Solution theory to semilinear hyperbolic stochastic partial differential equations with polynomially bounded coefficients

Abstract

We study mild solutions of a class of stochastic partial differential equations, involving operators with polynomially bounded coefficients. We consider semilinear equations under suitable hyperbolicity hypotheses on the linear part. We provide conditions on the initial data and on the stochastic terms, namely, on the associated spectral measure, so that mild solutions exist and are unique in suitably chosen functional classes. More precisely, function-valued solutions are obtained, as well as a regularity result. 1 2 ALESSIA ASCANELLI, SANDRO CORIASCO, AND ANDRÉ SÜSS uniformly bounded coefficients depending on x ∈ R d , d ≥ 1. There, sufficient conditions on the stochastic terṁ Ξ and on the coefficients of A are given, in order to find a unique function-valued solution. In the present paper we show existence and uniqueness of a function-valued solution to a wider class of semilinear hyperbolic SPDEs, with possibly unbounded coefficients depending on (t,A disadvantage of the function(al-spaces)-valued solution u of (1.1) sketched above is that, in general, it cannot be evaluated in the spatial argument (usually, it is a random element in the t (that is, time) parameter, taking values in a L p (R d )-modeled Hilbert or Banach space). Then, an alternative approach focuses instead on the concept of stochastic integral with respect to a martingale measure. That is, the stochastic integral in (1.2) is defined through the martingale measure derived from the random noiseΞ. Here one obtains a so-called random-field solution, that is, u is defined as a map associating a random variable to each (t, x) ∈ [0, T 0 ]×R d , where T 0 > 0 is the time horizon of the equation. For more details, see, e.g., the classical references [10,17,34], and the recent papers [2,7], where the existence of a random-field solution in the case of linear hyperbolic SPDEs with (t, x)-dependent coefficients has been shown. On the other hand, a disadvantage of such random-field solution u of (1.1) is that its construction for non-linear equations is based on the stationarity condition Λ = Λ(t − s, x − y), which is fulfilled by SPDEs with constant coefficients, but cannot be assumed if we want to deal with more general linear operators L in (1.1), indeed admitting variable coefficients. Namely, we have shown in [2] how random-field solutions can be constructed for arbitrary order, linear (weakly) hyperbolic SPDEs, with (possibly unbounded) coefficients smoothly depending onIt is remarkable that, in many cases, the theory of integration with respect to processes taking values in functional spaces, and the theory of integration with respect to martingale measures, lead to the same solution u (in some sense) of an SPDE, see [18] for a precise comparison. We conclude the paper showing a result of that kind, comparing the function-valued solutions to (1.1) obtained in the present paper, in the special case of the linear equations, with the random-field solutions of the same equation found in [2].We remark that in the present paper, as well as in [2,7], the...

Solutions of Hyperbolic Stochastic PDEs on Bounded and Unbounded Domains

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“…The points are arranged according to their p-values 2. The shaded area indicates the smoothed kernel density estimator of the p-values.…”