Existence, uniqueness, and regularity for stochastic evolution equations with irregular initial values
Introduction
There are a number of existence, uniqueness, and regularity results for mild solutions of semilinear stochastic evolution equations (SEEs) in the literature; see, e.g., [10], [11], [4], [28], [16], [18], [21], [27] and the references mentioned therein. In this work we extend the above cited results by adding singularities in the initial condition and by introducing singularities at the initial time of the time-dependent coefficients of the considered SEE; see also Chen & Dalang [7], [8] for related results. To be more specific, in the first main result of this work (see Proposition 2.7) we establish a general perturbation estimate (see (3) below) for a general class of stochastic processes which allows us to derive a priori bounds (see, e.g., (5) below) for solutions and numerical approximations of SEEs with singularities at the initial time. This perturbation estimate, in turn, is used to prove the second main result of this article (see Theorem 2.9) which establishes existence, uniqueness, and regularity properties for solutions of SEEs with singularities at the initial time (see (4) and (5) below). As an application of our perturbation estimate and this second main result of our article, we reveal a regularity barrier (see (8) below) for the initial condition of the considered SEE under which the considered SEE has a unique solution which is Lipschitz continuous with respect to initial values (see Corollary 2.10). By means of several counterexamples (see Proposition 3.2, Proposition 3.4, Proposition 3.5) we also demonstrate that this regularity barrier can in general not essentially be improved (cf. (10) and (11) below). We illustrate the above findings in the case of possibly nonlinear stochastic heat equations on an interval such as the continuous version of the parabolic Anderson model on an interval (cf. Corollary 3.1, Proposition 3.2, and Proposition 3.3). Existence, uniqueness, and regularity results for possibly nonlinear stochastic heat equations on the whole real line with rough initial values, that is, signed Borel measures with exponentially growing tails over as initial values can be found in Chen & Dalang [7], [8] (see Theorem 2.4 in Chen & Dalang [8] for an existence and uniqueness result and a priori estimates and see Theorem 3.1 in Chen & Dalang [7] for a Hölder regularity result). Moreover, Proposition 2.11 in Chen & Dalang [8] disproves the existence of a solution of the considered stochastic heat equation in the case of a specific rough initial value, that is, the derivative of the Dirac delta measure at zero as the initial value.
To illustrate the results of this article in more details, we assume the following setting throughout this introductory section. Let and be nontrivial separable -Hilbert spaces. Let , , , , , , , , satisfy . Let be a stochastic basis. Let be an -cylindrical -Wiener process. Let be a generator of a strongly continuous analytic semigroup with . Let , , be a family of interpolation spaces associated to (cf., e.g., [26, Section 3.7]). Let be a /-measurable mapping, let be a /-measurable mapping, and assume for all , that
In displays (3)–(11) below we illustrate the above framework through several examples and applications. Our first result is a suitable perturbation estimate for predictable stochastic processes. We employ the following additional notation to formulate this perturbation estimate. For every and every sufficiently regular predictable stochastic process let be a predictable stochastic process which satisfies for all -a.s. that . Proposition 2.7 below then proves that there exists a function such that for all , and a wide class of -predictable stochastic processes it holds that We also note that we explicitly specify the function in Proposition 2.7 below. Estimate (3) follows from an appropriate application of a generalized Gronwall-type inequality (see the proof of Proposition 2.7 below for details).
We use inequality (3) to establish an existence, uniqueness, and regularity result for SEEs with singularities at the initial time. More precisely, in Theorem 2.9 below we prove that there exists a function such that for all suitable , it holds (i) that there exists a suitable up-to-modifications unique -predictable stochastic process which satisfies for all -a.s. that and (ii) that In Theorem 2.9 we also explicitly specify the function . We would like to point out that inequality (5) under the generality of (1) and (2) is a crucial ingredient to establish essentially sharp weak convergence rates for numerical approximations of SEEs with possibly smooth initial values (see the last paragraph in this introductory section for more details). Inequality (5) follows from the perturbation estimate (3) (with and in the notation of (3)).
We now illustrate Theorem 2.9 and (4)–(5), respectively, by some examples. In particular, in Corollary 2.10 below we prove by an application of Theorem 2.9 that for all , , it holds (i) that there exist up-to-modifications unique -predictable stochastic processes , , , which fulfill for all , , , that , that , and -a.s. that and (ii) that Here and below for two -Banach spaces and we denote by the set of all Lipschitz continuous functions from V to W and for two -Banach spaces and and a function we denote by the Lipschitz semi-norm associated to f (see (13) in Subsection 1.1 below for details). The finiteness of the second element in the set in the maximum in (7) follows from the perturbation estimate (3) (with and for , in the notation of (3)) and the finiteness of the first element in the set in the maximum in (7) is a consequence from (5), which, in turn, also follows from the perturbation estimate (3) (see above and the proof of Corollary 2.10 for details). Roughly speaking, Corollary 2.10 establishes the existence of mild solutions of the SEE (6) and also establishes the Lipschitz continuity of the solutions with respect to the initial conditions for any initial condition in and any (see (7)). In Corollary 3.1, Proposition 3.2, Proposition 3.4, and Proposition 3.5 below we demonstrate that the regularity barrier for the regularity of the initial conditions revealed in Corollary 2.10 (and Proposition 2.7 and Theorem 2.9, respectively) can, in general, not essentially be improved. In particular, Corollary 3.1 and Proposition 3.2 below prove in the case where , where , where is the Laplacian with periodic boundary conditions on H, and where satisfies (B is not a constant function) that it holds (i) that there exist up-to-modifications unique -predictable stochastic processes , , , which fulfill for all , , , that , that , and -a.s. that (ii) that and (iii) that The SEE (9) is sometimes referred to as a continuous version of the parabolic Anderson model in the literature (see, e.g., Carmona & Molchanov [6]). In addition, Proposition 3.2 below disproves the existence of square integrable solutions of the SEE (9) with initial conditions in . The noise in the counterexample SEE (9) is spatially very rough and one might question whether the regularity barrier (8) can be overcome in the case of more regular spatially smooth noise. In Proposition 3.4 below we answer this question to the negative by presenting another counterexample SEE with a non-constant diffusion coefficient but a spatially smooth noise for which we disprove the existence of square integrable solutions with initial conditions in (cf., however, also Proposition 3.3 below). Proposition 3.5 below also provides a further counterexample SEE which illustrates the sharpness of the regularity barrier (8) in the case where B is a constant function.
Proposition 2.7, Theorem 2.9, and Corollary 2.10 outlined above (see (3)–(7)) are of particular importance for establishing regularity properties for Kolmogorov backward equations associated to parabolic semilinear SEEs and, thereby, for establishing essentially sharp probabilistically weak convergence rates for numerical approximations of parabolic semilinear SEEs (cf., e.g., Lemmas 4.4–4.6 in Debussche [12], Lemma 3.3 in Wang & Gan [30], (4.2)–(4.3) in Andersson & Larsson [1], Propositions 5.1–5.2 and Lemma 5.4 in Bréhier [2], Lemma 3.3 in Wang [29], (79) in Conus et al. [9], Proposition 7.1, Lemma 10.5, and Lemma 10.10 in Kopec [20], and (183)–(184) in Jentzen & Kurniawan [17]). The analytically weak norm for the initial condition in (7) as well as the singularities in the nonlinear coefficients of the SEE in (1) and (2) above translate in an analytically weak norm for the approximation errors in the probabilistically weak error analysis which, in turn, results in essentially sharp probabilistically weak convergence rates (cf., e.g., Theorem 2.2 in Debussche [12], Theorem 2.1 in Wang & Gan [30], Theorem 1.1 in Andersson & Larsson [1], Theorem 1.1 in Bréhier [2], Theorem 5.1 in Bréhier & Kopec [3], Corollary 1 in Wang [29], Corollary 5.2 in Conus et al. [9], Theorem 6.1 in Kopec [20], and Corollary 8.2 in [17]). The perturbation inequality in Proposition 2.7 (see (3) above) is also useful to establish essentially sharp probabilistically strong convergence rates for numerical approximations and perturbations of SEEs (cf., e.g., Proposition 4.1 in Conus et al. [9] and Proposition 4.3 in [17]).
Throughout this article the following notation is used. For two measurable spaces and we denote by the set of all /-measurable functions. For a set A we denote by the power set of A and we denote by the counting measure on A. For a Borel measurable set we denote by the Lebesgue-Borel measure on A. For a real number and a probability space with a normal filtration (see, e.g., Definition 2.1.11 in [23]) we call the quadruple a stochastic basis. For a real number and a filtered probability space we denote by the sigma-algebra given by (the predictable sigma-algebra associated to ). We denote by , , the functions which satisfy for all , that , . For -Banach spaces and we denote by and the functions which satisfy1 for all that and we denote by the set given by . We denote by the function with the property that for all it holds that (Beta function). We denote by , , the functions which satisfy for all , that (generalized exponential function; cf. Lemma 7.1.1 in Chapter 7 in Henry [14], (1.0.3) in Chapter 1 in Gorenflo et al. [13], and Lemma 2.6 below). For a separable -Hilbert space , real numbers , , , , , , , and a generator of a strongly continuous analytic semigroup with we denote by the real numbers given by and (cf., e.g., [24, Lemma 11.36]) and we denote by the function which satisfies for all that For a measure space , a measurable space , and an /-measurable function we denote by the set given by and, as usual, we often do not distinguish between an /-measurable function and its equivalence class .
Throughout this article the following setting is frequently used. Consider the notation in Section 1.1, let and be separable -Hilbert spaces with >1, let , , let be a stochastic basis, let be an -cylindrical -Wiener process, let be a generator of a strongly continuous analytic semigroup with , let , , be a family of interpolation spaces associated to .
Section snippets
Stochastic evolution equations (SEEs) with singularities at the initial time
In the main result of this section, see Theorem 2.9 in Subsection 2.4 below, we establish existence, uniqueness, and regularity properties for solutions of certain SEEs with time-dependent coefficients and singularities at the initial time. In Subsection 2.1 below we formulate the precise framework which we employ to state Theorem 2.9 in Subsection 2.4 below. The framework in Subsection 2.1 is similar to the hypothesis used in the introductory section above.
Examples and counterexamples for SEEs with irregular initial values
Corollary 2.10 in Subsection 2.4 above establishes existence, uniqueness, and regularity properties for solutions of parabolic SEEs. In this section we first illustrate the statement of Corollary 2.10 in the case of semilinear stochastic heat equations with space-time white noise and periodic boundary conditions; see Corollary 3.1 in Subsection 3.2 below. Roughly speaking, Corollary 3.1 shows existence and uniqueness of solutions of the considered stochastic heat equation provided that the
Acknowledgments
Stig Larsson and Christoph Schwab are gratefully acknowledged for some useful comments. This project has been supported through the SNSF-Research project 200021_156603 “Numerical approximations of nonlinear stochastic ordinary and partial differential equations”.
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