The author develops the theory of solvability of the Cauchy problem for linear and some quasilinear stochastic partial differential equations like \[ du= (a^{ij} u_{x^ix^j}+ b^i u_{x^i}+ f) dt+ (u_{x^i} \sigma^{ik} u_{x^i}+ g^k) dW^k_t,\quad t>0,\tag{1} \] (summation w.r.t. repeated indices \(i\), \(j\), \(k\), where \(1\leq i\leq j\), \(k\geq 1\), and \(W^1,W^2,\dots\) are independent Brownian motions) in the spaces of Bessel potentials (or Sobolev spaces with fractional derivative) \(H^n_p(R^d)= \{u: (\text{Id}- \Delta)^{n/2}u\in L^p\}\), with exponent \(p\geq 2\). If \(p=2\), this space is noting else the Sobolev space \(W^n_2(R^d)\). The integration by parts formula as well as approaches on semigroup methods are here the main tools of a rather complete and satisfactory theory. But if one needs some regularity for the solution without having \(n\) sufficiently large, it is important to have \(p>2\) great. The main tools used by the author to study the solvability for \(p\geq 2\) are the (deterministic) theory of spaces \(H^n_p(R^d)\) borrowed from H. Triebel (1983, 1992), in particular an estimate of the \(L^p\)-norms of the derivatives of the solution of the heat equation by O. A. Ladyshenskaya, V. A. Solonnikov and N. N. Uraltseva (1967), as well as some analog to the maximal regularity property of stochastic convolutions in Hilbert spaces obtained by G. Da Prato (1983).
The present paper is a major revision and improvement of an earlier one from the author from 1996. The paper is organized as follows: After discussing a general scheme of proving the solvability of PDEs in Section 2, the author introduces the stochastic Banach space of predictable \(L^p\)-processes with values in \(H^n_p(R^d)\) in which the solvability of SPDEs is studied then, first for a particular “perturbated heat” equation (Section 4), then for general equations with coefficients not depending on \(x\) (Section 4) and equations with variable coefficients where, in (1), \(a\) and \(\sigma\) depend only on \((t,x)\) and \(f\), \(g\) can depend also on \(u\). Section 7 is devoted to embedding theorems for the stochastic Banach spaces, Section 8 to some applications (filtering equations, equations with space-time white noise, study of non-explosion for the linear equation) and, finally, in Section 9 some open problems are presented.
For the entire collection see [Zbl 0904.00017].