By Robert Dalang, Davar Khoshnevisan, Carl Mueller, David Nualart, Yimin Xiao, Firas Rassoul-Agha
In may possibly 2006, The college of Utah hosted an NSF-funded minicourse on stochastic partial differential equations. The aim of this minicourse used to be to introduce graduate scholars and up to date Ph.D.s to varied sleek subject matters in stochastic PDEs, and to assemble numerous specialists whose examine is based at the interface among Gaussian research, stochastic research, and stochastic partial differential equations. This monograph includes an updated compilation of lots of these lectures. specific emphasis is paid to showcasing vital principles and exhibiting a number of the many deep connections among the pointed out disciplines, forever retaining a pragmatic speed for the coed of the subject.
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Additional info for A Minicourse on Stochastic Partial Differential Equations
J. Math. 60(4), 897–936 The Stochastic Wave Equation Robert C. Dalang Summary. These notes give an overview of recent results concerning the non-linear stochastic wave equation in spatial dimensions d ≥ 1, in the case where the driving noise is Gaussian, spatially homogeneous and white in time. We mainly address issues of existence, uniqueness and H¨ older–Sobolev regularity. We also present an extension of Walsh’s theory of stochastic integration with respect to martingale measures that is useful for spatial dimensions d ≥ 3.
5]) to have t ds 0 R2 dy G2 (t − s , x − y) < +∞. (36) The integral is equal to t ds 0 |y−x| 4 The Wave Equation in Spatial Dimension 2 We shall consider the following form of the stochastic wave equation in spatial dimension d = 2: ∂2u − Δu (t , x) = σ(u(t , x)) F˙ (t , x), (t , x) ∈ ]0 , T ] × R2 , (54) ∂t2 with vanishing initial conditions. By a solution to (54), we mean a jointly measurable adapted process (u(t, x)) that satisﬁes the associated integral equation u(t , x) = [0,t]×R2 G(t − s , x − y) σ(u(s , y)) M (ds , dy), (55) where M is the worthy martingale measure associated with F˙ .
4 The Wave Equation in Spatial Dimension 2 We shall consider the following form of the stochastic wave equation in spatial dimension d = 2: ∂2u − Δu (t , x) = σ(u(t , x)) F˙ (t , x), (t , x) ∈ ]0 , T ] × R2 , (54) ∂t2 with vanishing initial conditions. By a solution to (54), we mean a jointly measurable adapted process (u(t, x)) that satisﬁes the associated integral equation u(t , x) = [0,t]×R2 G(t − s , x − y) σ(u(s , y)) M (ds , dy), (55) where M is the worthy martingale measure associated with F˙ .