| Abstract: |
| When approximating hyperbolic SPDEs such as the stochastic Schroedinger equation in time, challenges arise due to the lack of regularising behaviour of the underlying semigroup. In this talk, we present convergence rates for time discretization schemes for such equations, where the leading operator is the generator of a $C_0$-semigroup.
The first main result are optimal bounds for the uniform strong error
\[
E_k^\infty:= \Big(\mathbb{E}\max_{1\le j \le M}\|U(t_j)-U_j\|_X^p\Big)^{1/p}
\]
on a Hilbert space $X$ for $p\in [2,\infty)$, a time step $k>0$ and $T=Mk>0$. Under conditions on the globally Lipschitz nonlinearity and multiplicative noise, we show $E_k^\infty\lesssim\sqrt{k\log(T/k)}$
for a large class of time discretisation schemes. For equations such as Maxwell`s or Schroedinger, our results provide
the first results known for rational approximations of $(S(t))_{t\ge 0}$ and improve mean-square error estimates to pathwise uniform ones.
The second main result concerns polynomially growing nonlinearities and noise, which imposes additional challenges since only a local Lipschitz condition is satisfied. For a tamed exponential Euler scheme, stability and convergence $E_k^\infty\lesssim\sqrt{k}$ are shown under a coercivity assumption using a stochastic Gronwall inequality. This extension of the abstract framework developed in the first part allows to consider, e.g., the nonlinear Schroedinger equation.
This is based partly on joint work with Mark Veraar and partly on ongoing work by the author. |
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