Potential Analysis (2022) 57:243–261 https://doi.org/10.1007/s11118-021-09914-3 Absolute Continuity of Solutions to Reaction-Diffusion Equations with Multiplicative Noise Carlo Marinelli1 · Lluı́s Quer-Sardanyons2 Received: 23 May 2019 / Accepted: 29 December 2020 / Published online: 6 April 2021 © The Author(s) 2021 Abstract We prove absolute continuity of the law of the solution, evaluated at fixed points in time and space, to a parabolic dissipative stochastic PDE on L2 (G), where G is an open bounded domain in Rd with smooth boundary. The equation is driven by a multiplicative Wiener noise and the nonlinear drift term is the superposition operator associated to a real function that is assumed to be monotone, locally Lipschitz continuous, and growing not faster than a polynomial. The proof, which uses arguments of the Malliavin calculus, crucially relies on the well-posedness theory in the mild sense for stochastic evolution equations in Banach spaces. Keywords Stochastic PDEs · Reaction-diffusion equations · Malliavin calculus Mathematics Subject Classification (2010) 60H07 · 60H15 1 Introduction Let G be a bounded domain of Rd , d > 1, with smooth boundary. Consider a semilinear stochastic equation of the type du(t) + Au(t) dt = f (u(t)) dt + σ (u(t))B dW (t), u(0) = u0 , (1) where A is the negative generator of an analytic semigroup on q ≥ 2, f : R → R Lq (G), is a locally Lipschitz continuous decreasing function with polynomial growth, σ : R → R is a Lipschitz continuous function, B is a γ -Radonifying operator from L2 (G) to Lq (G), and W is a cylindrical Wiener process on L2 (G) (precise assumptions on the data of the problem are provided in Section 2 below). Then (1) admits a unique mild solution which is continuous in space and time. Our aim is to prove that the law of the random variable u(t, x) Carlo Marinelli c.marinelli@ucl.ac.uk 1 Department of Mathematics, University College London, Gower Street, London WC1E 6BT, UK 2 Departament de Matemàtiques, Universitat Autònoma de Barcelona, 08193 Cerdanyola del Vallès (Barcelona), Catalonia, Spain 244 C. Marinelli, L. Quer-Sardanyons is absolutely continuous with respect to Lebesgue measure for every fixed (t, x) ∈ R+ × G. It seems that, somewhat surprisingly, this natural question has not been addressed in the literature. In fact, all results of which we are aware about existence (and regularity) of the density of solutions to SPDEs with multiplicative noise deal with the case where G is the whole space, −A is the Laplacian, and the drift coefficient f is (globally) Lipschitz contin- uous (see, e.g., [13, 15, 16, 19] and references therein). Our results do not rely on any one of these assumptions. In particular, we essentially just assume that the semigroup generated by −A is self-adjoint and given by a family of kernel operators, so that, for instance, very large classes of elliptic second-order operators are allowed, and the function f can be of polyno- mial type. Another major difference with respect to the above-mentioned works is that we rely almost exclusively on the interpretation of (1) as an equation for an Lq (G)-valued pro- cess, and that we view the pointwise Malliavin derivative of its solution as a process taking values in Lq (G; H ), where H is a suitably chosen Hilbert space. This point of view, which allows us to rely on powerful techniques of the functional-analytic approach to stochastic evolution equations on UMD Banach spaces, is probably the most interesting aspect of this work. The more common random field interpretation of (1), that seems the only one used in previous work, at least in connection with techniques of the Malliavin calculus, is used here very sparingly, essentially only to take the pointwise Malliavin derivative of the solution to (1). Existence and regularity of the density of solutions to semilinear heat equations with additive noise, i.e. for the easier case where σ = 1 and −A is the Laplacian, were obtained in [9]. Those results, however, depend heavily on the noise being additive, and cannot be extended to the general setting considered here. In fact, if the noise is additive, then the Malliavin derivative of the solution satisfies a deterministic equation with random coeffi- cients, which yields quite strong estimates using pathwise arguments. On the other hand, if the noise is multiplicative, then the Malliavin derivative is only expected to satisfy a fur- ther stochastic evolution equation with quite singular initial condition, which is much more difficult to handle than the deterministic PDE arising in the case of additive noise. As a con- sequence, while in [9] we obtained existence as well as regularity of the density, here we can only show existence. As it is natural to expect, regularity could be obtained also in the case of multiplicative noise and Lipschitz continuous drift. However, we concentrate here only on the existence issue, and we shall deal with the regularity problem somewhere else, hopefully also in the general case where f is monotone and polynomially bounded. Let us briefly describe the main content of the paper. We first show existence and unique- ness of a unique mild solution u to (1) which is continuous in space and time. This follows by relatively recent results on well-posedness in the mild sense for stochastic evolution equations in Banach spaces (see Section 2). Assuming that the semigroup generated by −A is a family of kernel operators, the mild solution can be interpreted also in the sense of random fields. Considering first the case where f is Lipschitz continuous, so that the mild solution is the unique fixed point of an operator , this reformulation allows to compute the Malliavin derivative of applied to a class of sufficiently regular processes. Using esti- mates for stochastic convolutions in Banach spaces, we show that the fixed-point operator leaves invariant a subspace of Malliavin differentiable processes with finite moment. This yields, by closability properties of the Malliavin derivative, that the unique mild solution to (1) is pointwise Malliavin differentiable. As a second step, we provide sufficient condi- tions ensuring that the Malliavin derivative is non-degenerate, adapting a method used in [16, Theorem 5.2] for equations on Rd (see Section 3). This yields, as is well known, the pointwise absolute continuity of the law of the solution. As mentioned above, the results Absolute Continuity of Solutions to Reaction-Diffusion... 245 should be interesting in their own right, as equations in domains (in dimension higher than one) do not appear to have been considered in the literature. Finally, in the general case of equations of reaction-diffusion type, the pointwise absolute continuity of the law of the solution is treated by localization techniques, i.e. by means of the Bouleau-Hirsch criterion (see Section 4), and by convergence results for stochastic evolution equations with locally Lipschitz continuous coefficients in spaces of continuous functions. 2 Well-Posedness in the Space of Continuous Functions We are going to establish well-posedness in the mild sense for the stochastic equation (1) in a space of continuous functions, using general well-posedness results for stochastic evolution equations in UMD Banach spaces (see [6, 21]). Assuming that the semigroup generated by −A is a family of integral operators, we shall also show that the solution thus obtained can be viewed as a solution in the sense of random field (cf. [3, 22]). 2.1 Preliminaries Let us consider the following stochastic evolution equation, posed on a general Banach space X: du(t) + Au(t) dt = f (u(t)) dt + B(u(t)) dW (t), u(0) = u0 , (2) where W is a cylindrical Wiener process on a Hilbert space U , and all other coefficients are specified below. The following well-posedness result is a slightly simplified version of [6, Theorem 4.9]. The space of γ -Radonifying operators from a Hilbert space K to a Banach space E will be denoted by γ (K, E). Theorem 2.1 Let E be a UMD Banach space with type 2, such that X is densely and continuously embedded in E, and A be a sectorial, accretive operator on E such that the semigroup S on E generated by −A restricts to a C0 -semigroup of contractions on X. Assume that f : X → X is locally Lipschitz continuous and there exists m > 0 such that f (x + y) − f (y), x ∗ 1+ y m − x m , f (y) 1+ y m for all x, y ∈ X and x ∗ ∈ ∂ x . Let p > 2 and assume that there exists a number η ∈ R+ , with 1 1 η< − , 2 p such that Eη := D((I + A)η ) is densely and continuously embedded in X. If B : X → γ (U, E) is locally Lipschitz continuous with linear growth, and u0 ∈ Lp ( ; X), then there exists a unique X-valued mild solution to (2), which satisfies p p E sup u(t) X 1 + E u0 X. t≤T Here ∂ x stands for the subdifferential at x, in the sense of convex analysis, of the convex function · , that is, denoting the dual of X by X , ∂ x = x ∗ ∈ X : x ∗ = 1, x ∗ , x = 1 . 246 C. Marinelli, L. Quer-Sardanyons Moreover, the notation a b means that there exists a constant N such that a ≤ N b. To emphasize the dependence of N on parameters p1 , . . . , pn , we shall write a p1 ,...,pn b. Remark 2.2 In [6] the authors also require that −Ax + f (x + y), x ∗ 1+ y m + x for every x ∈ D(A|X ) and x, y ∈ X. Since we are assuming that A is accretive in X, it follows that −Ax, x ∗ ≤ 0. Moreover, f (x + y), x ∗ = f (x + y) − f (y), x ∗ + f (y), x ∗ 1+ y m + | f (y), x ∗ | 1+ y m , hence their condition, under our assumptions, is automatically satisfied. Remark 2.3 Further well-posedness results in Lq spaces for semilinear parabolic SPDEs of accretive type, with more natural assumptions on the nonlinear drift term f , can be found in [7, 8, 10–12]. See also [2] for related results in spaces of continuous functions. We shall also need some basic facts on interpolation. The real and the complex interpola- tion functors are denoted by (·, ·) and [·, ·], respectively. Moreover, we shall write X → Y to mean that X is continuously embedded in Y . Lemma 2.4 Let X and Y be two Banach spaces forming an interpolation pair, A a positive operator on X, and θ, θ ∈ ]0, 1[, q, q ∈ [1, ∞] be constants. The following statements hold true: (a) if X ⊂ Y and θ < θ , then (X, Y )θ,q → (X, Y )θ ,q ; (b) (X, Y )θ,1 → (X, Y )θ,∞ ; (c) (X, Y )θ,1 → [X, Y ]θ → (X, Y )θ,∞ ; (d) X, D(A) θ,1 → D(Aθ ) → X, D(A) θ,∞ . Proof All statements can be found in [20]. Specific references are provided for each result: (a) and (b) are parts of Theorem 1.3.3, p. 25; (c) is a consequence of Theorem 1, p. 64, taking into account Definition 1.10.1, p. 61; (d) is part of Theorem 1.15.2, p. 101. 2.2 Existence of a Unique Mild Solution Let us now turn to equation (1), about which the following standing assumptions are assumed from now on. Hypothesis 1 (a) The operator A is the realization on Lq (G), q ≥ 2, of a second-order strongly elliptic operator with C ∞ coefficients, with Dirichlet boundary conditions. (b) The function f : R → R is an odd polynomial of degree m > 0 with negative leading coeffi- cient. (c) W is a cylindrical Wiener process on L2 (G) defined on a filtered probability space ( , F , (Ft )t∈[0,T ] , P), with T ∈ R+ , where (Ft )t∈[0,T ] is the completion of the filtration generated by W . It follows by (b) that |f (x)| 1 + |x|m for all x ∈ R. Absolute Continuity of Solutions to Reaction-Diffusion... 247 Proposition 2.5 Assume that d 1 1 < − , 2q 2 p σ : R → R is locally Lipschitz continuous with linear growth, and B ∈ γ (L2 (G), Lq (G)). If u0 ∈ Lp ( ; C(G)), then (1) admits a unique C(G)-valued mild solution u, which satisfies the estimate p p E sup u(t) 1 + E u0 . C(G) C(G) t≤T Here C(G) denotes the space of continuous functions on G, the closure of G. Proof We are going to verify that the assumptions of Theorem 2.1 are satisfied. It follows from Hypothesis 1 that, for any q ≥ 2, A is a sectorial, accretive operator on Lq (G), and that the semigroup S generated by −A restricts to a C0 -semigroup on C(G) (see, e.g., [17, Theorem 3.5, pp. 213-214 and Theorem 3.7, p. 217]). Moreover, denoting the evaluation operator on C(G) associated to f by the same symbol, it is not difficult to see that f satisfies the assumptions of Theorem 2.1 (detail can be found in [6, Examples 4.2 and 4.5]). Moreover, one easily verifies that u → σ (u)B is locally Lipschitz continuous and has linear growth as a map from C(G) to γ (U, Lq (G)). Let θ < θ be such that d 1 1 <θ <θ < − . 2q 2 p Setting E := Lq := Lq (G), let us show that Eθ → C(G) densely: recall that, by Lemma 2.4, Eθ → Lq , D(A) θ,∞ → Lq , D(A) θ ,1 → Lq , D(A) θ , where, by the characterization of D(A) in [20, Theorem 4.9.1, p. 334], D(A) = Hq,D 2 (G) := φ ∈ Hq2 (G) : φ|∂G = 0 . Moreover, thanks to [20, Theorem 3.3.4, p. 321], one has 2 Lq , Hq,D θ = Hq,D 2θ if 2θ = 1/q. Since d > 1 and 2θ > d by hypothesis, the latter condition is obviously satisfied, hence Eθ → Hq,D 2θ ⊂ H 2θ . Finally, the Sobolev embedding theorem (cf. [20, q Theorem 4.6.1, p. 328]) yields Hq2θ → C(G), assuming that 2θ > d/q, which is satisfied by hypothesis. We have thus shown that all assumptions of Theorem 2.1 are met, hence the claim is proved. Note that p > 2 imply that, for q large enough, the hypothesis d/(2q) < 1/2 − 1/p is always satisfied. Remark 2.6 Instead of assuming that f is an odd polynomial with negative leading coeffi- cient, one could also assume that f : R → R is locally Lipschitz continuous, polynomially bounded, and quasi-monotone, i.e. that there exists λ > 0 such that x → λx − f (x) is increasing. In fact, assume that there exists m > 0 such that |f (x)| 1 + |x|m . By dissipativity of f − λI , f (x + y) − λ(x + y) − (f (y) − λy), x ∗ ≤ 0, hence f (x + y) − f (y), x ∗ ≤ λ x, x ∗ ≤ λ, 248 C. Marinelli, L. Quer-Sardanyons and f (x + y), x ∗ ≤ λ + | f (y)x ∗ | λ+1+ y m . 2.3 Mild Solution as Random Field We assume from now on, in addition to Hypothesis 1, the following condition on the semigroup S generated by −A. Hypothesis 2 The semigroup S = (S(t))t≥0 is sub-Markovian (i.e. S(t) is positive and contracting in L∞ (G) for all t ≥ 0) and admits a kernel, in the sense that there exists a function K : R+ × G2 → R+ such that S(t)φ (x) = Kt (x, y)φ(y) dy G for every φ ∈ Lq (G), q ≥ 1. Let Q := BB ∗ , which is a symmetric and non-negative definite bounded operator. Recall that a cylindrical Q-Wiener process on L2 := L2 (G) is a Gaussian family of random vari- ables W := {Wh (t), h ∈ L2 , t ≥ 0} such that, for all s, t ≥ 0 and h, g ∈ L2 , E(Wh (t)) = 0 and E(Wh (t)Wg (s)) = (t ∧ s) Qh, g L2 (in spite of the slight abuse of notation, no confusion should arise with the cylindrical Wiener process W ). Let L2Q be the Hilbert space defined as the completion of L2 with respect to the scalar product h, g L2 := Qh, g L2 . Note that, denoting the pseudoinverse Q of Q1/2 by Q−1/2 , if (ek )k∈N is a basis of L2 , then (ēk ) := (Q−1/2 ek ) is a basis of L2Q . One can define stochastic integrals with respect to W as follows (see, e.g., [4, Sec. 2]): let {X(t, x) : (t, x) ∈ [0, T ] × G} be a predictable process in L2 ( × [0, T ]; L2Q ). Then T ∞ T X(t, x) W (dt, dx) := X(t, ·), ēk L2Q dWēk (t), (3) 0 G k=1 0 and the isometry property reads T 2 T E X(t, x) W (dt, dx) =E X(t, ·) 2 L2Q dt. 0 G 0 In order to prove that the Malliavin derivative of the solution u of (1) satisfies a stochastic equation, we need to verify that u can be interpreted as a mild solution to (1) in the sense of random fields (see, e.g., [3, 4, 22]). This is indeed the case (cf. the analogous result for equations with additive noise in [9]). Proposition 2.7 Let the assumptions of Proposition 2.5 be satisfied. For any (t, x) ∈ [0, T ] × G, set u(t, x) := [u(t)](x), where u is the unique C(G)-valued mild solution to (1). Then, for any (t, x) ∈ ]0, T ] × G, t u(t, x) = Kt (x, y)u0 (y) dy + Kt−s (x, y)f (u(s, y)) dy ds G 0 G t + Kt−s (x, y)σ (u(s, y)) W (ds, dy). (4) 0 G Absolute Continuity of Solutions to Reaction-Diffusion... 249 Proof As in the proof of [9, Proposition 3.1], it suffices to show that, for every t ∈ ]0, T ] and for almost every x ∈ G, the process (s, y) → Kt−s (x, y)σ (u(s, y)) belongs to L2 ( × [0, T ]; L2Q ) and that t t S(t − s)σ (u(s))B dW (s) = Kt−s (·, y)σ (u(s, y))W (ds, dy) (5) 0 0 G as an equality in L2 . Recalling that (ēk ) = (Q−1/2 ek ), is a basis of the Hilbert space L2Q , one easily verifies that ∞ 2 Kt−s (x, ·)σ (u(s, ·)) 2 L2Q = [S(t − s)σ (u(s))](ẽk )(x) , k=1 where (ẽk ) := (Q1/2 ek ) is a basis of Q1/2 (L2 ). Note that t ∞ E [S(t − s)σ (u(s))](ẽk ) 2 L2 ds < ∞, 0 k=1 because the stochastic integral on the left-hand side of (5) is well defined. Thus, for almost all x ∈ G, t E Kt−s (x, ·)σ (u(s, ·)) 2 L2Q < ∞, 0 so the stochastic integral on the right-hand side of (5) is well defined. Using the standard formal expansion of the cylindrical Wiener process W as ∞ W (t) = ek wk (t), k=1 where wk := Wēk , k ≥ 1, form a family of independent standard one-dimensional Wiener processes, one has t ∞ t S(t − s)σ (u(s))B dW (s) = Kt−s (·, y)[σ (u(s))Bek ](y) dy dwk (s). 0 k=1 0 G Then (5) follows taking into account the definition (3) and that BB ∗ = Q. 3 Equations with Lipschitz Continuous Coefficients We assume throughout this section that the coefficients f and σ in equation (1) are Lipschitz continuous. We are going to prove that, for any fixed (t, x) ∈ (0, T ] × G, the law of the solution u(t, x) to (1) is absolutely continuous with respect to the Lebesgue measure. For this, note that the Gaussian space where we will make use of the Malliavin calculus is determined by the isonormal Gaussian process on the Hilbert space H := L2 (0, T ; L2Q ) that can be naturally associated to the cylindrical Q-Wiener process W defined in the previous section (see [14]). We will first deal with the Malliavin differentiability of the solution, and then we shall provide sufficient conditions implying that the pointwise Malliavin derivative is non-degenerate. We need further assumptions, that will be assumed to hold from now on. 250 C. Marinelli, L. Quer-Sardanyons Hypothesis 3 One has d 1 1 < − . 2q 2 p Moreover, B ∈ γ (L2 , Lq ) and u0 ∈ C(G). Hypothesis 4 The semigroup S is self-adjoint and Markovian. Recall also that we assume that Hypotheses 1 and 2 are in force throughout. By Propo- sition 2.5, it follows that (1) admits a unique C(G)-valued mild solution u, and that (1) can also be written as an equality of random fields. 3.1 Pointwise Malliavin Differentiability of the Solution The main result of this section is the following. Theorem 3.1 Let u ∈ Lp ( ; C([0, T ]; C(G))) be the unique mild solution to (1). Then u ∈ L∞ ([0, T ] × G; D1,p ) and the family of Malliavin derivatives {Du(t, x)}(t,x)∈[0,T ]×G satisfies the following linear equation in H : t Du(t, x) = v0 (t, x) + Kt−s (x, y)F (s, y)Du(s, y) dy ds 0 G t + Kt−s (x, y) (s, y)Du(s, y) W (ds, dy) (6) 0 G where v0 (t, x) := (τ, z) → Kt−τ (x, z)σ (u(τ, z)) 1[0,t] (τ ), and F , : × [0, T ] × G → R are adapted bounded random fields. The stochastic integral in (6) must be interpreted as an H -valued integral with respect to the cylindrical Q-Wiener process W (see, e.g., [16, Section 3]). The following estimate plays an important role in the proof Theorem 3.1 as well as in q several further developments. We shall write Eη , for any q ≥ 1 and η > 0, to denote (I + A)−η Lq . Lemma 3.2 Let v ∈ Lp ( ; C([0, T ]; C(G))) be adapted and w : × [0, T ] × G → H be the process defined as w(t, x) := (τ, z) −→ Kt−τ (x, z)σ (v(τ, z)) 1[0,t] (τ ). For any η ∈ ]d/(2q), 1/2 − 1/p[ one has t sup w(t, x) 2 H S(t − s)σ (v(s))B 2 q ds. γ (L2 ,Eη ) x∈G 0 Absolute Continuity of Solutions to Reaction-Diffusion... 251 Proof Since H = L2 (0, T ; L2Q ) and Qh, h = B ∗ h 2L2 for every h ∈ L2Q , denoting a complete orthonormal basis of L2 by (ek )k∈N , it follows by Plancherel’s theorem that t w(t, x) 2 H = B ∗ Kt−τ (x, ·)σ (v(τ, ·)) 2 L2 dτ 0 t = Kt−τ (x, ·)σ (v(τ, ·)), Bek 2 dτ 0 k∈N t 2 = Kt−τ (x, z)σ (v(τ, z))[Bek ](z) dz dτ 0 k∈N G t = S(t − τ )σ (v(τ ))Bek (x)2 dτ, 0 k∈N where we have used the integral representation of the semigroup S in the last step. Let (γk )k∈N be a sequence of independent standard Gaussian random variable on an auxiliary probability space . Then 2 t w(t, x) 2 H = E γk S(t − τ )σ (v(τ ))Bek (x) dτ, 0 k∈N hence also, by Minkowski’s inequality and the embedding Eη → L∞ , q 2 t sup w(t, x) 2 H E γk S(t − τ )σ (v(τ ))Bek dτ x∈G 0 k∈N q Eη t = S(t − τ )σ (v(τ ))B 2 q dτ . γ (L2 ,Eη ) 0 The proof of Theorem 3.1 uses a maximal inequality for stochastic convolutions that is a special (simpler) case of [21, Proposition 4.2]. We shall use the notation R F to denote the process t R F: t → R(t − s)F (s) dW (s), 0 where R is an analytic semigroup of contractions on a UMD Banach space E and F : × R+ → L (L2 , E) is an L2 -strongly measurable and adapted process. Denoting the generator of R by −C, we shall write Eη , for any η > 0, to denote D((I + C)η ). Proposition 3.3 Let α ∈ ]0, 1/2[, p > 2, θ ≥ 0 be such that 1 θ <α− , p and T > 0. There exists ε > 0 such that T E s → (t − s)−α F (s) p p E R F C([0,T ];Eη ) T pε γ (L2 (0,t;L2 ),E) . 0 We shall also need a deep result by Pisier (see [18, Theorem 1.2 and Remark 1.8] as well as [23, p. 5730]) on vector-valued extensions of analytic semigroup, according to which Hpothesis 4 implies that (S(t) ⊗ IH )t≥0 , where IH denotes the identity of H , admits a 252 C. Marinelli, L. Quer-Sardanyons (unique) extension from Lq ⊗ H to Lq (H ), denoted by SH , which is again analytic. Let AH denote the negative generator of SH and (λ + AH )−1 λ>0 its resolvent. The Laplace transform identity ∞ (λ + AH )−1 = e−λt SH (t) dt 0 implies that (λ + AH )−1 coincides with the unique continuous linear extension of (λ + A)−1 ⊗ IH to Lq (H ). By Hypothesis 3 there exists η ∈ ]d/(2q), 1/2 − 1/p[ such that D(Aη ) → L∞ , hence (I + A)−η ∈ L (Lq , L∞ ). Since (I + A)−η is positivity preserving by Hypothesis 2, (I + A)−η admits a unique extension to a continuous linear operator from Lq (H ) := Lq (G; H ) to L∞ (H ) := L∞ (G; H ), with the same norm (see, e.g., [5, Theorem 12.2]). By the above, recalling well-known expressions for fractional powers of closed operators (see, e.g., [17, §2.6]), this extension coincides with (I +AH )−η . Therefore, setting Eη (H ) := (I + A)−η Lq (H ), we have Eη (H ) → L∞ (H ). q q Proof of Theorem 3.1 Let be the fixed-point operator associated to equation (1), i.e. t t : v −→ S(t)u0 + S(t − s)f (v(s)) ds + S(t − s)σ (v(s))B dW (s). 0 0 It follows by the (the proof) of Theorem 2.1 that the operator , or a suitable power of it, is a contractive endomorphism of Lp ( ; C([0, T ]; C(G))). We are going to show that, for any p > 2, there exists T0 > 0, a positive constants c < 1 depending on T0 , and a positive constant N depending on the Lp ( ; C([0, T ]; C(G))) norm of v, such that D (v) L∞ ([0,T0 ]×G;Lp ( ;H )) ≤ N + c Dv L∞ (0,T0 ;Lp ( ;L∞ (G;H ))) . (7) Let v ∈ Lp ( ; C([0, T ]; C(G))) be such that Dv ∈ L∞ (0, T ; Lp ( ; L∞ (G; H ))). Writing t (v) (t, x) = Kt (x, y)u0 (y) dy + Kt−s (x, y)f (v(s, y)) dy ds G 0 G t + Kt−s (x, y)σ (v(s, y)) W (dy, ds), 0 G well-known criteria of Malliavin calculus imply that the Malliavin derivatives of all terms on the right-hand side exist, so that D (v) (t, x) can be written as the right-hand side of (6) with u replaced by v. The proof of (7) will be split in several steps, where each term appearing in the expression of D (v) is estimated. STEP 1. Let us set, for every (t, x), (τ, z) ∈ [0, T ] × G, w0 (t, x) := (τ, z) → Kt−τ (x, z)σ (v(τ, z)) 1[0,t] (τ ). Let η ∈ ]d/(2q), 1/2 − 1/p[. Lemma 3.2 yields t w0 (t, ·) 2 L∞ (H ) S(t − τ )σ (v(τ ))B 2 q dτ, γ (L2 ,Eη ) 0 where S(t − τ )σ (v(τ ))B γ (L2 ,Eη ) q (t − τ )−η σ (v) C([0,T ];C(G)) B γ (L2 ,Lq ) . Absolute Continuity of Solutions to Reaction-Diffusion... 253 This implies t p/2 (t − τ )−2η dτ p p p E w0 L∞ ([0,T ]×G;H ) 1+E v C([0,T ];C(G)) B γ (L2 ,Lq ) sup t≤T 0 p p 1+E v B T p(1−2η)/2 , C([0,T ];C(G)) γ (L2 ,Lq ) where the last term on the right-hand side is finite by assumption. STEP 2. Let α < 1/2 be such that η < α − 1/p. Recalling that Eη (H ) → L∞ (H ), q Minkowski’s and Jensen’s inequalities yield t 2 t S(t − s)F (s)Dv(s) ds T S(t − s)F (s)Dv(s) 2 q Eη (H ) ds 0 L∞ (H ) 0 t (t − s)−2η Dv(s) 2 Lq (H ) ds. 0 Since η < α − 1/p by assumption, we have −2η > −2α + 2/p, hence −2η = −2α + 2/p + ε, with ε > 0. Then t t (t − s)−2η Dv(s) 2 Lq (H ) ds = (t − s)−2α (t − s)2/p+ε Dv(s) 2 Lq (H ) ds 0 0 t ≤ t 2/p+ε (t − s)−2α Dv(s) 2 Lq (H ) ds 0 t T s −2α Dv(t − s) 2 Lq (H ) ds. 0 As the measure μ on [0, t] defined as 1 − 2α −2α μ(ds) := s ds t 1−2α is a probability measure, it follows by Jensen’s inequality that t p/2 p/2 t 1−2α t s −2α Dv(t − s) 2 Lq (H ) ds = Dv(t − s) 2 Lq (H ) μ(ds) 0 1 − 2α 0 t p t (1−2α)p/2 Dv(t − s) Lq (H ) μ(ds) 0 t s −2α Dv(t − s) p t (1−2α)(p/2−1) Lq (H ) ds 0 t (t − s)−2α Dv(s) p T Lq (H ) ds. 0 Therefore t p t (t − s)−2α E Dv(s) p E S(t − s)F (s)Dv(s) ds T L∞ (H ) ds. 0 L∞ (H ) 0 254 C. Marinelli, L. Quer-Sardanyons STEP 3. Using again the continuous embedding Eη (H ) → L∞ (H ), we have q p p E S ( DvB) C([0,t];L∞ (H )) E S ( DvB) q C([0,t];Eη (H )) t (τ − ·)−α p T E Dv B γ (L2 (0,τ ;L2 ),Lq (H )) dτ 0 t E (τ − ·)−α p Dv B L2 (0,τ ;γ (L2 ,Lq (H ))) dτ, 0 where the third inequality follows by Proposition 3.3, as Lq (H ) is a UMD Banach space and η < α − 1/p, and the fourth estimate follows by Fubini’s theorem and the embedding L2 (0, τ ; γ (L2 , Lq (H ))) → γ (L2 (0, τ ; L2 ), Lq (H )), which holds because Lq (H ) has type 2. Since Dv(s) ∈ L∞ (H ) by assumption and ∈ L∞ ([0, T ] × G) by the Lipschitz continuity of σ , it follows that (τ − s)−α (s)Dv(s)B γ (L2 ,Lq (H )) −α ≤ (τ − s) L∞ ([0,T ]×G) Dv(s) L∞ (H ) B γ (L2 ,Lq ) , hence (τ − ·)−α Dv B 2 L2 (0,τ ;γ (L2 ,Lq (H ))) τ ≤ B 2 γ (L2 ,Lq ) 2 L∞ ([0,T ]×G) (τ − s)−2α Dv(s) 2 L∞ (H ) ds. 0 Proceeding as in the previous step, we obtain τ E (τ − ·)−α (τ − s)−2α E Dv(s) p p Dv B L2 (0,τ ;γ (L2 ,Lq (H ))) T L∞ (H ) ds, 0 therefore, by Tonelli’s theorem, t τ (τ − s)−2α E Dv(s) p p E S ( DvB) C([0,t];L∞ (H )) T L∞ (H ) ds dτ 0 0 t t (τ − s)−2α dτ ds, p = E Dv(s) L∞ (H ) 0 s where t t−s 1 (τ − s)−2α dτ = τ −2α dτ = (t − s)1−2α , s 0 1 − 2α hence t (t − s)−2α E Dv(s) p p E S ( DvB) C([0,t];L∞ (H )) T L∞ (H ) ds. 0 STEP 4. Setting p p φ(t) := E Dv(t) L∞ (H ) , ψ(t) := E D (v)(t) L∞ (H ) , p N := 1 + E v C([0,T ];C(G)) , the estimates in the previous steps can be written as t ψ(t) T N+ (t − s)−2α φ(s) ds, 0 Absolute Continuity of Solutions to Reaction-Diffusion... 255 hence, using the notation h∗ (s) := supr≤s |h(r)| for any function h : R → R for which it makes sense, t 1 ψ(t) T N + φ ∗ (t) (t − s)−2α ds = N + t 1−2α φ ∗ (t), 0 1 − 2α thus also 1 ψ ∗ (t) T N+ t 1−2α φ ∗ (t), 1 − 2α from which (7) follows. Let u0 be identified with the process equal to u0 for all t ∈ [0, T ], which clearly belongs to Lp ( ; C([0, T ]; C(G))) and is such that Du0 ∈ L∞ (0, T ; Lp ( ; L∞ (G; H ))), and introduce the sequence of processes (un ), un := (un−1 ). Then un converges to u in Lp ( ; C([0, T ]; C(G))), possibly along a subsequence of the type (kn), with constant k (if is not a contraction, but k is). In particular, (un ) is bounded in Lp ( ; C([0, T ]; C(G))). This in turn implies, thanks to (7), that (Dun ) is bounded in L∞ ([0, T0 ] × G; Lp ( ; H )). Let us show that this actually implies that (Dun ) is bounded in L∞ ([0, T ] × G; Lp ( ; H )). In fact, setting p p φn (s) := E Dun (s) L∞ (H ) , φ0 := 1 + sup E un < ∞, C([0,T ];C(G)) n∈N we have already shown that t φn+1 (t) T φ0 + (t − s)−2α φn (s) ds ∀t ∈ [0, T ], 0 and that (φn∗ (T0 ))n is bounded. We now proceed by induction: assuming that (φn∗ (j T0 ))n is bounded, let us show that (φn∗ ((j + 1)T0 ))n is also bounded. Let j T0 < t ≤ (j + 1)T0 . We have t φn+1 (t) T φ0 + (t − s)−2α φn (s) ds 0 j T0 t = φ0 + (t − s)−2α φn (s) ds + (t − s)−2α φn (s) ds, 0 j T0 where t > j T0 implies t − s > j T0 − s and (t − s)−2α < (j T0 − s)−2α , hence j T0 j T0 (j T0 )1−2α ∗ (t − s)−2α φn (s) ds < (j T0 − s)−2α φn (s) ds ≤ φ (T0 ), 0 0 1 − 2α n so that (j T0 )1−2α ∗ t φn+1 (t) T φ0 + φ (j T0 ) + (t − s)−2α φn (s) ds 1 − 2α n j T0 (j T0 )1−2α ∗ t T φ0 + φ (j T0 ) + φn∗ ((j + 1)T0 ) (t − s)−2α ds, 1 − 2α n j T0 where t t−j T0 T0 T01−2α (t − s)−2α ds = s −2α ds ≤ s −2α ds = . j T0 0 0 1 − 2α This in turn implies, taking the supremum over [0, (j + 1)T0 ], ∗ (j T0 )1−2α ∗ T 1−2α ∗ φn+1 ((j + 1)T0 ) T φ0 + φn (j T0 ) + 0 φ ((j + 1)T0 ). 1 − 2α 1 − 2α n 256 C. Marinelli, L. Quer-Sardanyons Since φn∗ (j T0 ) is bounded uniformly with respect to n by the inductive assumption, we deduce that φn∗ ((j + 1)T0 ) is bounded uniformly over n as well, thus completing the induc- tive argument. This implies, by a standard argument based on the closure of the Malliavin derivative, that u ∈ L∞ ([0, T ] × G; D1,p ). Finally, the equation for Du follows immediately by differentiating equation (4), upon applying the chain rule for theMalliavin derivative (see, e.g., [14, Proposition 1.2.4]). 3.2 Non-Degeneracy of the Malliavin Derivative This section is devoted to study, for any fixed (t, x) ∈ ]0, T ] × G, the norm of the Malliavin derivative of u(t, x). Together with the results of the previous section, we will deduce the existence of the density for the law of the random variable u(t, x). Recall that throughout the section we are assuming that f and σ are globally Lipschitz continuous functions. We will need an estimate for the norm of Du(t, x) in H (a, b) := L2 (a, b; L2Q ), 0 ≤ a < b ≤ T. Proposition 3.4 Let 0 ≤ a < b ≤ T , p > 2, and η ∈ ]d/(2q), 1/2 − 1/p[. There exists a positive constant N , independent of a and b, such that p sup E Du(t, x) H (a,b) ≤ N (b − a)p(1/2−η) . (t,x)∈[a,b]×G Proof Repeating the proof of Theorem 3.1 with H replaced by H (a, b), we get p p sup E Du(t, x) H (a,b) ≤N sup E v0 (t, x) H (a,b) , (t,x)∈[0,T ]×G (t,x)∈[0,T ]×G and, by Lemma 3.2, t∧b sup v0 (t, x) 2 H (a,b) S(t − s)σ (u(s))B 2 γ (L2 ,Eη ) ds, x∈G a where S(t − s)σ (u(s))B γ (L2 ,Eη ) (t − s)−η 1 + u C([0,T ×G) B γ (L2 ,Lq ) . Therefore t p/2 (t − s)−2η ds p p sup E Du(t, x) 1+ u sup H (a,b) Lp ( ;C([0,T ×G)) (t,x)∈[0,T ]×G t≤b a (b − a)(1−2η)p/2 . In the next result we establish sufficient conditions under which the norm of the Malliavin derivative of u(t, x) does not vanish, almost surely. Proposition 3.5 Assume that there exists a constant c > 0 such that |σ (z)| ≥ c for all z ∈ R and that Q is positivity preserving. Let (t, x) ∈ ]0, T ] × G, α ∈ ]0, 1/2[, and η ∈ ]d/(2q), α − 1/p[. If there exist β ∈ ]0, 1 − α − η] such that δβ lim = 0, (8) δ→0 K(x, ·) H (0,δ) then Du(t, x) H > 0 almost surely. Absolute Continuity of Solutions to Reaction-Diffusion... 257 Proof We are going to estimate P( Du(t, x) H ≤ 1/n) for n ∈ N and pass to the limit as n → ∞. Let δ ∈ ]0, 1[, and set, for compactness of notation, Hδ := H (t −δ, t). The obvious inequality a + b ≥ a − b applied to the expression of Du given by Theorem 3.1 yields Du(t, x) H ≥ v0 (t, x) Hδ − S ∗ (F Du)(t, x) + S ( DuB)(t, x) Hδ . Hence, simplifying the notation a bit and denoting the second term within the norm on the right-hand side by Y , P Du(t, x) H ≤ 1/n ≤ P v0 (t, x) Hδ − Y Hδ ≤ 1/n = P Y Hδ ≥ v0 (t, x) Hδ − 1/n . Since Q as well as the semigroup S is positivity preserving, hence K is positive, and σ : R → R is continuous, we have t v0 (t, x) 2 Hδ = Kt−s (x, ·)σ (u(s, ·)) 2 L2Q ds t−δ t = Kt−s (x, y)σ (u(s, y)) Q[Kt−s (x, ·)σ (u(s, ·))](y) dy ds t−δ G t = Kt−s (x, y)|σ (u(s, y))| Q[Kt−s (x, ·)|σ (u(s, ·))|](y) dy ds t−δ G t ≥ c 2 Kt−s (x, y)Q[Kt−s (x, ·)](y) dy ds t−δ G δ = c2 Ks (x, ·) 2 L2Q ds = c2 K· (x, ·) 2 H (0,δ) . 0 This implies that we can use Chebyshev’s inequality to write, for n sufficiently large, P Du(t, x) H ≤ 1/n ≤ P Y Hδ ≥ c K· (x, ·) H (0,δ) − 1/n p E Y Hδ ≤ p, c K· (x, ·) H (0,δ) − 1/n where, thanks to Theorem 3.1 and Proposition 3.4, p p E Y Hδ = E S ∗ (F Du)(t, x) + S ( DuB)(t, x) Hδ p δ p(1/2−α) Du L∞ ([0,T ]×G;Lp ( ;Hδ )) δ p(1−α−η) . Taking the limit as n → ∞, we are left with p δ 1−α−η P Du(t, x) H =0 K· (x, ·) H (0,δ) Since this inequality holds for every δ ∈ ]0, 1[, and the limit of the right-hand side as δ → 0 is zero by assumption, it follows that P Du(t, x) H = 0 = 0. As an immediate consequence of the above result and of Theorem 3.1 we obtain suffi- cient conditions for the pointwise absolute continuity of the law of the mild solution to (1), thanks to well-known criteria of the Malliavin calculus (see, e.g., [14, Theorem 2.1.3]). Theorem 3.6 Let u ∈ Lp ( ; C([0, T ]; C(G))) be the unique mild solution to equation (1), with f and σ Lipschitz continuous and u0 ∈ C(G). Assume that there exists c > 0 such that |σ (z)| ≥ c > 0 for all z ∈ R and Q = BB ∗ is positivity preserving. Let (t, x) ∈ ]0, T ] × G, 258 C. Marinelli, L. Quer-Sardanyons α ∈ ]0, 1/2[, and η ∈ ]d/(2q), α − 1/p[. If there exist β ∈ ]0, 1 − α − η] such that (8) is fulfilled, then the law of the random variable u(t, x) is absolutely continuous with respect to Lebesgue measure. Example 3.7 Assume that A has compact resolvent in L2 . Since A is accretive and self- adjoint, there exist an orthonormal basis (ek )k∈N of L2 and a sequence (λk )k∈N ≥ 0 such that ek ∈ D(A), Aek = λk ek and limk→∞ λk = +∞. Moreover, let B = (I + A)−m , with m ∈ N, and fix (t, x) ∈ ]0, T ] × G. Since Q = (I + A)−2m , one has, for any δ ∈ ]0, 1[, δ K· (x, ·) 2 H (0,δ) = Ks (x, y)[QKs (x, ·)](y) dy ds 0 G δ = (1 + λk )−2m Ks (x, ·), ek 2 L2 ds 0 k≥0 δ = (1 + λk )−2m e−2sλk |ek (x)|2 ds 0 k≥1 1 = (1 + λk )−2m λ−1 k (1 − e −2δλk )|ek (x)|2 . 2 k≥1 Moreover, we have that 2δλk 2δλk 1 − e−2δλk ≥ ≥ . 1 + 2δλk 1 + 2λk Hence K· (x, ·) 2 H (0,δ) ≥δ (1 + λk )−2m (1 + 2λk )−1 |ek (x)|2 . k≥1 Assuming that x ∈ G is such that there exists k ∈ N for which ek (x) = 0, the quantity Cx := (1 + λk )−2m (1 + 2λk )−1 |ek (x)|2 k≥1 is strictly positive. Therefore we have K· (x, ·) 2 H (0,δ) ≥ Cx δ, i.e. δ 1/2 −1/2 ≤ Cx , K· (x, ·) H (0,δ) which implies that condition (8), hence also the assumptions of Theorem 3.6, are satisfied if we can find α and η such that 1 − α − η > 1/2. This is possible if m is sufficiently large, so that B ∈ γ (L2 , Lq ) with q large and d/(2q) is smaller than, say, 1/4. 4 Reaction-Diffusion Equations Let us now consider equation (1) in the general case, i.e. assuming that f : R → R is an odd polynomial with negative leading coefficient. As already observed, we could also assume that x → f (x) − λx is decreasing for some λ ≥ 0, locally Lipschitz continuous, and with polynomial growth. Absolute Continuity of Solutions to Reaction-Diffusion... 259 Let u0 ∈ C(G), and u ∈ Lp ( ; C([0, T ]; C(G))) be the unique mild solution to equa- tion (1), the existence of which is guaranteed by Proposition 2.5. For every n ∈ N, consider the function fn : R → R defined as f (x), |x| ≤ n, fn (x) = f (nx/|x|), |x| > n. Then fn is Lipschitz continuous, and the equation dun (t) + Aun (t) dt = fn (un (t)) dt + σ (un (t))B dW (t), u(0) = u0 , admits a unique mild solution un ∈ ; C([0, T ]; C(G))). Moreover, by construction of Lp ( u (see [6]), un coincides with u on the stochastic interval [[0, Tn ]], where the stopping time Tn is defined as Tn := inf t ≥ 0 : un (t) C(G) ≥ n ∧ T , and limn→∞ Tn = T almost surely. In particular, un → u in Lr ( ; C([0, T ]; C(G))) for all r ∈ [1, p[. Let t ∈ ]0, T ] be arbitrary but fixed and set, for every n ∈ N, n := ω ∈ : t ≤ Tn (ω) . Since (Tn ) is a sequence of stopping times monotonically increasing to T as n → ∞, ( n ) is a sequence in F monotonically increasing to as n → ∞. Clearly {t} × n ⊂ [[0, Tn ]], hence u(t) = un (t) on n , as an identity in C(G). This implies that u(t, x) = un (t, x) on n for every x ∈ G. Moreover, as fn is Lipschitz continuous, Theorem 3.1 implies that un (t, x) ∈ D1,p for every x ∈ G, for all p ≥ 1. We have thus shown that u(t, x) ∈ 1,p Dloc , with localizing sequence ( n , un (t, x)) (cf. [1, §III] or [14, §1.3.5]). This implies that u(t, x) is Malliavin differentiable, i.e. that there exists a random variable Du(t, x), independent of the chosen localizing sequence, such that Du(t, x) = Dun (t, x) on n . We are now in the position to state and prove the main result of the paper. Theorem 4.1 Let u ∈ Lp ( ; C([0, T ]; C(G))) be the unique mild solution to (1) with initial datum u0 ∈ C(G). Assume that Q = BB ∗ is positivity preserving and that there exists c > 0 such that |σ (z)| ≥ c > 0 for all z ∈ R. Let (t, x) ∈ ]0, T ] × G, α ∈ ]0, 1/2[, and η ∈ ]d/(2q), α − 1/p[. If there exist β ∈ ]0, 1 − α − η] such that δβ lim = 0, δ→0 K(x, ·) H (0,δ) then the law of the random variable u(t, x) is absolutely continuous with respect to the Lebesgue measure on R. Proof Let (t, x) ∈ GT be arbitrary but fixed. Then, by the Bouleau-Hirsch criterion (see [1, Proposition 7.1.4]), it suffices to prove that Du(t, x) H > 0 almost surely. Since fn is Lipschitz continuous for all n ∈ N, Du(t, x) H > 0 on n for all n ∈ N. This readily implies that Du(t, x) H > 0 almost surely: assume by contradiction that there exists ⊂ with strictly positive probability such that Du(t, x) H = 0 on . Since n increases monotonically to , there exists n0 ∈ N such that P > 0, where := n0 ∩ . In particular, by definition of n0 , one has Du(t, x) H > 0 on because ⊂ n0 . This is clearly a contradiction, because ⊂ . The claim is thus proved. Remark 4.2 Very minor adjustments allow to consider the case where σ : R → R is locally Lipschitz continuous with linear growth. In fact, the construction of a unique global solution 260 C. Marinelli, L. Quer-Sardanyons is obtained again by récollement of local solutions (see [6]), and the above reasoning can be repeated almost verbatim. Remark 4.3 The setting of Example 3.7 obviously satisfies the assumptions of Theorem 4.1. Acknowledgments The authors thank the anonymous referees for a careful reading of the manuscript. The first-named author is sincerely grateful to Prof. S. Albeverio for several very pleasant stays at the Inter- disziplinäres Zentrum für Komplexe Systeme, Universität Bonn, where most of the work for this paper was done. The second-named author is supported by the grants MTM2015-67802P and PGC2018-097848-B-I00 (Ministerio de Economı́a y Competitividad). Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. 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