We study some classes of pseudo-differential operators with symbols a admitting anisotropic exponential growth at infinity and we prove mapping properties for these operators on Gelfand-Shilov spaces of type S . Moreover, we deduce algebraic and certain invariance properties of these classes.

We deduce one-parameter group properties for pseudo-differential operators Op(a), where a belongs to the class Γ (ω 0 ) * of certain Gevrey symbols. We use this to show that there are pseudo-differential operators Op(a) and Op(b) which are inverses to each others, where a ∈ ΓWe apply these results to deduce lifting property for modulation spaces and construct explicit isomorpisms between them. For each weight functions ω, ω 0 moderated by GRS submultiplicative weights, we prove that the Toeplitz operator (or localization operator) Tp(ω 0 ) is an isomorphism from M p,q(ω) onto M p,q(ω/ω 0 ) for every p, q ∈ (0, ∞].

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We study a class of hyperbolic Cauchy problems, associated with linear operators and systems with polynomially bounded coefficients, variable multiplicities and involutive characteristics, globally defined on R n . We prove wellposedness in Sobolev-Kato spaces, with loss of smoothness and decay at infinity. We also obtain results about propagation of singularities, in terms of wave-front sets describing the evolution of both smoothness and decay singularities of temperate distributions. Moreover, we can prove the existence of random-field solutions for the associated stochastic Cauchy problems. To this aim, we first discuss algebraic properties for iterated integrals of suitable parameter-dependent families of Fourier integral operators, associated with the characteristic roots, which are involved in the construction of the fundamental solution. In particular, we show that, also for this operator class, the involutiveness of the characteristics implies commutative properties for such expressions. 2010 Mathematics Subject Classification. Primary: 58J40; Secondary: 35S05, 35S30, 47G30, 58J45. Key words and phrases. Fourier integral operator, multi-product, hyperbolic Cauchy problem, involutive characteristics, stochastic PDEs. of operators such that $ & % LEpt, sq " 0 0 ď s ď t ď T, Eps, sq " I, s P r0, Tq, so obtaining by Duhamel's formula the solution Uptq " Ept, 0qG`i ż t 0Ept, sqFpsqdsto the system and then the solution uptq to the corresponding higher order Cauchy problem (1.1) by the reduction procedure.In the present paper we consider the Cauchy problem (1.1) in the SG setting, that is, under the growth condition (1.3), the hyperbolicity condition (1.4) and the involutiveness assumption (1.5) for suitable parameter-dependent, real-valued symbols b jk , d jk P C 8 pr0, Ts; S 0,0 pR 2n qq, j, k P N. The involutiveness of the characteristic roots of the operator, or, equivalently, of the eigenvalues of the (diagonal) principal part of the corresponding first order system is here the main assumption (see Assumption I in Section 3.3 for the precise statement of such condition).We obtain several results concerning the Cauchy problem (1.1): ‚ we prove, in Theorem 5.8, the existence, for every f P C 8 pr0, Ts; H r,ρ pR n qq and g k P H r`m´1´k,ρ`m´1´k pR n q, k " 0, . . . , m´1, of a uniquefor a suitably small 0 ă T 1 ď T, solution of (1.1); we also provide, in (5.15), the explicit expression of u, using SG Fourier integral operators (SG FIOs); ‚ we give a precise description, in Theorem 5.21, of a global wave-front set of the solution, under a (mild) additional condition on the operator L; namely, we prove that, when L is SG-classical, the (smoothness and decay) singularities of the solution of (1.1) with f " 0 are points of unions of arcs of bicharacteristics, generated by the phase functions of the SG FIOs appearing in the expression (5.15), and emanating from (smoothness and decay) singularities of the Cauchy data g k , k " 0, . . . , m´1; ‚ we deal, in Theorem 6.3, with a stochastic version of the Cauchy proble...