Hölder continuity for spatial and path processes via spectral analysis
Let \(\nu( d \theta)\) be a \(\sigma\)-finite Borel measure on \(\mathbb R^d\), and \(L^2( \nu(d\theta))\) the usual \(L^2\)-space of all measurable functions \(f(\theta)\) on \(\mathbb R^d\) with respect to \(d \nu\) with the norm \(\|\cdot \|\). \(\lambda(\theta)\geq \Lambda >0\) denotes a Borel measurable function of \(\theta \in\mathbb R^d\). \(m(t,\theta)\), \(\theta \in \mathbb R^d\), is a continuous martingale with quadratic variation process \[ [ m(\cdot, \theta)](t) = \int_0^t g(s,\theta) ds \quad \text{for fixed } \theta \in \mathbb R^d, \] where \(g(w, s, \theta)\) is measurable in \((w,s,\theta)\) with \(w\in \Omega\) of a probability space \((\Omega, {\mathcal F}, P)\), satisfying that \[ \sup_{0 \leq u \leq t} E[ g(u,\theta)^r ]^{1/r} \leq G(r,t,\theta). \] Here \(G\) is a measurable function in \(\theta\) for each \(t \leq T\) and \(r \geq 1\), such that \[ \int_{\lambda(\theta) \leq a} G(r,T,\theta) \nu(d \theta) < \infty \quad \text{and} \quad \int_{\lambda(\theta)> a} G(r, T, \theta) \lambda^{\delta-1}(\theta) \nu(d\theta) < \infty \] for \(\exists a \geq 0\) and \(\exists \delta \in (0,1]\). Then the \(L^2( \nu(d\theta))\)-valued stochastic process \(Y(t)\) is defined by \(Y(t)= y(t, \cdot)\) with \(E \|Y(t) \|^2< \infty\) for each \(t \geq 0\). The authors prove timewise Hölder continuity and maximal inequalities for \(Y\). Namely, Theorem A. Let \(r \delta > 1\). Then \(Y\) has a modification \(\widetilde{Y}\) satisfying \[ E \Bigl[ \sup_{0 \leq s < t \leq T} ( \|\widetilde{Y}(t) - \widetilde{Y}(s) \|/ |t-s |^{\beta})^{2r} \Bigr] < \infty \] for any \(\beta \in [ 0, (\delta/2) - (1/2r))\). Theorem B. (a) For \(r \delta > 1\), \(0 \leq s \leq t \leq T\), \[ E \Bigl[ \sup_{s \leq u, v \leq t} \|Y(v) - Y(u) \|^{2r} \Bigr] \leq C(T, \delta, r, a, \Lambda) |t-s |^{ r \delta}. \] (b) Furthermore, assume \(\sup_{0 \leq u < \infty} E[ g(u,\theta)^r ]^{1/r}\leq G(r,\theta)\) where \(G\) satisfies some conditions. Then \[ E\Bigl[ \sup_{0 \leq t \leq T} \|Y(t) \|^{2r} \Bigr] \leq C(a,r,\delta) (T^r + T^{r \delta}). \] (c) In addition, if \(\int_{R^d} G(r,\theta) \lambda^{\delta - 1}(\theta) \nu(d\theta)< \infty\) for \(\delta \in (0,1]\), then \[ E\Bigl[ \sup_{0 \leq t \leq T} \|Y(t) \|^{2r} \Bigr] \leq C(r,\delta) T^{r \delta}. \] Moreover, the authors use these results to derive the standard Sobolev space regularity results for a class of \((\alpha, d, 1)\)-superprocesses [cf. \textit{D. A. Dawson}, in: Ecole d'Eté de probabilités de Saint-Flour XXI -- 1991. Lect. Notes Math. 1541, 1-260 (1993; Zbl 0799.60080)] as well as a class of stochastic evolutions of the form \(dX= A_{\alpha} X dt+\Phi dW\), where \(A_{\alpha}\) is the generator of a symmetric stable process on \(\mathbb R^d\) of index \(\alpha \in\) \((0,2]\), \(\Phi = \Phi(s)\) is a random operator and \(W\) is a cylindrical Hilbert space valued Brownian motion [cf. \textit{G. Da Prato} and \textit{J. Zabczyk}, ``Stochastic equations in infinite dimensions'' (1992; Zbl 0761.60052)]. Their method for the proof is greatly due to the result on a maximal inequality from \textit{M. A. Kouritzin} and \textit{A. J. Heunis} [Ann. Probab. 22, No. 2, 659-679 (1994; Zbl 0806.60017)]. For other related works, see e.g. \textit{D. Blount} and \textit{A. Bose} [Ann. Probab. 28, No. 1, 104-131 (2000)].
stochastic partial differential equations
scientific article; zbMATH DE number 1618290