Open Access

Pascal Schoppe

• We consider (damped) stochastic wave type equations on R≥0 ×Tq, ∂ttu(t,x) = Au(t,x) + ∂tDu(t,x) + f (t,x) + σ(t)W(t,x), where by wave-type we mean that the equations are second order in time and the spatial differential operator A, aside from a reasonable lower order perturbation, is a essentially (−1)m−1Δm (m ∈ N∗) and σW is an additive space-time white noise. The aim of this thesis is to study how these equations emerge as limiting equations for discrete equations of the form ∂ttXN(t, l) = ANXN(t, l) + ∂tDNXN(t, l) + gN(t, l,XN(t, l)) + σBN(t, l), whose spatial variables live on an equidistant, cubic, Bravais lattice, that is a discretization of the spatial torus Tq. Where (AN)∗ N∈N and (DN)N∈N∗ constitute sequences of discrete difference quotient operators, whose continuous realizations converge to A respectively D in the strong operator topology and BN is a spatial L1-average of W, with respect to the step size of the lattice. We first consider linear discrete equations, hereWe consider (damped) stochastic wave type equations on R≥0 ×Tq, ∂ttu(t,x) = Au(t,x) + ∂tDu(t,x) + f (t,x) + σ(t)W(t,x), where by wave-type we mean that the equations are second order in time and the spatial differential operator A, aside from a reasonable lower order perturbation, is a essentially (−1)m−1Δm (m ∈ N∗) and σW is an additive space-time white noise. The aim of this thesis is to study how these equations emerge as limiting equations for discrete equations of the form ∂ttXN(t, l) = ANXN(t, l) + ∂tDNXN(t, l) + gN(t, l,XN(t, l)) + σBN(t, l), whose spatial variables live on an equidistant, cubic, Bravais lattice, that is a discretization of the spatial torus Tq. Where (AN)∗ N∈N and (DN)N∈N∗ constitute sequences of discrete difference quotient operators, whose continuous realizations converge to A respectively D in the strong operator topology and BN is a spatial L1-average of W, with respect to the step size of the lattice. We first consider linear discrete equations, here (gN)N∈N = (fN)∗ N∈N is a sequence approximating f , in a space of continuous functions with values in a suitable Sobolev space and we show that, maximal regularity aside, the solutions of the discrete equations approximate the solution of the continuous equation, and in turn get approximated by spatially averaged, orthogonal projections of the continuous solution. Thus proving a Cauchy-Born rule under the influence of space-time white noise. While in a second step we consider renormalized continuum limits of non-linear equations. Here gN(t, l,XN(t, l)) = fN(t, l) + b(XN(t, l)), is the sum of the function fN from the linear part and a polynomial non-linearity, rescaled to compensate the divergence in the limit, that will either disappear in the limit or spawn a linear term, depending on the strength of the rescaling compared to the non-linearity’s speed of divergence. Here we determine a candidate for the scaling under which the renormalization into a linear term occurs and show that for stronger rescaled non-linearities the term vanishes in the limit, for odd powered non-linearities a small gap between the critical scaling and the vanishing scalings can be observed.…