Random-field solutions of weakly hyperbolic stochastic partial differential equations with polynomially bounded coefficients

Abstract

We study random-field solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider linear equations under suitable hyperbolicity hypotheses, and we provide conditions on the initial data and on the stochastic term, namely, on the associated spectral measure, so that these kind of solutions exist in suitably chosen functional classes. We also give a regularity result for the expected value of the solution.

Appendix A

We collect in this Appendix, for the convenience of the reader, some additional results concerning the SG-calculus and its applications to hyperbolic problems, which we mentioned along the main text. This material appeared, sometimes in slightly different form, in [4] and the references quoted therein.

The continuity property of the elements of ${{\text {Op}}} (S^{m,\mu })$ on the scale of spaces $H^{z,\zeta }({\mathbb {R}}^d)$, $(m,\mu ),(z,\zeta )\in {\mathbb {R}}^2$, is precisely expressed in the next Theorem A.1 (see [13] and the references quoted therein for the result on more general classes of SG-symbols).

Theorem A.1

Let $a\in S^{m,\mu }({\mathbb {R}}^d)$, $(m,\mu )\in {\mathbb {R}}^2$. Then, for any $(z,\zeta )\in {\mathbb {R}}^2$, ${{\text {Op}}} (a)\in {\mathcal {L}}(H^{z,\zeta }({\mathbb {R}}^d),H^{z-m,\zeta -\mu }({\mathbb {R}}^d))$, and there exists a constant $C>0$, depending only on $d,m,\mu ,z,\zeta$, such that

$$\begin{aligned} \Vert {{\text {Op}}} (a)\Vert _{{\mathscr {L}}(H^{z,\zeta }({\mathbb {R}}^d), H^{z-m,\zeta -\mu }({\mathbb {R}}^d))}\le C\Vert a \Vert _{\left[ \frac{d}{2}\right] +1}^{m,\mu }, \end{aligned}$$

(A.1)

where [t] denotes the integer part of $t\in {\mathbb {R}}$.

The following characterization of the class ${\mathcal {O}}(-\infty ,-\infty )$ is often useful, see [13].

Theorem A.2

The class ${\mathcal {O}}(-\infty ,-\infty )$ coincides with ${{\text {Op}}} (S^{-\infty ,-\infty }({\mathbb {R}}^d))$ and with the class of smoothing operators, that is, the set of all the linear continuous operators $A:{\mathcal {S}}^\prime ({\mathbb {R}}^d)\rightarrow {\mathcal {S}}({\mathbb {R}}^d)$. All of them coincide with the class of linear continuous operators A admitting a Schwartz kernel $k_A$ belonging to ${\mathcal {S}}({\mathbb {R}}^{2d})$.

An operator $A= {{\text {Op}}} (a)$ and its symbol $a\in S ^{m,\mu }$ are called elliptic (or $S ^{m,\mu }$-elliptic) if there exists $R\ge 0$ such that

$$\begin{aligned} C\langle x\rangle ^{m} \langle \xi \rangle ^{\mu }\le |a(x,\xi )|,\qquad |x|+|\xi |\ge R, \end{aligned}$$

for some constant $C>0$. If $R=0$, $a^{-1}$ is everywhere well-defined and smooth, and $a^{-1}\in S ^{-m,-\mu }$. If $R>0$, then $a^{-1}$ can be extended to the whole of ${\mathbb {R}}^{2d}$ so that the extension ${\widetilde{a}}_{-1}$ satisfies ${\widetilde{a}}_{-1}\in S ^{-m,-\mu }$. An elliptic SG operator $A \in {{\text {Op}}} (S ^{m,\mu })$ admits a parametrix $A_{-1}\in {{\text {Op}}} (S ^{-m,-\mu })$ such that

$$\begin{aligned} A_{-1}A=I + R_1, \quad AA_{-1}= I+ R_2, \end{aligned}$$

for suitable $R_1, R_2\in {{\text {Op}}} (S^{-\infty ,-\infty })$, where I denotes the identity operator. In such a case, A turns out to be a Fredholm operator on the scale of functional spaces $H^{z,\zeta }({\mathbb {R}}^d)$, $(z,\zeta )\in {\mathbb {R}}^2$.

The study of the composition of $M\ge 2$SG FIOs of type I ${{\text {Op}}} _{\varphi _j}(a_j)$ with regular SG-phase functions $\varphi _j\in {\mathfrak {P}}_\delta (\lambda _j)$ and symbols $a_j\in S^{m_j,\mu _j}({\mathbb {R}}^{d})$, $j=1,\ldots ,M$, has been done in [4]. The result of such composition is still an SG-FIO with a regular SG-phase function $\varphi$ given by the so-called multi-product$\varphi _1\sharp \cdots \sharp \varphi _M$ of the phase functions $\varphi _j$, $j=1,\ldots ,M$, and symbol a as in Theorem A.3 here below.

Theorem A.3

Consider, for $j=1,2, \dots , M$, $M\ge 2$, the SG FIOs of type I ${{\text {Op}}} _{\varphi _j}(a_j)$ with $a_j\in S^{m_j,\mu _j}({\mathbb {R}}^{d})$, $(m_j,\mu _j)\in {\mathbb {R}}^2$, and $\varphi _j\in {\mathfrak {P}}_\delta (\lambda _j)$ such that $\lambda _1+\cdots +\lambda _M\le \lambda \le \frac{1}{4}$ for some sufficiently small $\lambda >0$. Then, there exists $a\in S^{m,\mu }({\mathbb {R}}^{d})$, $m=m_1+\cdots +m_M$, $\mu =\mu _1+\cdots +\mu _M$, such that, setting $\phi =\varphi _1\sharp \cdots \sharp \varphi _M$, we have

$$\begin{aligned} {{\text {Op}}} _{\varphi _1}(a_1) \circ \cdots \circ {{\text {Op}}} _{\varphi _M}(a_M)= {{\text {Op}}} _{\phi }(a).\end{aligned}$$

Moreover, for any $\ell \in {\mathbb {N}}_0$ there exist $\ell ^\prime \in {\mathbb {N}}_0$, $C_\ell >0$ such that

$$\begin{aligned} \Vert a \Vert _\ell ^{m,\mu } \le C_\ell \prod _{j=1}^M \Vert a_j \Vert _{\ell ^\prime }^{m_j,\mu _j}. \end{aligned}$$

(A.2)

Theorem A.3 is a corollary of the main Theorem in [4]. There, the multi-product of regular SG-phase functions is defined and its properties are studied, parametrices and compositions of regular SG FIOs with amplitude identically equal to 1 are considered, leading to the general composition ${{\text {Op}}} _{\varphi _1}(a_1) \circ \cdots \circ {{\text {Op}}} _{\varphi _M}(a_M)$. It is needed for the determination of the fundamental solutions of the hyperbolic operators (1.3), involved in (1.1), in the case of involutive roots with non-constant multiplicities, see [1].

The next one is a key result in the analysis of SG-hyperbolic Cauchy problems by means of the corresponding class of Fourier operators. Given a symbol $\varkappa \in C([0,T]; S^{1,1})$, set $\Delta _{T_0}=\{(s,t)\in [0,T_0]^2:0\le s\le t\le T_0\}$, $0<T_0\le T$, and consider the eikonal equation

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t\varphi (t,s,x,\xi )=\varkappa (t,x,\varphi '_x(t,s,x,\xi )),&{} t\in [s,T_0], \\ \varphi (s,s,x,\xi )=x\cdot \xi ,&{} s\in [0,T_0), \end{array}\right. } \end{aligned}$$

(A.3)

with $0<T_0\le T$. By an extension of the theory developed in [15], it is possible to prove that the following Proposition A.4 holds true.

Proposition A.4

For any small enough $T_0\in (0,T]$, equation (A.3) admits a unique solution $\varphi \in C^1(\Delta _{T_0},$$S^{1,1}({\mathbb {R}}^d))$, satisfying $J\in C^1(\Delta _{T_0},S^{1,1}({\mathbb {R}}^d))$ and

$$\begin{aligned} \partial _s\varphi (t,s,x,\xi )=-\varkappa (s,\varphi '_\xi (t,s,x,\xi ),\xi ), \end{aligned}$$

(A.4)

for any $(t,s)\in \Delta _{T_0}$. Moreover, for every $\ell \in {\mathbb {N}}_0$ there exists $\delta >0$, $c_\ell \ge 1$ and ${\widetilde{T}}_\ell \in [0,T_0]$ such that $\varphi (t,s,x,\xi )\in {\mathfrak {P}}_\delta (c_\ell |t-s|)$, with $\Vert J\Vert _{2,\ell }\le c_\ell |t-s|$ for all $(t,s)\in \Delta _{{\widetilde{T}}_\ell }$.

Remark A.5

Of course, if additional regularity with respect to $t\in [0,T]$ is fulfilled by the symbol $\varkappa$ in the right-hand side of (A.3), this reflects in a corresponding increased regularity of the resulting solution $\varphi$ with respect to $(t,s)\in \Delta _{T_0}$. Since here we are not dealing with problems concerning the t-regularity of the solution, we assume smooth t-dependence of the coefficients of L. Some of the results below will anyway be formulated in situations of lower regularity with respect to t.

The fundamental solution of a first order SG-hyperbolic system with diagonal principal part, E(t, s), has the following properties, which actually hold for the broader class of symmetric first order system of the type (3.13), of which systems with real-valued, diagonal principal part are a special case, see [13], Ch. 6, §3, and [15].

Theorem A.6

Let the system (3.13) be hyperbolic with diagonal principal part $\kappa _1\in C^1([0,T], S^{1,1}$$({\mathbb {R}}^d))$, and lower order part $\kappa _0\in C^1([0,T], S^{0,0}({\mathbb {R}}^d))$. Then, for any choice of $W_0\in H^{z,\zeta }({\mathbb {R}}^d)$, $Y\in C([0,T], H^{z,\zeta }({\mathbb {R}}^d))$, there exists a unique solution $W\in C([0,T], H^{z,\zeta }({\mathbb {R}}^d))\cap C^1([0,T], H^{z-1,\zeta -1}({\mathbb {R}}^d))$ of (3.13), $(z,\zeta )\in {\mathbb {R}}^2$, given by Duhamel’s formula

$$\begin{aligned} W(t)=E(t,s)W_0+i\displaystyle \int _s^t E(t,\vartheta )Y(\vartheta )d\vartheta ,\quad t\in [0,T]. \end{aligned}$$

Moreover, the solution operatorE(t, s) has the following properties:

• (1) $E(t,s):{\mathcal {S}}^\prime ({\mathbb {R}}^d)\rightarrow {\mathcal {S}}^\prime ({\mathbb {R}}^d)$ is an operator belonging to ${\mathcal {O}}(0,0)$, $(t,s)\in [0,T]^2$; its first order derivatives, $\partial _t E(t,s)$, $\partial _s E(t,s)$, exist in the strong operator convergence of ${\mathscr {L}}(H^{z,\zeta }({\mathbb {R}}^d),H^{z-1,\zeta -1}({\mathbb {R}}^d))$, $(z,\zeta )\in {\mathbb {R}}^2$, and belong to ${\mathcal {O}}(1,1)$;

• (2) E(t, s) is bounded and strongly continuous from $[0,T]^2_{ts}$ to ${\mathscr {L}}(H^{z,\zeta }({\mathbb {R}}^d),H^{z,\zeta }({\mathbb {R}}^d))$, $(z,\zeta )\in {\mathbb {R}}^2$; $\partial _t E(t,s)$ and $\partial _s E(t,s)$ are bounded and strongly continuous from $[0,T]^2_{ts}$ to ${\mathscr {L}}(H^{z,\zeta }({\mathbb {R}}^d),H^{z-1,\zeta -1}({\mathbb {R}}^d))$, $(z,\zeta )\in {\mathbb {R}}^2$;

• (3) for $t,s,t_0\in [0,T]$ we have

  $$\begin{aligned} E(t_0,t_0)=I, \quad E(t,s)E(s,t_0)=E(t,t_0), \quad E(t,s)E(s,t)=I; \end{aligned}$$

• (4) E(t, s) satisfies, for $(t,s)\in [0,T]^2$, the differential equations

  $$\begin{aligned} D_tE(t,s) - ( {{\text {Op}}} (\kappa _1(t)) + {{\text {Op}}} (\kappa _0(t)))E(t,s)&= 0, \end{aligned}$$

  (A.5)
  $$\begin{aligned} D_sE(t,s) + E(t,s)( {{\text {Op}}} (\kappa _1(s)) + {{\text {Op}}} (\kappa _0(s)))&= 0; \end{aligned}$$

  (A.6)

• (5) the operator family E(t, s) is uniquely determined by the properties (1)-(3) here above, and one of the differential equations (A.5), (A.6).

Corollary A.7

• (1) Under the hypotheses of Theorem A.6, E(t, s) is invertible on ${\mathcal {S}}({\mathbb {R}}^d)$, ${\mathcal {S}}^\prime ({\mathbb {R}}^d)$, and $H^{z,\zeta }({\mathbb {R}}^d)$, $(z,\zeta )\in {\mathbb {R}}^2$, with inverse given by E(s, t), $s,t\in [0,T]$.

• (2) If, additionally, one assumes $\kappa _1\in C^m([0,T], S^{1,1}({\mathbb {R}}^d))$, $\kappa _0\in C^m([0,T], S^{0,0}({\mathbb {R}}^d))$, $m\ge 2$, the partial derivatives $\partial _t^j\partial _s^k E(t,s)$ exist in strong operator convergence of ${\mathcal {S}}({\mathbb {R}}^d)$ and ${\mathcal {S}}^\prime ({\mathbb {R}}^d)$, and $\partial _t^j\partial _s^k E(t,s)\in {\mathcal {O}}(j+k,j+k)$, $j+k\le m$. Moreover, $\partial _t^j\partial _s^k E(t,s)$ is strongly continuous from $[0,T]^2_{ts}$ to every ${\mathscr {L}}(H^{z,\zeta }({\mathbb {R}}^d),$$H^{z-j-k,\zeta -j-k}({\mathbb {R}}^d))$, $(z,\zeta )\in {\mathbb {R}}^2$, $j+k\le m$.

In [4] we have proved the next Theorem A.8, concerning the structure of E(t, s), in the spirit of the approach followed in [24].

Theorem A.8

Under the same hypotheses of Theorem A.6, if $T_0$ is small enough, for every fixed $(t,s)\in \Delta _{T_0}$, E(t, s) is a limit of a sequence of matrices of SG FIOs of type I, with regular phase functions $\varphi _{jk}(t,s)$ belonging to ${\mathfrak {P}}_\delta (c_h|t-s|)$, $c_h\ge 1$, of class $C^1$ with respect to $(t,s)\in \Delta _{T_0}$, and amplitudes belonging to $C^1(\Delta _{T_0}, S^{0,0}({\mathbb {R}}^{d}))$.

The next results are employed to switch from (4.3) to a first order linear system of the form (3.13).

Proposition A.9

Let L be a hyperbolic operator with constant multiplicities $l_{j}$, $j=1,\dots ,n \le m$. Denote by $\theta _j\in G_j$, $j=1,\dots ,n$, the distinct real roots of ${{\mathcal {L}}}_m$ in (1.4). Then, it is possible to factor L as

$$\begin{aligned} L = L_{n} \cdots L_{1} + \sum _{j=1}^m {{\text {Op}}} (r_{j}(t)) D_{t}^{m-j} \end{aligned}$$

(A.7)

with

$$\begin{aligned} L_{j}&= (D_{t} - {{\text {Op}}} (\theta _{j}(t)))^{l_{j}} + \sum _{k=1}^{l_{j}} {{\text {Op}}} (h_{jk}(t)) \, (D_{t} - {{\text {Op}}} (\theta _{j}(t)))^{l_{j}-k},\end{aligned}$$

(A.8)

$$\begin{aligned} h_{jk}&\in C^\infty ([0,T], S^{k-1, k-1}({\mathbb {R}}^d)), \quad r_{j} \in C^\infty ([0,T],S^{-\infty ,-\infty }({\mathbb {R}}^d)), \quad \nonumber \\&\quad j=1, \dots , n, k = 1, \dots , l_{j}. \end{aligned}$$

(A.9)

The following corollary is an immediate consequence of Proposition A.9, and is proved by means of a reordering of the distinct roots $\theta _{j}$, $j=1,\dots ,n$.

Corollary A.10

Let ${\varpi }_{j}$, $j=1, \dots , n$, denote the reordering of the n-tuple $(1, \dots , n)$, given, for $k = 1, \dots , n$, by

$$\begin{aligned} {\varpi }_{j}(k) = {\left\{ \begin{array}{ll} j + k -1 &{} \text{ for } j + k \le n+1, \\ j + k - n - 1 &{} \text{ for } j + k > n+1, \end{array}\right. } \end{aligned}$$

(A.10)

That is, for $n\ge 2$, ${\varpi }_{1} = (1, \dots , n), {\varpi }_{2} = (2, \dots , n,1),\dots , {\varpi }_{n} = (n, 1, \dots , n-1)$. Then, under the same hypotheses of Proposition A.9, we have, for any $p=1, \dots , n$,

$$\begin{aligned} L = L^{(p)}_{{\varpi }_{p}(n)} \dots L^{(p)}_{{\varpi }_{p}(1)} + \sum _{j=1}^m {{\text {Op}}} (r^{(p)}_{j}(t)) D_{t}^{m-j} \end{aligned}$$

(A.11)

with

$$\begin{aligned}&L^{(p)}_{j}= (D_{t} - {{\text {Op}}} (\theta _{j}(t)))^{l_{j}} + \sum _{k=1}^{l_{j}} {{\text {Op}}} (h^{(p)}_{jk}(t)) \, (D_{t} - {{\text {Op}}} (\theta _{j}(t)))^{l_{j}-k},\quad \end{aligned}$$

(A.12)

$$\begin{aligned}&h^{(p)}_{jk} \in C^\infty ([0,T], S^{k-1, k-1}({\mathbb {R}}^d)), j=1, \dots , n, k = 1, \dots , l_{j}, \quad \nonumber \\&\quad r^{(p)}_{j} \in C^\infty ([0,T],S^{-\infty ,-\infty }({\mathbb {R}}^d)), j=1, \dots , m. \end{aligned}$$

(A.13)

Remark A.11

Of course, for $n=1$, we only have the single “reordering” $\varpi _1=(1)$, $l_1=l=m$, and

$$\begin{aligned} L = L^{(1)}_{1} + \sum _{j=1}^m {{\text {Op}}} (r^{(1)}_{j}(t)) D_{t}^{m-j} \end{aligned}$$

with

$$\begin{aligned}&L^{(1)}_{1}= (D_{t} - {{\text {Op}}} (\theta _{1}(t)))^m + \sum _{k=1}^{m} {{\text {Op}}} (h^{(1)}_{1k}(t)) \, (D_{t} - {{\text {Op}}} (\theta _{1}(t)))^{m-k}, \\&h^{(1)}_{1k} \in C^\infty ([0,T], S^{k-1, k-1}({\mathbb {R}}^d)), k = 1, \dots , m, \quad \\&\quad r^{(1)}_{j} \in C^\infty ([0,T],S^{-\infty ,-\infty }({\mathbb {R}}^d)), j=1,\dots ,m. \end{aligned}$$

With inductive procedures similar to those performed in [9, 10] and [27], respectively, it is possible to prove the following Lemma A.12.

Lemma A.12

Under the same hypotheses of Proposition A.9, for all $k=0, \dots , m-1$, it is possible to find symbols $\varsigma _{kpq} \in C^\infty ([0,T],S^{k-q+l_{p}-n,k-q+l_p-n}({\mathbb {R}}^d))$, $p = 1, \dots , n$, $q = 0, \dots , l_{p}-1$, such that, for all $t\in [0,T]$,

$$\begin{aligned} \theta ^k = \sum _{p=1}^n \left[ \sum _{q=0}^{l_{p}-1} \varsigma _{kpq}(t) (\theta - \theta _{p}(t))^q\right] \cdot \left[ \prod _{\genfrac{}{}{0.0pt}1{1\le j \le n}{j\not =p}} (\theta - \theta _{j}(t))^{l_{j}} \right] . \end{aligned}$$

In the case of strict hyperbolicity, or, more generally, hyperbolicity with constant multiplicities, we can actually “decouple” the equations in (3.13) into n blocks of smaller dimensions, by means of the so-called perfect diagonalizer, an element of $C^\infty ([0,T], {{\text {Op}}} (S^{0,0}))$. Thus, the solution of (3.13) can be reduced to the solution of n independent smaller systems. The principal part of the coefficient matrix of each one of such decoupled subsystems admits then a single distinct eigenvalue of maximum multiplicity, so that it can be treated, essentially, like a scalar SG hyperbolic equations of first order. Explicitely, see, e.g., [15, 24],

Theorem A.13

Assume that the system (3.13) is hyperbolic with constant multiplicities $\nu _j$, $j=1,\dots ,N$, $\nu _1+\cdots +\nu _n=\nu$, with diagonal principal part $\kappa _1\in C^\infty ([0,T],S^{1,1}({\mathbb {R}}^d))$ and $\kappa _0\in C^\infty ([0,T], S^{0,0}({\mathbb {R}}^d))$, both of them ($\nu \times \nu$)-dimensional matrices. Then, there exist ($\nu \times \nu$)-dimensional matrices $\omega \in C^\infty ([0,T],S^{0,0}({\mathbb {R}}^d))$ and ${\widetilde{\kappa }}_0\in C^\infty ([0,T],S^{0,0}({\mathbb {R}}^d))$ such that

$$\begin{aligned}&\det (\omega )\asymp 1\Rightarrow \omega ^{-1}\in C^\infty ([0,T],S^{0,0}({\mathbb {R}}^d)), \quad \nonumber \\&\quad {\widetilde{\kappa }}_0=\mathrm {diag}({\widetilde{\kappa }}_{01}, \dots , {\widetilde{\kappa }}_{0n}), \; {\widetilde{\kappa }}_{0j} (\nu _j\times \nu _j)\text {-dimensional matrix}, \end{aligned}$$

and

$$\begin{aligned}&(D_t- {{\text {Op}}} (\kappa _1(t))- {{\text {Op}}} (\kappa _0(t))) {{\text {Op}}} (\omega (t)) \nonumber \\&\quad - {{\text {Op}}} (\omega (t))(D_t- {{\text {Op}}} (\kappa _1(t))\nonumber \\&\quad - {{\text {Op}}} ({\widetilde{\kappa }}_0(t))) \in C^\infty ([0,T], {{\text {Op}}} (S^{-\infty ,-\infty }({\mathbb {R}}^d)). \end{aligned}$$

(A.14)