefficient b. In line with the previously used notation we define  Proof. In order to prove the existence of a strong solution we use Picard iteration and proceed similar to the well-known case of finite dimensional SDEs. More precisely, we define inductively the sequence Y 0 := x + B and for all n ≥ 1 We show next that {Y n } n≥0 is a Cauchy sequence in L 2 ([0, T ] × Ω). Indeed, due to monotone convergence we get for every n ≥ 1 and t ∈ [0, T ] By induction we obtain for every n ≥ 0 a constant A depending on C, λ and L such that Hence, for every m, n ≥ 0 Since B(n, m) is bounded by T Using the Lipschitz continuity of b, we get Hence, (X t ) t∈[0,T ] is a strong solution of SDE (3).
In order to show pathwise uniqueness, let X and Y be two strong solutions on the same stochastic basis (Ω, F , P, B) with the same initial condition. Then for all t ∈ [0, T ] we get similar to (30) that Using Grönwall's inequality yields that E X t − Y t 2 H = 0 for all t ∈ [0, T ], and therefore X t = Y t P-a.s. for all t ∈ [0, T ]. But since X and Y are almost surely continuous we get Next we investigate under which conditions the unique strong solution is Malliavin differentiable. But let us start with a definition of Malliavin differentiability of a random variable in the space H. We say a function f : H → H is in the space L 0 (H) if there exist sequences of constants L, M ∈ ℓ 2 such that for all k ≥ 1 and x, y ∈ H Lemma 4.6 Let f ∈ L 0 (H) with associated Lipschitz sequences L, M ∈ ℓ 2 and Y ∈ D 1,2 (H). Then, f (Y ) ∈ D 1,2 (H) and there exists a double-sequence {G Moreover, Proof. First, consider the case f : Recall the change of basis operator τ : H → ℓ 2 defined in (7). Let now f : Then g is Lipschitz continuous in the sense of (31) with associated Lipschitz sequences L, M ∈ ℓ 2 and due to equality (33) we get the identity Thus, equation (32) where the last equality holds due to (34) and dominated convergence.
and Y 0 = x + B. We denote the k-th dimension of the infinite dimensional system (35) by Y n,(k) := Y n , e k H . Using the Picard iteration (35), we show that for every step n ≥ 0 the process Y n is Malliavin differentiable. We prove this using induction. For n = 0 we have that for all t ∈ [0, T ] using (15) Thus, we get for Y n+1 that Hence, Y n+1 is Malliavin differentiable in the sense of Definition 4.5. Moreover, we can find a positive constant A depending on L, M, λ and T such that Consequently, Y n t 2 D 1,2 (H) is uniformly bounded in n ≥ 0 and therefore, since Y n → X in L 2 ([0, T ] × Ω) and the Malliavin derivative is a closable operator, also X is Malliavin differentiable in the sense of Definition 4.5.
Let us finally put the previous results together and show that SDE (24) has a unique Malliavin differentiable strong solution. (26 (27).

Corollary 4.8 Let b d,ε : [0, T ] × H → H be defined as in
Note that we can find sequences {L k } 1≤k≤d and {M i } 1≤i≤d such that for all 1 ≤

Iterating this step yields
Further note that Thus, we get for every n ≥ 1 where η 0 = k and η n = l and consequently, representation (36) Proof. Using the Wiener transform , it suffices to show for any arbitrary f ∈ L 2 ([0, T ]; H) that So, let f ∈ L 2 ([0, T ]; H) be arbitrary, then by using Girsanov's theorem we get

RESTORATION OF WELL-POSEDNESS OF INF-DIM SINGULAR ODE'S VIA NOISE 24
Using the inequality |e x − e y | ≤ |x − y| (e x + e y ) ∀x, y ∈ R, we get For every k ≥ 1, we get with representation (18) that which is bounded by Consequently, we get for every d ≥ 1 using the Burkholder-Davis-Gundy inequality that Hence, by dominated convergence Equivalently, we have Thus, again by dominated convergence Similarly, one can show that A d,ε (f ) vanishes for every f ∈ L 2 ([0, T ]; H) as ε → 0 and d → ∞.
Proof. We are aiming at applying the compactness criterion given in Theorem A.3. Therefore, let 0 < α m < β m < 1 2 and γ m > 0 for all m ≥ 1 and define the sequence

RESTORATION OF WELL-POSEDNESS OF INF-DIM SINGULAR ODE'S VIA NOISE 26
Note first that (38) is fulfilled due to the uniform boundedness of {b d,ε } d≥1,ε>0 and the definition of the process (B t ) t∈[0,T ] , see (20). Next we show uniform boundedness of (39). Note first that under the assumption u ≤ s we have Using iteration we obtain the representation where by Corollary 4.8 Consequently, we get due to (37) that where I 1 := (K Hm (t, s) − K Hm (t, u)) e m , In the following we consider each I i , i = 1, 2, 3, separately starting with the first. Due to Lemma B.3 there exists β 1 ∈ 0, 1 2 and a constant K 1 > 0 such that Consider now I 2 . Define the density E d t by Then applying Girsanov's theorem 2.2, monotone convergence and noting that .
Using equation (9) yields that can be written as Repeating the application of (9) yields K y permits the use of Proposition B.2 with 4n j=1 ε j = 4, |α j | = 1 for all 1 ≤ j ≤ 4n and thus |α| = 4n. Consequently, we get using the assumptions on H and b that For n ≥ 1 we have due to the assumptions on H that Thus, we have for n sufficiently large that and therefore by the approximations in Remark B.7 Moreover, due to the assumptions on H there exists a finite constant K > 0 which is independent of d and H such that K d,H ≤ K, cf. (62). Consequently, there exists a constant C > 0 independent of d, ε and n such that for n sufficiently large and thus due to the comparison test n≥1 D n < ∞.
Hence, there exists a constant C 2 > 0 independent of d and ε such that and thus we can find a β 2 ∈ 0, 1 2 sufficiently small such that Equivalently, we can show for I 3 that there exists a β 3 ∈ 0, 1 2 such that where C Hm,T = C ·c Hm due to Lemma B.4. Here, c Hm is the constant in (14). Thus, we can find a constantC > 0 independent of H m such that sup H∈(0, 1 6 ) C H,T ≤ C < ∞. Finally, we get with β m := min{β 1 , β 2 , β 3 } that we can find γ m , m ≥ 1, such that Proof. Let (X t ) t∈[0,T ] be a weak solution of SDE (3) which is unique in law due to Proposition 3.5. Due to Lemma 4.9 we know that for every bounded globally Lipschitz continuous function φ : . Furthermore, by Theorem 4.10 there exist subsequences {d k } k≥1 and {ε n } n≥1 such that Uniqueness of the limit yields that X t is F W t -measurable for all t ∈ [0, T ]. Since F W = F B , we get that (X t ) t∈[0,T ] is a unique strong solution of SDE (3). Malliavin differentiability follows by (40) and noting that the estimate holds also for γ m ≡ 1.

Example
In this section we give an example of a drift function b ∈ B([0, T ] × H; H) to show that the class does not merely contain the null function. Let such that C f , D f ∈ ℓ 1 and define for every k ≥ 1 an operator This yields Due to the definition C f ∈ ℓ 1 and D f · D A ∈ ℓ 1 and thus b ∈ B([0, T ] × H; H).
A possible choice for f is where a, b ∈ R and A ⊂ H, which obviously fulfills the assumptions (41). The operator A k , k ≥ 1, can for example be chosen such that there exists a finite subset N k ⊂ N such that for all k ≥ 1 and we have for every x ∈ H Then A k is invertible on A k H for every k ≥ 1 and Appendix A. Compactness Criterion The following result which is originally due to [14] in the finite dimensional case and which can be e.g. found in [9], provides a compactness criterion of square integrable cylindrical Wiener processes on a Hilbert space. In this paper we aim at using a special case of the previous theorem, which is more suitable for explicit estimations. To this end we need the following auxiliary result from [14].
Then for α < β < 1 2 we have that Theorem A. 3 Let D k be the Malliavin derivative in the direction of the k-th component of (B t ) t∈[0,T ] . In addition, let 0 < α k < β k < 1 2 and γ k > 0 for all ≤ c, Then G is relatively compact in L 2 (Ω). holds.
Proof. For notational simplicity we consider merely the case θ = 0 and write Λ f α (t, z) := Λ f α (0, t, z). For any integrable function g : (R d ) n −→ C we have that where u (k) σ = u .

RESTORATION OF WELL-POSEDNESS OF INF-DIM SINGULAR ODE'S VIA NOISE 37
Moreover, we obtain for every σ ∈ S(n, n) that where u (k) := u (k) 1 , . . . , u (k) 2n ⊤ . For every 1 ≤ k ≤ d we have by using substitution that Considering a standard Gaussian random vector Z ∼ N (0, Id 2n ), we get that Using a Brascamp-Lieb type inequality which is due to Lemma C.1, we further get that , and S m denotes the permutation group of size m. Using an upper bound for the permanent of positive semidefinite matrices which is due to [3], we find that