Fourier integral operators in SG classes II application to SG hyperbolic cauchy problems

Solutions of Hyperbolic Stochastic PDEs on Bounded and Unbounded Domains

“…Op(h jk (t)) (D t − Op(θ j (t))) l j −k , (4.12) Under the hypotheses of weak SG-hyperbolicity with constant multiplicities or with involutive roots, plus a suitable Levi condition 3 , or of strict SG-hyperbolicity, it is possible to show that the Cauchy problem (4.14) can be solved recursively by induction on the length of the multiindex γ . This follows from the next Theorem 4.6, which summarizes some of the main results proved in [1,4,8,12,14], applied to L (0,0,...) .…”

“…The fundamental solution operator to such a system allows then to write the (unique) solution of the original Cauchy problem in terms of SG-FIOs (modulo remainder terms). A remarkable feature, typical for these classes of hyperbolic problems, is the well-posedness with loss of decay/gain of growth at infinity, observed, e.g., in [2,3,12]. We also mention that random-field solutions of hyperbolic SPDEs via Fourier integral operator methods have been recently studied in [5,8], while function-valued solutions for associated semilinear hyperbolic SPDEs have been obtained in [7].…”

“…with 0 < T 0 ≤ T . By the theory developed in [1,12], it is possible to prove that the following proposition holds true.…”

“…. , N , in the three cases considered here E(t, s) can actually be reduced, modulo smoothing operators, to a finite linear combination of (compositions of) SG Fourier integral operators, see [1,[12][13][14]. The next Theorem 4.10 is the analogue for systems of Theorem 4.6.…”

“…The second main tool we use in this paper is the SG calculus of Fourier integral operators (further abbreviated as SG-FIOs theory). Applications of the SG-FIOs theory to SG-hyperbolic Cauchy problems were initially given in [12,14]. Many authors have, since then, expanded the SG-FIOs theory and its applications to the solution of hyperbolic problems in various directions.…”