Strict positivity for stochastic heat equations
Let us consider a bilinear stochastic heat equation \[ dX(t,\xi) = \tfrac 12\Delta X(t,\xi) dt + X(t,\xi) dW(t), \quad X(0,\xi) = x(\xi), \quad \xi \in \mathbb R^{n}, \tag{1} \] where \(W\) denotes a spatially homogeneous Wiener process in the space \(\mathcal S '(\mathbb R^{n})\) of tempered distributions, defined on a filtered probability space \((\Omega ,\mathcal F, (\mathcal F_{t}),\mathbf P)\). That is, the covariance operator \(Q\) of the process \(W\) is of the form \(Q\varphi = \Gamma * \varphi \), \(\varphi \in \mathcal S(\mathbb R^{n})\), for a positive definite distribution \(\Gamma \). Hence \(\Gamma \) is a Fourier transform of a tempered measure \(\mu \) on \(\mathbb R^{n}\); it is assumed that \(\mu \) has a density \(\gamma \) with respect to the \(n\)-dimensional Lebesgue measure such that \(\gamma \in L^{p}(\mathbb R^{n})\) for some \(p\in [1,\infty ]\) with \((1-1/p)d/2 <1\). Then it is known that for any \(x\in L^{2}(\mathbb R^{n})\) there exists a unique mild solution \(X^{x}\) to (1), and \(X^{x}\) is an \(L^2(\mathbb R^{n})\)-valued process; see the paper by \textit{S. Peszat} and \textit{J. Zabczyk} [ibid. 72, No. 2, 187-204 (1997)] for a thorough discussion of the problem (1). In the paper under review, it is shown that \(X^{x}(t)>0\) \(\mathbf P\otimes \mu \)-almost everywhere for any \(t>0\), whenever \(x\in L^{2}(\mathbb R^{n})\), \(x\geq 0\), is such that the set \(\{\xi \in \mathbb R^{n}; x(\xi)\geq \varepsilon\}\) has a nonempty interior for some \(\varepsilon >0\). In the course of the proof, exponential tail estimates for stochastic convolutions in \(L^{q}(\mathbb R^{n})\) spaces are established, generalizing those obtained by \textit{S. Peszat} [Bull. Pol. Acad. Sci., Math. 40, No. 4, 323-333 (1992; Zbl 0818.60050)] in the case \(q=2\).
scientific article; zbMATH DE number 1364502