Some non-linear s.p.d.e.'s that are second order in time

We extend Walsh's theory of martingale measures in order to deal with hyperbolic stochastic partial differential equations that are second order in time, such as the wave equation and the beam equation, and driven by spatially homogeneous Gaussian noise. For such equations, the fundamental solution can be a distribution in the sense of Schwartz, which appears as an integrand in the reformulation of the s.p.d.e. as a stochastic integral equation. Our approach provides an alternative to the Hilbert space integrals of Hilbert-Schmidt operators. We give several examples, including the beam equation and the wave equation, with nonlinear multiplicative noise terms

Random field solutions to linear SPDEs driven by symmetric pure jump L\'evy space-time white noises

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric L\'evy white noise. We identify conditions for existence for these two kinds of solutions, and we identify conditions under which they are essentially equivalent. We establish a necessary condition for the existence of a random field solution to a linear SPDE, and we apply this result to the linear stochastic heat, wave and Poisson equations driven by a symmetric $\alpha$-stable noise.Comment: 26 page

Multiple points of the Brownian sheet in critical dimensions

It is well known that an $N$-parameter $d$-dimensional Brownian sheet has no $k$-multiple points when $(k-1)d>2kN$, and does have such points when $(k-1)d<2kN$. We complete the study of the existence of $k$-multiple points by showing that in the critical cases where $(k-1)d=2kN$, there are a.s. no $k$-multiple points.Comment: Published at http://dx.doi.org/10.1214/14-AOP912 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Hitting Probabilities for Systems of Non-Linear Stochastic Heat Equations with Additive Noise

We consider a system of $d$ coupled non-linear stochastic heat equations in spatial dimension 1 driven by $d$-dimensional additive space-time white noise. We establish upper and lower bounds on hitting probabilities of the solution $\{u(t, x)\}_{t \in \mathbb{R}_+, x \in [0, 1]}$, in terms of respectively Hausdorff measure and Newtonian capacity. We also obtain the Hausdorff dimensions of level sets and their projections. A result of independent interest is an anisotropic form of the Kolmogorov continuity theorem.Comment: 44 pages; submitted for publicatio

Hitting properties of parabolic s.p.d.e.'s with reflection

We study the hitting properties of the solutions $u$ of a class of parabolic stochastic partial differential equations with singular drifts that prevent $u$ from becoming negative. The drifts can be a reflecting term or a nonlinearity $cu^{-3}$, with $c>0$. We prove that almost surely, for all time $t>0$, the solution $u_t$ hits the level 0 only at a finite number of space points, which depends explicitly on $c$. In particular, this number of hits never exceeds 4 and if $c>15/8$, then level 0 is not hit.Comment: Published at http://dx.doi.org/10.1214/009117905000000792 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org