Large deviations for a spatial average of stochastic heat and wave equations

In this article, we study a class of stochastic partial differential equations with fractional differential operators subject to some time-independent multiplicative Gaussian noise. We derive sharp conditions, under which a unique global $$L^p(\Omega )$$ -solution exists for all $$p\ge 2$$ . In this case, we derive exact moment asymptotics following the same strategy as that in a recent work by Balan et al. (Inst Henri Poincaré Probab Stat. To appear, 2021). In the case when there exists only a local solution, we determine the precise deterministic time, $$T_2$$ , before which a unique $$L^2(\Omega )$$ -solution exists, but after which the series corresponding to the $$L^2(\Omega )$$ moment of the solution blows up. By properly choosing the parameters, results in this paper interpolate the known results for both stochastic heat and wave equations.

We consider a stochastic partial differential equation (SPDE) driven by a Lévy white noise, with Lipschitz multiplicative term σ. We prove that, under some conditions, this equation has a unique random field solution. These conditions are verified by the stochastic heat and wave equations. We introduce the basic elements of Malliavin calculus with respect to the compensated Poisson random measure associated with the Lévy white noise. If σ is affine, we prove that the solution is Malliavin differentiable and its Malliavin derivative satisfies a stochastic integral equation.

In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index $H$, with $1/4 < H < 1/2$. We assume that the diffusion coefficient is given by an affine function $\sigma(x)=ax+b$, and the initial value functions are bounded and Holder continuous of order $H$. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is $L^{2}(\Omega)$-continuous and its $p$-th moments are uniformly bounded, for any $p \geq 2$.

Dans ce papier, nous montrons un principe de comparaison trajectoriel pour l’equation de la chaleur stochastique, fractionnaire, nonlineaire sur $\mathbb{R}$ avec une donnee initiale a valeur mesure. Nous donnons des estimations quantitatives de la proximite a zero d’une solution. Ces resultats etendent le principe de comparaison de Mueller pour l’equation de la chaleur stochastique et permettent de considerer des donnees initiales plus generales telles que des mesures de Dirac et des mesures a queue plus lourde qu’une croissance exponentielle lineaire en ${\pm}\infty$. Ces resultats generalisent un travail recent par Moreno Flores (Ann. Probab. 42 (2014) 1635–1643), qui a prouve la stricte positivite de l’equation de la chaleur stochastique partant d’un Dirac. Comme application, nous etablissons la complete intermittence pour l’equation. Dans une etape intermediaire, nous prouvons la regularite Holder de solutions partant d’une donnee initiale a valeur mesure ce qui generalise, dans un certain sens, un travail recent de Chen and Dalang (Stoch. Partial Differ. Equ. Anal. Comput. 2 (2014) 316–352).

In this paper we establish a large deviation principle for the nonlinear one dimensional&#13;\nstochastic heat equation driven by a Gaussian noise which is white in time and&#13;\nwhich has the covariance of a fractional Brownian motion with Hurst parameter H ∈( 14 ; 12) in the space variable.

We consider a d-dimensional random field that solves a possibly non-linear system of stochastic partial differential equations, such as stochastic heat or wave equations. We present results, obtained in joint works with Davar Khoshnevisan and Eulalia Nualart, and with Marta Sanz-Sole, on upper and lower bounds on the probabilities that the random field visits a deterministic subset of \(\mathbb {R}^d\), in terms, respectively, of Hausdorff measure and Newtonian capacity of the subset. These bounds determine the critical dimension above which points are polar, but do not, in general, determine whether points are polar in the critical dimension. For linear SPDEs, we discuss, based on joint work with Carl Mueller and Yimin Xiao, how the issue of polarity of points can be resolved in the critical dimension.

In this note, we establish optimal lower and upper Gaussian bounds for the density of the solution to a class of stochastic integral equations driven by an additive spatially homogeneous Gaussian random field. The proof is based on the techniques of the Malliavin calculus and a density formula obtained by Nourdin and Viens. Then, the main result is applied to the mild solution of a general class of SPDEs driven by a Gaussian noise which is white in time and has a spatially homogeneous correlation. In particular, this covers the case of the stochastic heat and wave equations in ℝd with d ≥ 1 and d ∈ {1, 2, 3}, respectively. The upper and lower Gaussian bounds have the same form and are given in terms of the variance of the stochastic integral term in the mild form of the equation.

We consider one-dimensional stochastic heat equations of the following form: where X(t) is an C([-m, m])-valued Stochastic Process and {Bt t≥0} denotes the so–called cylindrical Brownian motion on the real, separable Hilbert space H = L 2[-m m]. For the case that σ is a multiplication operator we prove the weak convergence of solutions of a stochastic heat equation on the interval [-m m] to the solution of the corresponding equation on the whole real line as m→∞ For the case that σ is a particular operator (depending only on m) we show convergence of the solutions to a stationary Gaussian limit process that can serve as model of stationary freeway traffic flow

We prove Girsanov's theorem for continuous orthogonal martingale measures. We then define space-time SDEs, and use Girsanov's theorem to establish a oneto- one correspondence between solutions of two space-time SDEs differing only by a drift coefficient. For such stochastic equations, we give necessary conditions under which the laws of their solutions are absolutely continuous with respect to each other. Using Girsanov's theorem again, we prove additional existence and uniqueness results for space-time SDEs. The same one-to-one correspondence and absolute continuity theorems are also proved for the stochastic heat and wave equations

In this paper, we first establish some sufficient conditions for the existence and construction of a random exponential attractor for a continuous cocycle on a separable Banach space. Then we mainly consider the random attractor and random exponential attractor for stochastic non-autonomous damped wave equation driven by linear multiplicative white noise with small coefficient when the nonlinearity is cubic. First step, we prove the existence of a random attractor for the cocycle associated with the considered system by carefully decomposing the solutions of system in two different modes and estimating the bounds of solutions. Second step, we consider an upper semicontinuity of random attractors as the coefficient of random term tends zero. Third step, we show the regularity of random attractor in a higher regular space through a recurrence method. Fourth step, we prove the existence of a random exponential attractor for the considered system, which implies the finiteness of fractal dimension of random attractor. Finally we remark that the stochastic non-autonomous damped cubic wave equation driven by additive white noise also has a random exponential attractor.

To model wave propagation in inhomogeneous media with frequency-dependent power-law attenuation, it is needed to use the fractional powers of symmetric coercive elliptic operators in space and the Caputo tempered fractional derivative in time. The model studied in this paper is semilinear stochastic space-time fractional wave equations driven by infinite dimensional multiplicative Gaussian noise and fractional Gaussian noise, because of the potential fluctuations of the external sources. We first give a representation of the mild solution and some stability estimates for the homogeneous problem, and then prove the existence and uniqueness of the mild solution by using fixed point theorem. Finally, the decay and regularity theory of the solution are provided.

In this paper we study the Large Deviations Principle (LDP in abbreviation) for a class of Stochastic Partial Differential Equations (SPDEs) in the whole space [Formula: see text], with arbitrary dimension [Formula: see text], under random influence which is a Gaussian noise, white in time and correlated in space. The differential operator is a fractional derivative operator. We prove a Large deviations principle for our equation, using a weak convergence approach based on a variational representation of functionals of infinite-dimensional Brownian motion. This approach reduces the proof of LDP to establishing basic qualitative properties for controlled analogues of the original stochastic system.

The Arnol’d-Thorn classification of caustics for the Burgers equation suggests that there should be an analogous one for the wavefronts of the corresponding heat equation. We present a general theorem for Hamiltonian systems char acterizing how the level surfaces of Hamilton’s principal function meet the caustic surface in both the deterministic and stochastic cases. Such a characterization allows one to give a fairly detailed description of the behaviour of the solution of the heat equation in the vicinity of the wavefront and caustic. It allows one to propose some reasons for the “blow-up” of the Burgers velocity field on the caustic. In the case of small noise the shapes of the random wavefront and random caustic may easily be obtained, and to first order the caustic is merely displaced. In the stochastic case we have the possibility of “rapid” changes in the caustic-wavefront intersection. This will engender stochastic turbulence in the Burgers velocity field and, due to its stochasticity, may be of an intermittent nature. There is no analogue of this in the deterministic case. Throughout our studies much use has been made of computer algebra packages in building an understanding of the archetypal cases. Numerical simulations and numerical solutions of the partial differential equations involved have been immensely useful in clarifying conjectures and determining apt characterizations.

The Arnol'd-Thom classification of caustics (shockwaves) for Burgers equation suggests a similar one for the wavefronts of the corresponding Heat equation. We give here a general theorem for Hamiltonian systems characterising how the level surfaces of Hamilton's principal function (wavefronts) meet the caustic surface in both the deterministic and stochastic cases. We further show how these results can be applied to the stochastic Burgers equation. The generic example of a caustic, appearing in the two-dimensional case, is the semicubical parabolic cusp with the corresponding zero level surface being a combination of a generalised hypocycloid and a line pair. Our results explain in terms of classical mechanics the properties of the caustic and wavefront for this archetypal example and characterise the caustic-wavefront intersection for the general stochastic case. We discuss the application of these results to turbulence for the Burgers velocity field.

We establish a version of the Feynman–Kac formula for the multidimensional stochastic heat equation with a multiplicative fractional Brownian sheet. We use the techniques of Malliavin calculus to prove that the process defined by the Feynman–Kac formula is a weak solution of the stochastic heat equation. From the Feynman–Kac formula, we establish the smoothness of the density of the solution and the Hölder regularity in the space and time variables. We also derive a Feynman–Kac formula for the stochastic heat equation in the Skorokhod sense and we obtain the Wiener chaos expansion of the solution.