Path properties of the solution to the stochastic heat equation with Lévy noise
Tóm tắt
We consider sample path properties of the solution to the stochastic heat equation, in
$${\mathbb {R}}^d$$
or bounded domains of
$${\mathbb {R}}^d$$
, driven by a Lévy space–time white noise. When viewed as a stochastic process in time with values in an infinite-dimensional space, the solution is shown to have a càdlàg modification in fractional Sobolev spaces of index less than
$$-\frac{d}{2}$$
. Concerning the partial regularity of the solution in time or space when the other variable is fixed, we determine critical values for the Blumenthal–Getoor index of the Lévy noise such that noises with a smaller index entail continuous sample paths, while Lévy noises with a larger index entail sample paths that are unbounded on any non-empty open subset. Our results apply to additive as well as multiplicative Lévy noises, and to light- as well as heavy-tailed jumps.
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