Random-field solutions to linear hyperbolic stochastic partial differential equations with variable coefficients

Abstract

In this article we show the existence of a random-field solution to linear stochastic partial differential equations whose partial differential operator is hyperbolic and has variable coefficients that may depend on the temporal and spatial argument. The main tools for this, pseudo-differential and Fourier integral operators, come from microlocal analysis. The equations that we treat are second-order and higher-order strictly hyperbolic, and second-order weakly hyperbolic with uniformly bounded coefficients in space. For the latter one we show that a stronger assumption on the correlation measure of the random noise might be needed. Moreover, we show that the well-known case of the stochastic wave equation can be embedded into the theory presented in this article.

Solution theory to semilinear hyperbolic stochastic partial differential equations with polynomially bounded 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 main tools used to construct and study the solutions, namely, pseudodifferential and Fourier integral operators, come from microlocal analysis. To our best knowledge, in [7] their full potential has been rigorously applied for the first time within the theory of random-field solutions to hyperbolic SPDEs.…”

“…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.…”

Solutions of Hyperbolic Stochastic PDEs on Bounded and Unbounded Domains

“…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].…”

“…The time-derivative of the Brownian motion exists in the generalized sense and belongs to the Kondratiev space (S) −1,− p for p > 5 12 [23, p. 21]. We refer to it as to white noise and its formal expansion is given by…”

“…Example 5. 5 In this example we provide some instances of operators with variable (time-space depending) coefficients that provide an insight into the fine difference between strict and weak hyperbolicity. The examples are based on [8].…”