Regularity of the density for the stochastic heat equation

Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department of Mathematics University of Kansas Lawrence, Kansas, 6645 USA email: nualart@math.ku.edu January 8, 28 Abstract We study the smoothness of the density of a semilinear heat equation with multiplicative spacetime white noise. Using Malliavin calculus, we reduce the problem to a question of negative moments of solutions of a linear heat equation with multiplicative white noise. Then we settle this question by proving that solutions to the linear equation have negative moments of all orders. 1 Supported by NSA and NSF grants. 2 Supported by NSF grant DMS-6427. Key words and phrases. heat equation, white noise, Malliavin calculus, stochastic partial differential equations. 2 Mathematics Subject Classification. Primary, 6H15; Secondary, 6H7. 1

1 Introduction Consider the one-dimensional stochastic heat equation on [, 1] with Dirichlet boundary conditions, driven by a two-parameter white noise, and with initial condition u : t = 2 u x bt, x, ut, x σt, x, ut, W 2 x 2 t x. 1 Assume that the coefficients bt, x, u, σt, x, u have linear growth in t, x and are Lipschitz functions of u, uniformly in t, x. In [5] Pardoux and Zhang proved that ut, x has an absolutely continuous distribution for all t, x such that t > and x, 1, if σ, y, u y for some y, 1. Bally and Pardoux have studied the regularity of the law of the solution of Equation 1 with Neumann boundary conditions on [, 1], assuming that the coefficients bu and σu are infinitely differentiable functions, which are bounded together with their derivatives. Let ut, x be the solution of Equation 1 with Dirichlet boundary conditions on [, 1] and assume that the coefficients b and σ are infinitely differentiable functions of the variable u with bounded derivatives. The aim of this paper is to show that if σ, y, u y for some y, 1, then ut, x has a smooth density for all t, x such that t > and x, 1. Notice that this is exactly the same nondegeneracy condition imposed in [5] to establish the absolute continuity. In order to show this result we make use of a general theorem on the existence of negative moments for the solution of Equation 1 in the case bt, x, u = Bt, xu and σt, x, u = Ht, xu, where B and H are some bounded and adapted random fields. 2 Preliminaries First we define white noise W. Let W = {W A, A a Borel subset of R 2, A < } be a Gaussian family of random variables with zero mean and covariance E [ W AW B ] = A B, where A denotes the Lebesgue measure of a Borel subset of R 2, defined on a complete probability space Ω, F, P. Then W t, x = W [, t] [, x] defines a two-parameter Wiener process on [, 2. 2

We are interested in the following one-dimensional heat equation on [, [, 1] t = 2 u x bt, x, ut, x σt, x, ut, W 2 x 2 t x, 2 with initial condition u, x = u x, and Dirichlet boundary conditions ut, = ut, 1 =. We will assume that u is a continuous function which satisfies the boundary conditions u = u 1 =. This equation is formal because the partial derivative 2 W does not exist, and 2 is usually replaced t x by the evolution equation ut, x = 1 G t x, yu ydy G t s x, ybs, y, us, yus, ydyds G t s x, yσs, t, us, yus, yw dy, ds, 3 where G t x, y is the fundamental solution of the heat equation on [, 1] with Dirichlet boundary conditions. Equation 3 is called the mild form of the equation. If the coefficients b and σ are have linear growth and are Lipschitz functions of u, uniformly in t, x, there exists a unique solution of Equation 3 see Walsh [8]. The Malliavin calculus is an infinite dimensional calculus on a Gaussian space, which is mainly applied to establish the regularity of the law of nonlinear functionals of the underlying Gaussian process. We will briefly describe the basic criteria for existence and smoothness of densities, and we refer to Nualart [3] for a more complete presentation of this subject. Let S denote the class of smooth random variables of the the form F = fw A 1,..., W A n, 4 where f belongs to Cp R n f and all its partial derivatives have polynomial growth order, and A 1,..., A n are Borel subsets of R 2 with finite Lebesgue measure. The derivative of F is the two-parameter stochastic process defined by n f D t,x F = W A 1,..., W A n 1 Ai t, x. x i i=1 In a similar way we define the iterated derivative D k F. The derivative operator D resp. its iteration D k is a closed operator from L p Ω into 3

L p Ω; L 2 R 2 resp. L p Ω; L 2 R 2k for any p > 1. For any p > 1 and for any positive integer k we denote by D p,k the completion of S with respect to the norm { [ k F k,p = E F p E Dz1 D zj F p ]} 1 p 2 2 dz1 dz j. R 2j j=1 Set D = k,p D k,p. Suppose that F = F 1,..., F d is a d-dimensional random vector whose components are in D 1,2. Then, we define the Malliavin matrix of F as the random symmetric nonnegative definite matrix DF σ F = i, DF j. L 2 R 2 1 i,j d The basic criteria for the existence and regularity of the density are the following: Theorem 1 Suppose that F = F 1,..., F d is a d-dimensional random vector whose components are in D 1,2. Then, 1. If det σ F > almost surely, the law of F is absolutely continuous. 2. If F i D for each i = 1,..., d and E [det σ F p ] < for all p 1, then the F has an infinitely differentiable density. 3 Negative moments Theorem 2 Let ut, x be the solution to the stochastic heat equation = 2 u t x Bu W 2 Hu 2 t x, 5 u, x = u x on x [, 1] with Dirichlet boundary conditions. Assume that B = Bt, x and H = Ht, x are bounded and adapted processes. Suppose that u x is a nonnegative continuous function not identically zero. Then, for all p 2, t > and < x < 1. E[ut, x p ] < 4

For the proof of this theorem we will make use of the following large deviations lemma, which follows from Proposition A.2, page 53, of Sowers [7]. Lemma 3 Let wt, x be an adapted stochastic process, bounded in absolute value by a constant M. Let ɛ >. Then, there exist constants C, C 1 > such that for all λ > and all T > P sup t T sup x 1 G t s x, yws, yw ds, dy > λ C exp C 1λ 2 T 1 2 ɛ We also need a comparison theorem such as Corollary 2.4 of [6]; see also Theorem 3.1 of Mueller [4] or Theorem 2.1 of Donati-Martin and Pardoux [2]. Shiga s result is for x R, but it can easily be extended to the following lemma, which deals with x [, 1] and Dirichlet boundary conditions. Lemma 4 Let u i t, x : i = 1, 2 be two solutions of i = 2 u i t x B 2 W iu 2 i Hu i t x, 6 u i, x = u i x where B i t, x, Ht, x, u i x satisfy the same conditions as in Theorem 2. Also assume that with probability one for all t, x [, 1] B 1 t, x B 2 t, x u 1 x u 2 x. Then with probability 1, for all t, x [, 1]. u 1 t, x u 2 t, x. Proof of Theorem 2. We shall repeatedly use the comparison lemma, Lemma 4, along with the following argument. Observe that if < wt, x ut, x with probability one, and p >, then E [ ut, x p] E [ wt, x p]. Thus, to bound E[ut, x p ], it suffices to find a nonnegative function wt, x ut, x and to prove a bound for E[wt, x p ]. Such a function wt, x might be found using the comparison lemma, Lemma 4. 5.

Suppose that Bt, x K almost surely for some constant K >. By the comparison lemma, Lemma 4, it suffices to consider the solution to the equation w = 2 w t x Kw Hw 2 W 2 t x w, x = u x 7 on x [, 1] with Dirichlet boundary conditions. Indeed, the comparison lemma implies that a solution wt, x of 7 will be less than or equal to a solution ut, x of 5. Then we can use the argument outlined in the previous paragraph to conclude that the boundedness of E [wt, x p ] implies the boundedness of E [ut, x p ]. Set ut, x = e Kt wt, x, where ut, x is not the same as earlier in the paper. Simple calculus shows that ut, x satisfies and we have = 2 u t x W 2 Hu 2 t x. 8 u, x = u x E [ wt, x p] = e Ktp E [ ut, x p]. So, we can assume that K =, that is that ut, x satisfies 8. The mild formulation of Equation 8 is ut, x = 1 G t x, yu ydy G t s x, yhs, yus, yw ds, dy. Suppose that u x δ > for all x [a, b], 1. Since 8 is linear, we may divide this equation by δ, and assume δ = 1, and also u x = 1 [a,b] x. Fix T >, and consider a larger interval [a, b] [c, d] of the form d = b γt and c = a γt, where γ >. We are going to show that EuT, x p < for x [c, d] and for any p 1. Define c = inf inf ts T, a γts x bγts bγs a γs G t x, ydy and note that < c < 1 for each γ > and a, b, 1. Next we inductively define a sequence {τ n, n } of stopping times and a sequence 6

of processes v n t, x as follows. Let v t, x be the solution of 8 with initial condition u = 1 [a,b] and let { τ = inf t > : inf v t, x = c a γt x bγt 2 or sup v t, x = 2 }. x 1 c Next, assume that we have defined τ n 1 and v n 1 t, x for τ n 2 t τ n 1. Then, {v n t, x, τ n 1 t} is defined by 8 with initial condition v n τ n 1, x = c 2 n 1 [a γτn 1,bγτ n 1 ]x. Also, let { τ n = inf c t > τ n 1 : inf v nt, x = a γt x bγt 2 or sup v n t, x = x 1 n1 It is not hard to see that τ n < almost surely. Notice that c n1 inf v n τ n, x. a γτ n x bγτ n 2 By the comparison lemma, we have that n= } n1 2. c ut, x v n t, x 9 for all t, x and all n. For all p 1 we have E [ ut, x p] P ut, x 1 np 2 [ c n1 c n P ut, x, c 2 2 n= np 2 c n 1 P ut, x <. 1 c 2 Taking into account 9, the event {ut, x < c 2 n } is included in A n = {τ n < T }. Set σ n = τ n τ n 1, for all n, with the convention τ 1 =. We have P σ i < 2 i1 2 n F τ i 1 P sup sup v i t, x > τ i 1 <t<τ i 1 2 n, x 1 c c i1 P inf inf v it, x > a γt x bγt 2 τ i 1 <t<τ i 1 2 n, 7

Notice that, for τ i 1 < t < τ i we have i 2 bγτi 1 v i t, x = G t τi 1 x, ydy c a γτ i 1 t [ 1 2 i G t s x, yhs, y v i s, y] τ i 1 c As a consequence, by Lemma 3 P σ i < 2 n F τ i 1 P C exp sup sup τ i 1 t τ i 1 2 n, x 1 2 c W ds, dy. N i t, x > 1 C 1 n 1 2 ɛ. 11 Next we set up some notation. Let B n be the event that at least half of the variables σ i : i =,..., n satisfy Note that τ i < 2T n A n B n since if more than half of the σ i : i = 1,..., n are larger than or equal to 2T/n then τ n > T. For convenience we assume that n = 2k is even, and leave the odd case to the reader. Let Ξ n be all the subsets of {1,..., n} of cardinality k = n/2. Using Stirling s formula, the reader can verify that as n n = O2 n 12 n/2 Then, P B n P k {i 1,...,i k } Ξ n j=1 {i 1,...,i k } Ξ n P k j=1 { σ ij σ ij < 2T n < 2T n } 8

Using the estimate 11 and 12 yields P B n C 2 n exp C 1 n 1/2 ε n C exp C 1 n 3/2 ε C 2 n C exp C 1 n 3/2 ε where the constants C, C 1 may have changed from line to line. Hence, c n P ut, x < C exp 2 C 1 n 3/2 ε 13 Finally, substituting 5 into 7 yields E [ut, x p ] <. 4 Smoothness of the density Let ut, x be the solution to Equation 2. Assume that the coefficients b and σ are continuously differentiable with bounded derivatives. Then ut, x belongs to the Soboev space D 1,p for all p > 1, and the derivative D θ,ξ ut, x satisfies the following evolution equation D θ,ξ ut, x = θ G t s x, y b s, y, us, yd θ,ξus, ydyds G t s x, y σ θ s, y, us, yd θ,ξus, yw dy, ds σuθ, ξg t θ x, ξ, 14 if θ < t and D θ,ξ ut, x = if θ > t. That is, D θ,ξ ut, x is the solution of the stochastic partial differential equation D θ,ξ u = 2 D θ,ξ u b t x 2 t, x, ut, xd θ,ξu σ t, x, ut, xd θ,ξu 2 W t x on [θ, [, 1], with Dirichlet boundary conditions and initial condition σuθ, ξδ x ξ. Theorem 5 Let ut, x be the solution of Equation 2 with initial condition u, x = u x, and Dirichlet boundary conditions ut, = ut, 1 =. We will assume that u is an α-hölder continuous function for some α >, which satisfies the boundary conditions u = u 1 =. Assume that the coefficients b and σ are infinitely differentiable functions with bounded derivatives. Then, if σ, y, u y for some y, 1, ut, x has a smooth density for all t, x such that t > and x, 1. 9

Proof. From the results proved by Bally and Pardoux in [1] we know that ut, x belongs to the space D for all t, x. Set C t,x = D θ,ξ ut, x 2 dξdθ. Then, by Theorem 1 it suffices to show that EC p t,x < for all p 2. Suppose that σ, y, u y >. By continuity we have that σ, y, u, y δ > for all y [a, b], 1. Then t b t b 2 C t,x D θ,ξ ut, x 2 dξdθ D θ,ξ ut, xdξ dθ. a Set Yt,x θ = b D a θ,ξut, xdξ. Fix r < 1 and ε > such that ε r < t. From ε ε r r Yt,x 2 Yt,x 2 Y θ 2 t,x dθ Ct,x we get ε r P C t,x < ε P P a Yt,x 2 Y θ Yt,x < 2ε 1 r 2 = P A 1 P A 2. t,x 2 dθ > ε Integrating equation 14 in the variable ξ yields the following equation for the process {Yt,x, θ t θ, x [, 1]} Y θ t,x = θ θ b a G t s x, y b s, y, us, yy θ s,ydyds G t s x, y σ θ s, y, us, yy s,yw dy, ds σuθ, ξg t θ x, ξdξ. 15 In particular, for θ =, the initial condition is Y,ξ = σ, ξ, u, ξ1 [a,b]ξ, and by Theorem 2 the random variable Yt,x has negative moments of all orders. Hence, for all p 1, P A 2 ε p 1

if ε ε. In order to handle the probability P A 1 we write [ P A 1 ε r 1q sup E Yt,x θ Y 2q] [ t,x E Yt,x θ Y 2q] 1/2 t,x. θ ε r We claim that and [ sup E Yx,t θ 2q] <, 16 θ t [ sup E Yt,x θ Y 2q] t,x < ε 2sq, 17 θ ε r for some s >. Property 16 follows easily from Equation 15. On the other hand, the difference Yt,x θ Yt,x satisfies Y θ t,x Y t,x = = θ θ θ 1 θ 1 b a G t s x, y b s, y, us, yy θ s,y Y x,tdyds G t s x, y σ θ s, y, us, yys,y Y x,tw dy, ds G t s x, y b s, y, us, yy s,ydyds G t s x, y σ s, y, us, yy s,yw dy, ds σuθ, ξg t θ x, ξ σu ξg t x, ξdξ 5 Ψ i θ. i=1 Applying Gronwall s lemma and standard estimates, to show 17 it suffices to prove that sup E Ψ i θ 2q < ε 2sq, 18 θ ε r for i = 3, 4, 5 and for some s >. The estimate 18 is clear for i = 3, 4 and for i = 5 we use the properties of the heat kernel and the Hölder continuity of the initial condition u. Finally, it suffices to choose r > 1 s and we get the desired estimate for P A 1. The proof is now complete. 11

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