59 research outputs found
Stochastic geometric wave equations with values in compact Riemannian homogeneous spaces
Let be a compact Riemannian homogeneous space (e.g. a Euclidean sphere).
We prove existence of a global weak solution of the stochastic wave equation
\mathbf D_t\partial_tu=\sum_{k=1}^d\mathbf
D_{x_k}\partial_{x_k}u+f_u(Du)+g_u(Du)\,\dot Wd\ge 1fgW\mathbb R^d$ with finite spectral measure. A
nonstandard method of constructing weak solutions of SPDEs, that does not rely
on martingale representation theorem, is employed
Maximal inequality of Stochastic convolution driven by compensated Poisson random measures in Banach spaces
Let be a Banach space such that, for some , the
function is of class and its first and second
Fr\'{e}chet derivatives are bounded by some constant multiples of -th
power of the norm and -th power of the norm and let be a
-semigroup of contraction type on . We consider the
following stochastic convolution process \begin{align*}
u(t)=\int_0^t\int_ZS(t-s)\xi(s,z)\,\tilde{N}(\mathrm{d} s,\mathrm{d} z), \;\;\;
t\geq 0, \end{align*} where is a compensated Poisson random measure
on a measurable space and is an -predictable function. We
prove that there exists a c\`{a}dl\`{a}g modification a of the
process which satisfies the following maximal inequality \begin{align*}
\mathbb{E} \sup_{0\leq s\leq t} \|\tilde{u}(s)\|^{q^\prime}\leq C\ \mathbb{E}
\left(\int_0^t\int_Z \|\xi(s,z) \|^{p}\,N(\mathrm{d} s,\mathrm{d}
z)\right)^{\frac{q^\prime}{p}}, \end{align*} for all and
with .Comment: This version is only very slightly updated as compared to the one
from September 201
Martingale solutions and invariant measures for stochastic evolution equations in Banach spaces
AbstractIn this paper we study the existence and uniqueness of weak solutions of stochastic differential equations on Banach spaces. We also study the existence of invariant measures for the corresponding Markovian semigroups. Our main tool is the factorization of stochastic convolutions. We close the paper with some examples
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