15 Alications to stochastic PE In this final lecture we resent some alications of the theory develoed in this course to stochastic artial differential equations. We concentrate on two secific examles: the wave equation and the heat equation. 15.1 Sace-time white noise It has been mentioned already in Lecture 6 that for H = L (), where is a domain in R d, H-cylindrical Brownian motions can be used to model sace-time white noise on. We begin by making this idea more recise. efinition 15.1. Let (A, A, µ) be a σ-finite measure sace and denote by A 0 the collection of all B A such that µ(b) <. Let (Ω, F, P) be a robability sace. A white noise on (A, A, µ) is a maing w : A 0 L (Ω) such that: (i) each w(b) is centred Gaussian with E(w(B)) = µ(b); (ii) if B 1 B N =, then w(b 1 ),..., w(b N ) are indeendent and ( N w B n ) = w(b n ). It follows from the general theory of Gaussian rocesses that such maings always exist. We shall not go into the details of this, since in all alications the white noise is assumed to be given. efinition 15.. A white noise w on [0, T], where is a domain in R d, will be called a sace-time white noise on.
1 15 Alications to stochastic PE Canonically associated with such w is an L ()-cylindrical Brownian motion W, defined by W(t)1 B := w([0, t] B), B B 0 (); this definition is extended to simle functions by linearity. To see that W is indeed an L ()-cylindrical Brownian motion note that for disjoint B 1,..., B N B 0 () and real numbers c 1,..., c N we have, by (i) and (ii), ( E W(t) ) N c n 1 Bn = c n E(w([0, t] B n)) = c n t B n = t c n 1 Bn L (). 15. The stochastic wave equation In this section we study the stochastic wave equation with irichlet boundary conditions, driven by multilicative sace-time white noise: u (t, ξ) = u(t, ξ) + B( u(t, ξ), (t, ξ)) w (t, ξ), ξ, t [0, T], u(t, ξ) = 0, ξ, t [0, T], u(0, ξ) = u 0 (ξ), ξ, (0, ξ) = v 0(ξ), ξ. (WE) Here w is a sace-time white noise on a bounded domain in R d with smooth boundary. In order to kee the technicalities at a minimum we discuss two secial cases in detail: the case where the oerator-valued function B is of rank one, which is equivalent to the formulation (WE1) below, and the case where is the unit interval in R and B = I. 15..1 Rank one multilicative noise We begin with the following secial case of (WE): u (t, ξ) = u(t, ξ) + b( u(t, ξ), (t, ξ)) W (t, ξ), u(t, ξ) = 0, u(0, ξ) = u 0 (ξ), ξ, (0, ξ) = v 0(ξ), ξ, ξ, t [0, T], ξ, t [0, T], (WE1)
15. The stochastic wave equation 13 where W is a standard Brownian motion. We assume that the diffusion term b : R R R satisfies the growth condition and the Lischitz condition b(ξ 1, ξ ) C 1( ξ 1 + ξ ) b(ξ 1, ξ ) b(η 1, η ) C ( ξ 1 η 1 + ξ η ). The initial values u 0 and v 0 are taken in W 1, () and L (), resectively. Writing the first equation as a system of two first order equations, (t, ξ) = v(t, ξ), ξ, t [0, T], v (t, ξ) = u(t, ξ) + b(u(t, ξ), v(t, ξ)) W (t, ξ), (15.1) we reformulate the roblem (WE) as a first order stochastic evolution equation as follows. Let denote the irichlet Lalacian on L () with domain ( ) = W, () W 1, 0 (); see Examles 7.1. On the Hilbert sace we define the oerator H := (( ) 1/ ) L () = W 1, () L () A := [ ] 0 I 0 with domain (A) := ( ) (( ) 1/ ) = (W, () W 1, 0 ()) W 1, (). As in Examle 7., this oerator is the generator of a bounded C 0 -grou on H, and we may reformulate the roblem (15.1) as an abstract stochastic evolution equation of the form { du(t) = AU(t)dt + B(U(t))dW(t), (15.) U(0) = U 0, where W is a Brownian motion and the function B : H H is the Nemytskii ma associated with b, [ [ ] [ f 0 f B :=, H, g] b(f, g) g] and U 0 := [ u0 ] H. v 0 Proosition 15.3. Under the above assumtions on b, the Nemytskii ma B : H H is well defined, Lischitz continuous with Li(B) Li(b), and of linear growth.
14 15 Alications to stochastic PE Proof. For all (f, g) H we have B(f, g) H = b(f(ξ), g(ξ)) dξ f(ξ) + g(ξ) dξ f + g (f, g) H. A similar estimate gives that B is Lischitz continuous from H to H with Li(B) Li(b). We say that a measurable adated rocess [ u : [0, ] T] Ω R is a u(t, ω, ) mild L -solution of (WE) if U(t, ω) := belongs to H for all u(t, ω, ) (t, ω) [0, T] Ω and the resulting rocess U : [0, T] Ω H is a mild L -solution of the roblem (15.). Theorem 15.4. Under the above assumtions, for all 1 < < the roblem (WE) admits a unique mild L -solution. Here, uniqueness is understood in the sense of L (Ω; γ(l (0, T), H )). Proof. By Proosition 15.3, the Nemytskii oerator B associated with b is Lischitz continuous. Moreover, as we have seen in Examle 7., the oerator A is the generator of a C 0 -grou on H. We have thus checked all assumtions of Corollary 14.14 (with H = R and E = H ) and conclude that for all 1 < < the roblem (WE) admits a unique mild L -solution. 15.. Additive sace-time white noise Our next examle concerns the stochastic wave equation with additive sacetime white noise on the unit interval (0, 1) in R: u w (t, ξ) = u(t, ξ) + (t, ξ), ξ (0, 1), t [0, T], u(t, 0) = u(t, 1) = 0, t [0, T], (WE) u(0, ξ) = u 0 (ξ), ξ (0, 1), (0, ξ) = v 0(ξ), ξ (0, 1). Here w is a sace-time white noise on (0, 1). We model this roblem as an abstract stochastic evolution equation on the Hilbert sace H = W 1, (0, 1) L (0, 1) as [ du(t) = AU(t)dt + d U(0) = U 0, 0 W L (t) ], (15.3)
15. The stochastic wave equation 15 [ ] 0 I where as before A =, with the irichlet Lalacian on L 0 (0, 1), and W L is the L (0, 1)-cylindrical Brownian motion canonically associated with w; see Section 15.1. To analyse the roblem (15.3) we use the functional calculus for self-adjoint oerators. Using this calculus it can be checked that the C 0 -grou S generated by A is of the form [ ] cos(t( ) S(t) = 1/ ) ( ) 1/ sin(t( ) 1/ ) ( ) 1/ sin(t( ) 1/ ) cos(t( ) 1/. ) By Theorem (8.6), the unique weak solution U of (15.3) is given by t [ ] t [ ] 0 ( ) U(t) = S(t s)d = 1/ sin((t s)( ) 1/ ) 0 W(s) 0 cos((t s)( ) 1/ dw ) L (s), (15.4) rovided both integrands are stochastically integrable with resect to W L. Noting that the trigonometric functions h n (ξ) := sin(nπξ), n 1, form an orthonormal basis of eigenfunctions for, by using Theorems 5.19 and 6.17 this is the case if and only if the following two conditions are satisfied: T 0 T 0 [sin (t( ) 1/ )h n, h n ] dt <, [cos (t( ) 1/ )h n, h n ] dt <. (15.5) But if these conditions hold, then by adding we obtain T 0 [h n, h n ] dt <, which is obviously false. We conclude that the roblem (15.3) fails to have a weak solution in H. Instead of looking for a solution in H, we could try to look for a solution in the larger sace G := L (0, 1) W 1, (0, 1), where W 1, (0, 1) denotes the comletion of L (0, 1) with resect to the norm f W 1, (0,1) := ( ) 1/ f. This definition of W 1, (0, 1) is motivated by the fact that W 1, (0, 1) can be characterised as the domain of ( ) 1/. The sace G is the so-called extraolation sace of H with resect to ( ) 1/ ; we refer to Exercise 5 for a more systematic discussion. As is easy to check, the semigrou S extends to a C 0 -semigrou on G, and the stochastic convolution (15.4) is well defined in G if and only if T 0 T 0 [( ) 1 sin (t( ) 1/ )h n, h n ] dt <, [( ) 1 cos (t( ) 1/ )h n, h n ] dt <. (15.6)
16 15 Alications to stochastic PE These conditions are indeed satified, as is clear from the identity ( ) 1 h n = (nπ) h n. Let us call a measurable adated rocess u : [0, [ T] Ω ] (0, 1) R an u(t, ω, ) extraolated weak solution of (WE) if U(t, ω) := belongs to G u(t, ω, ) for all (t, ω) [0, T] Ω and the resulting rocess U : [0, T] Ω G is a weak solution of the roblem (WE). Summarising the above discussion, we have roved: Theorem 15.5. The stochastic wave equation (WE) admits a unique extraolated weak solution. Here, uniqueness is understood in the sense of γ(l (0, T; L (0, 1)), G ). 15.3 The stochastic heat equation Next we consider two stochastic heat equations with irichlet boundary values, driven by multilicative sace-time white noise on a domain in R d : (t, ξ) = u(t, ξ) + B(u(t, ξ)) w(t, ξ), ξ, t [0, T], u(t, ξ) = 0, ξ, t [0, T], u(0, ξ) = u 0 (ξ), ξ, Again we discuss two articular cases of this roblem: multilicative rank one noise and additive sace-time white noise. In both cases, the roofs of the main results can only be sketched, as they deend on a fair amount of interolation theory and results from the theory of PE. We refer to the Notes for references on this material. 15.3.1 Rank one multilicative noise Let be a bounded domain in R d with smooth boundary. Our first examle concerns the following stochastic heat equation driven by a rank one multilicative noise: (t, ξ) = u(t, ξ) + b(u(t, ξ)) W (t), ξ, t [0, T], u(t, ξ) = 0, ξ, t [0, T], (HE1) u(0, ξ) = u 0 (ξ), ξ. Here W is standard real-valued Brownian motion. We assume that the function b : R R is Lischitz continuous. We fix 1 < < and assume that the initial value u 0 belongs to L (). We say that a measurable adated rocess u : [0, T] Ω R is a mild V θ - solution of (HE1) if ξ u(t, ω, ξ) belongs to L () for all (t, ω) [0, T] Ω
15.3 The stochastic heat equation 17 and the resulting rocess U : [0, T] Ω L () is a mild V θ -solution of the stochastic evolution equation { du(t) = AU(t)dt + B(U(t))dW(t), (15.7) U(0) = u 0. Here A is the irichlet Lalacian on L () and B : L () L () is the Nemytskii ma associated with b, (B(u))(ξ) := b(u(ξ)). Proosition 15.6. Under the above assumtions on b, the Nemytskii ma B : L () L () is well defined and γ-lischitz continuous with Li γ (B) C Li(b), where C is a constant deending only on. Proof. Let us first note that B(f) L () for all f L (), so B is well defined. It follows from the Kahane-Khintchine inequality that for all f 1,..., f N and g 1,..., g N in L (), ( E γ n (B(f n ) B(g n )) L () ) 1 (E γ n (B(f n ) B(g n )) ( = L () ) 1 E γ n (b(f n (ξ)) b(g n (ξ))) dξ ) 1 ( ( E γ n (b(f n (ξ)) b(g n (ξ))) ) ( = ( N ( Li(b) ( Li(b) E b(f n (ξ)) b(g n (ξ)) ) dξ ( N ) 1 f n (ξ) g n (ξ) ) ) 1 dξ γ n (f n g n ) L () ) 1, ) 1 dξ where the last equivalence is obtained by doing the same comutation backwards. Now we aly Proosition 14.1 (with H = R). Note that this result can be extended to Nemytskii mas on saces L (A), where (A, A, µ) is any σ-finite measure sace.
18 15 Alications to stochastic PE Let us say that an adated rocess u : [0, T] Ω R is a mild V θ -solution of the roblem (HE1) if if ξ u(t, ω, ξ) belongs to Vθ for all (t, ω) [0, T] Ω and the resulting rocess U : [0, T] Ω V θ is a mild -solution of the roblem (15.7). V θ Theorem 15.7. Let 1 < <, α 0, β 0, θ 0 be such that α + β + d/ < θ < 1. Then the roblem (HE1) has a unique mild V θ -solution u. This solution has a version with the roerty that u Su 0 has trajectories in C β ([0, T]; C α ()), where S denotes the semigrou generated by the irichlet Lalacian on L (). Proof (Sketch). We check the conditions of Theorem 14.16. The sace E = L () is UM and has Pisier s roerty and by Proosition 15.6, B is γ-lischitz continuous from E to E. The irichlet Lalacian A generates an analytic C 0 -semigrou S on E (see Exercise 3). Choose numbers 0 η < η < 1 such that α + d/ < η and η + β < θ. The fractional domain sace E η associated with A equals, u to an equivalent norm, the comlex interolation sace [E, (A)] η. Let W η, () be the Sobolev-Slobodetskii sace of all functions f : R such that f d W η, () := f L () + f(ξ) f(η) dξ dη <. ξ η d+η This sace equals, u to an equivalent norm, the real interolation sace (E, (A)) η,. By general results in interolation theory, we have a continuous embedding [E, (A)] η (E, (A)) η,. By the above identifications, this results in a continuous embedding E η W η, (). Now we aly Theorem 14.16, which tells us that U Su 0 has a version in with trajectories in C β ([0, T]; E η ). By the above, this sace embeds into C β ([0, T]; W η, ()). The roof is finished by an aeal to the Sobolev embedding theorem, which asserts that for 0 α < η d/ we have a continuous embedding W η, () C α (). 15.3. Additive sace-time white noise Our final examle is the stochastic heat equation driven by an additive sacetime white noise: w (t, ξ) = u(t, ξ) + (t, ξ), ξ (0, 1), t [0, T], u(t, 0) = u(t, 1) = 0, t [0, T], (HE) u(0, ξ) = u 0 (ξ), ξ (0, 1). Here w is a sace-time white noise on the unit interval (0, 1).
15.3 The stochastic heat equation 19 We formulate the roblem (HE) as an abstract stochastic evolution equation in L (0, 1) of the form { du(t) = AU(t)dt + dwl (t), t 0, (15.8) U(0) = u 0, where A is the irichlet Lalacian on L := L (0, 1) and W L is the L - cylindrical Brownian motion canonically associated with W. By a comutation similar to (15.9) below (see Exercise 3) it is easy to check that the assumtions of Theorem 8.6 are satisfied, and therefore for initial values u 0 L we obtain the existence of a unique weak solution U of (15.8) in L. Note that in contrast to the situation for the wave equation, here it is not necessary to ass to an extraolation sace. The reason behind this is that the regularising effect of the heat semigrou takes us back into L ; the wave semigrou does not have any such effect. It is nevertheless useful to consider the equation in a suitable extraolation scale, as this enables us to obtain recise Hölder regularity results. To this end we shall aly Theorem 10.19 in a suitable extraolation sace of L := L (0, 1). Fix δ > 1 4 and let L δ denote the extraolation sace of order δ associated with the irichlet Lalacian A on L, that is, L δ is the comletion of L with resect to the norm x δ := ( A ) δ x. Since A is invertible on L (see Exercise 4), ( A ) δ acts as an isomorhism from L onto L δ. We will show next that the identity oerator I on L extends to a bounded embedding from L into L δ which is γ-radonifying. Then, we will exloit the regularising effect of the semigrou S to get back into a suitable Sobolev sace contained in L and use this to deduce regularity roerties of the solution. As is well known, and H 1 := (A) = W, (0, 1) W 1, 0 (0, 1) E 1 := (A ) = W, (0, 1) W 1, 0 (0, 1) with equivalent norms. The functions h n (ξ) := sin(nπξ), n 1, form an orthonormal basis in L of eigenfunctions for A with eigenvalues λ n, where λ n = (nπ). If we endow H 1 with the equivalent Hilbert norm f H1 := Af, the functions λ 1 n h n form an orthonormal basis for H 1 and we have E n=m γ n λ 1 n h n L 1 δ = E = E n=m γ n λ 1 n ( A ) 1 δ h n γ n (nπ) δ h n n=m ( ) (nπ) 4δ, n=m (15.9)
0 15 Alications to stochastic PE where ( ) follows from a square function estimate as in the roof of Proosition 15.6 together with the fact that h n. The right hand side of (15.9) tends to 0 as M, N since we took δ > 1 4. It follows that the identity oerator on H 1 extends to a continuous embedding from H 1 into L 1 δ which is γ-radonifying. enoting this embedding by i δ, we obtain a commutative diagram H A 1 H 1 I δ L δ A i δ L 1 δ where the to maing I δ : H L δ is injective and γ-radonifying by the ideal roerty. We are now in a osition to aly Theorem 10.19. As before we assume that u 0 L. We say that a measurable adated rocess u : [0, T] Ω (0, 1) R is a weak solution of (HE) if ξ u(t, ω, ξ) belongs to L for all (t, ω) [0, T] Ω and the resulting rocess U : [0, T] Ω L is a weak solution of the roblem (15.8). Theorem 15.8. The roblem (HE) admits a unique weak solution u. For all α 0 and β 0 satisfying α+β < 1, the rocess u Su 0 has a version with trajectories in C β ([0, T]; C α 0 [0, 1]), where S denotes the semigrou generated by the irichlet Lalacian on L (0, 1). Proof (Sketch). Fix arbitrary real numbers α 0 and β 0 satisfying α + β < 1. Relacing δ by a smaller number if necessary, we can find θ 0 such that 1 4 < δ < θ, β+θ < 1, and α+δ < θ. Put η := θ δ. As is easy to check, (the extraolation of) A generates an analytic C 0 -semigrou in L δ. Hence we may aly Theorem 10.19 in the sace L δ to obtain a weak solution U of the roblem { du(t) = AU(t)dt + I δ dw H (t), t [0, T], U(0) = 0, with aths in the sace C β( [0, T]; (L δ ) θ) ( = C β [0, T]; Lη) ; the identity (L δ ) θ = L η is a generalisation of Lemma 10.8. Along the embedding L L δ, this solution is consistent with the weak solution U of this roblem in L. Noting that α < η we choose so large that α + 1 < η. We have L η = W η, 0 (0, 1) with equivalent norms, and by the Sobolev embedding theorem, W η, (0, 1) C α [0, 1]
15.4 Exercises 1 with continuous inclusion. We denote C α 0 [0, 1] = {f Cα [0, 1] : f(0) = f(1) = 0}. Putting things together we obtain a continuous inclusion L η Cα 0 [0, 1]. In articular it follows that U takes values in L. Almost surely, the trajectories of U belong to C β ([0, T]; C α 0 [0, 1]). If we comare Theorems 15.7 and 15.8 (for d = 1 and = (0, 1)), we notice that we get better Hölder regularity for the former (α + β < 1 in the limit ) than for the latter (α + β < 1 ). The exlanation for this is the additional δ > 1 4 needed in Theorem 15.8 to get the γ-radonification of B := I δ. In Theorem 15.7, γ-radonification came for free. 15.4 Exercises 1. Let (A, A, µ) be a σ-finite measure sace and ut H := L (A). Let (W H (t)) t [0,T] be an H-cylindrical Brownian motion. Show that w([0, t] B) := W H (t)1 B, t [0, T], B A, uniquely defines a sace-time white noise w on A.. Check the comutations leading to the conditions (15.5) and (15.6). Hint: A bounded oerator R : H 1 H, where H 1 and H are searable Hilbert saces, is Hilbert-Schmidt if and only if RR : H H has finite trace. 3. In this exercise we take a look at the following stochastic heat equation with additive sace-time white noise on the domain = (0, 1) d in R d. (t, ξ) = u(t, ξ) + W (t, ξ), ξ, t [0, T], u(t, ξ) = 0, ξ, t [0, T], u(0, ξ) = u 0 (ξ), ξ. We model this roblem as a stochastic evolution equation of the form (15.8). a) Prove that the irichlet Lalacian generates an analytic C 0 -semigrou on L (). b) Show that the roblem (15.8) has a weak solution in L () if and only if d = 1. Hint for a) and b): Find an orthonormal basis of eigenvectors. 4. Show that the heat semigrou generated by the irichlet Lalacian on L (0, 1) extends to an analytic C 0 -semigrou on L (0, 1), 1 < <, and show that its generator is invertible.
15 Alications to stochastic PE 5. In this exercise we take a closer look at extraolation saces. Let A be a densely defined closed oerator on a Banach sace E and denote by G (A) its grah, G (A) = {(x, Ax) E E : x (A)}. efine the extraolation sace of E with resect to A as the quotient sace E 1 := (E E)/G (A). a) Show that the maing x (0, x) defines a bounded dense embedding E E 1. b) Show that A 1 : x ( x, 0) defined a bounded oerator from E to E 1 which extends A. c) Show that if λ (A), then the identity ma on E extends to an isomorhism of Banach saces E 1 E 1 λ, where the latter is defined as the comletion of E with resect to the norm x E λ 1 := R(λ, A)x. Notes. The literature on stochastic artial differential equations is enormous and various aroaches are ossible. The functional analytic aroach taken here, where the equation is reformulated as a stochastic evolution equation on some infinite-dimensional state sace, goes back to Hille and Phillis in the deterministic case and give rise to the theory of C 0 -semigrous. In the setting of Hilbert saces, the theory of stochastic evolution equations was ioneered by a Prato and Zabczyk and their schools. We refer to their monograh [7] for further references. See also Curtain and Pritchard [5] for some earlier references. Our definition of a sace-time white noise in Section 15. follows the lecture notes of Walsh [107], where also Theorem 15.5 can be found. The resentation of Section 15.. follows a Prato and Zabczyk [7, Examle 5.8]. Concerning roblem (HE), the existence of a solution in C α ([0, T] [0, 1]) for 0 α < 1 4 was roved by a Prato and Zabczyk by very different methods; see [7, Theorem 5.0]. Theorem 15.8 was obtained by Brzeźniak [14] under more general assumtions. The aroach taken here is from [34]. The results on interolation theory needed in the roofs of Theorems 15.8 and 15.7 can be found in the book of Triebel [103].