On the local linearization of the one-dimensional stochastic wave equation with a multiplicative space-time white noise forcing

In this note, we establish a bi-parameter linear localization of the one-dimensional stochastic wave equation with a multiplicative space-time white noise forcing.


Introduction
We consider the following stochastic wave equation (SNLW) on R × R: where F : R → R is a Lipschitz continuous function and ξ denotes the (Gaussian) space-time white noise on R × R whose space-time covariance is formally given by The expression (1.2) is merely formal but we can make it rigorous by testing it against a test function.
Definition 1.1.A two-parameter white noise ξ on R 2 is a family of centered Gaussian random variables {ξ(ϕ) : ϕ ∈ L 2 (R 2 )} such that In [17], Walsh studied the Ito solution theory for (1.1) and proved its well-posedness.See, for example, [17,p.323,Exercise 3.7] and [7, p.45], where the fundamental properties of solutions to (1.1) are stated (implicitly).For readers' convenience, we state and prove basic properties of solutions to (1.1) in Appendix A. Our main goal in this note is to study the local fluctuation property of solutions to (1.1).
Let us first consider the following stochastic heat equation: It is well known that, under suitable assumptions on F and u 0 , the solution to (1.3) locally linearizes; namely by letting Z heat denote the linear solution satisfying ∂ t Z heat − ∂ 2 x Z heat = ξ with Z heat | t=0 = 0, the solution u to (1.3) satisfies where, as ε → 0, the remainder term R ε (t, x) tends to 0 much faster than Z heat (t, x + ε) − Z heat (t, x).See, for example, [11,15,8,12].The relation (1.4) states that, for fixed t, local fluctuations (in x) of the solution u(t) are essentially given by those of Z heat (t).In other words, if we ignore precise regularity conditions, then (1.4) states that u(t) is controlled by Z heat (t) in the sense of controlled paths due to Gubinelli [10]; see [9,Definition 4.6].
In [13], Khoshnevisan and the first author studied an analogous issue for SNLW (1.1).In particular, they showed that the solution to (1.1) with initial data (u 0 , u 1 ) ≡ (0, 1) does not locally linearize (for fixed t), which shows a sharp contrast to the case of the stochastic heat equation.In this note, we change our viewpoint and study the local linearization issue for SNLW (1.1) from a bi-parameter point of view.
In [17], Walsh studied the well-posedness issue of (1.1) by first switching to the null coordinates: (1.5) In the null coordinates, the Cauchy problem (1.1) becomes where with the latter interpreted in a suitable sense.Note that this change of coordinates is via an orthogonal transformation (which in particular preserves the L 2 -inner product on R 2 ) and thus ξ is also a two-parameter white noise in the sense of Definition 1.1.By integrating in x 1 and x 2 , we can rewrite (1.6) as where x = (x 1 , x 2 ), y = (y 1 , y 2 ), and Under the Lipschitz assumption on F , one can then interpret the last term on the right-hand side of (1.8) as a two-parameter stochastic integral ( [3,4]) and prove well-posedness of (1.8) (and hence of the original SNLW (1.1)); see [17,7].
In the following, we study the local linearization property of the solution v to (1.6) in the variable x = (x 1 , x 2 ) in a bi-parameter manner.For this purpose, let us introduce some notations.Let Z be the linearization of v in (1.6); namely, Z is the solution to (1.6) with F (v) ≡ 1 and (u 0 , u 1 ) = (0, 0): By a direction integration, we then have which is to be interpreted as a two-parameter stochastic integral.Given ε ∈ R, define the difference operator δ (j) ε , j = 1, 2, by setting (1.11) Then, from (1.11) and (1.8), we have (1.12) Similarly, from (1.10), we have Thus, from the Wiener isometry (see, for example, [14, (20) on p. 7]), we have which shows that the decay rate of |δ ε Z(x)| is ∼ |ε| on average.The following lemma shows that the decay rate (as ε → 0) of |δ Then, for any κ > 0, we have almost surely.
Lemma 1.2 follows from a simple application of the Borel-Cantelli lemma.We present the proof of Lemma 1.2 in the next section.
Let us now turn to the local linearization property of solutions to SNLW (1.1).In [13], Khoshnevisan and the first author investigated the local linearization issue for SNLW (1.1) by studying local fluctuations (in x) of u(t) for fixed t.While such an approach is suitable for the stochastic heat equation (1.3), it does not seem to be appropriate for the wave equation.We instead propose to study bi-parameter fluctuations of u with respect to the null coordinates For this purpose, let us first state the local linearization result for SNLW (1.6) in the null coordinates.Given (x 1 , x 2 ) ∈ R 2 and (small) ε ∈ R, define the , let v be the solution to SNLW (1.6) in the null coordinates and Z be as in (1.10).Then, given any x = (x 1 , x 2 ) ∈ R 2 and finite p ≥ 2, we have uniformly in small ε > 0.
Theorem 1.3 establishes bi-parameter local linearization for the solution v to (1.6) in the following sense; the remainder term R ± ε (x 1 , x 2 ) decays like ∼ ε 3 2 as ε → 0 on average, and hence, in view of Lemma 1.2, R ± ε (x 1 , x 2 ) tends to 0 much faster than δ ε Z(x 1 , x 2 ).As an immediate corollary to Theorem 1.3 and (1.7), we obtain the following bi-parameter local linearization for the solution u to SNLW (1.1) in the original space-time coordinates.
, let u be the solution to SNLW (1.1) and Z be the linear solution, satisfying Then, given any (t, x) ∈ R 2 and finite p ≥ 2, we have uniformly in small ε > 0, where ∆ ε and ∆ ε are defined by ).We also have the following claim as a direct corollary to Lemma 1.2 and (1.7); given any κ > 0 and (t, x) ∈ R 2 , we have almost surely.Hence, from Theorem 1.4 and (1.19), we have where the remainder term R ε Z(t, x), j = 1, 2, thus establishing a bi-parameter local linearization for the solution u to SNLW (1.1) in the original space-time coordinates.
Remark 1.5.In Theorems 1.3 and 1.4, we established local linearizability of SNLW in a biparameter sense.Such a bi-parameter point of view is natural in studying the one-dimensional (stochastic) wave equation.See, for example, [16,6,2] and the references therein.
Remark 1.6.(i) If f is a smooth function, we have ∆ (1) ).Thus, if both the solution u to (1.1) and Z satisfying (1.18) were smooth (in both t and x), then we would formally have ∆ (1)  ε u − F (u)∆ (1)  ε Z = O(ε 3 ) and ∆ (2) ).The main point of Theorem 1.4 is to justify such heuristics when both u and Z are non-smooth functions.
(ii) As shown in [13], local linearization in x (for fixed t) fails for (1.1).By switching the role of t and x, we also see that local linearization in t (for fixed x) fails for (1.1).
One may also study local linearization properties in x 1 (for fixed x 2 ) of solutions to (1.6) in the null coordinates.Let x 2 > x 1 .From (1.8) and (1.10), we have (1.20) On the other hand, from the Lipschitz continuity of F and (2.3), we have which is O(1) in the domain of integration for the second term on the right-hand side of (1.20) . Namely, the second term on the right-hand side of (1.20) does not decay faster than δ ε Z in general, and hence local linearization in x 1 (for fixed x 2 ) fails for (1.6).By symmetry, we also see that local linearization in x 2 (for fixed x 1 ) fails for (1.6).This is the reason that we need to consider the second order difference in studying local linearization for SNLW.
Next, we present the proof of Theorem 1.3.
Proof of Theorem 1.3.We only consider R + ε since the same proof applies to R − ε .As before, we use the short-hand notations x = (x 1 , x 2 ) and y = (y 1 , y 2 ).
We first recall the Hölder continuity of the solution v to (1.8).In particular, it follows from Proposition A.2 and (1.7) that, for any L > 0 and 2 ≤ p < ∞, we have where V 0 is as in (1.9).It is easy to see from (1.9) and (1.11) that Next, we estimate the term II in (2.4).To ensure adaptedness of the integrand, we first decompose the domain of integration (which is a square) into two triangular regions; see [5, p. 21] for a similar decomposition to recover adaptedness.For fixed x ∈ R 2 and small ε > 0, define the sets Namely, D 1 (x) is the triangular region with vertices: and while D 2 (x) is the triangular region with vertices: P 1 (x), P 3 (x), and By undoing the change of variables (1.7), we divide II into three parts as follows: From the Burkholder-Davis-Gundy inequality ([14, Theorem 5.27]), Minkowski's inequality, the Lipschitz continuity of F , and (2.3) with (2.6) and (1.5), we have (2.7) A similar calculation yields that where the X t,2 -norm is as in (A.3), it follows from (A.1) and (A.5) that Since H(0) = 0, Gronwall's inequality yields that H(t) = 0 for any t ∈ R + .This proves uniqueness of a solution.
Next, we prove existence.Define a sequence {u (n) } ∞ n=0 by setting u (0) (t, x) = ∂ t ˆR G(t, x − y)u 0 (y)dy + ˆR G(t, x − y)u 1 (y)dy and for n ∈ N.Then, Let T > 0 and 2 ≤ p < ∞.Then, from the Burkholder-Davis-Gundy inequality, Hölder's inequality, and the Lipschitz continuity of F , we have Hence, by defining H n by where the X t,p -norm is as in (A.3), it follows from (A.1) and (A.7) that there exists a constant C = C(T, p) > 0 such that Then, a Gronwall-type argument (see Lemma 6.5 in [14], for example) yields 1)! for any n ∈ N and t ∈ [0, T ].By summing over n ∈ N, we conclude that, given any T > 0 and finite p ≥ 2, there exists C 0 (T, p) > 0 such that for any t ∈ [0, T ].This implies that u (n) converges to some limit, denoted by u, with respect to the X T,p -norm for each T > 0 and finite p ≥ 2. In particular, u is the limit of u In view of the predictability of u (n) , we conclude that the limit u is also predictable.As a result, the limit u belongs to the class X defined in (A.4).Furthermore, from (A.6), we conclude that the limit u almost surely satisfies (A.2) for any (t, x) ∈ R + × R.
Proof.We have For 0 ≤ t ′ ≤ t ≤ T , we have Similar computations yield . This concludes the proof of Proposition A.2.