Second order PDE’s in finite and infinite dimension.

*(English)*Zbl 0983.60004
Lecture Notes in Mathematics. 1762. Berlin: Springer. ix, 330 p. (2001).

Let \((X,\mathbf P_{x})\) be the Markov process on \(\mathbb R^{m}\) defined by a stochastic differential equation
\[
dX = b(X) dt + \sigma(X) dW, \tag{1}
\]
where \(W\) is an \(m\)-dimensional Wiener process, and consider the associated Kolmogorov equation
\[
\partial_{t} u = \frac 12\text{Tr}\bigl(\sigma(x) \sigma^{*}(x)D^2u\bigr) + \langle b(x),Du\rangle, \quad u(0,\cdot) = \varphi, \tag{2}
\]
and for \(\lambda>0\) also the elliptic equation
\[
\lambda \xi - \frac 12\text{Tr}\bigl(\sigma(x) \sigma^{*}(x)D^2\xi\bigr) - \langle b(x),D\xi\rangle = f(x)\quad \text{on \(\mathbb R^{m}\)}. \tag{3}
\]
Nowadays it is well known that under suitable smoothness and boundedness hypotheses on coefficients of (1) the unique classical solution to (2) has the form \(u(t,x) = \mathbf E_{x}\varphi(X_{t})\), that is,
\[
u(t,\cdot) = P_{t}\varphi,\tag{4}
\]
\((P_{t})\) denoting the transition semigroup of (1), and the solution to (3) is given by \(\xi(x) = \int^{+\infty}_{0}e^{-\lambda t}P_{t}f(x) dt\).

The main goal of the first part of the book under review is to extend this interplay between (1), (2) and (3) to problems with coefficients \(b\) and \(\sigma\) that are only locally Lipschitz, of a polynomial growth, and satisfy certain dissipativity assumptions. This task is highly nontrivial. For example, to show the mean square differentiability of solutions to (1) with respect to initial conditions, which is the first step in the proof of (4) in the “bounded” case, is no longer easy. This problem is studied in Chapter 1, the results obtained then being used to establish the Bismut-Elworthy formula for derivatives of the function \(P_{t}\varphi\). This in turn yields smoothing properties of the semigroup \((P_{t})\), and existence and uniqueness of classical solutions to (2) and (3) follow. Moreover, it is shown that the solutions satisfy Schauder type estimates in the space of Hölder continuous functions.

Smoothing properties of the semigroup \((P_{t})\) are closely related to its ergodic properties. In Chapter 2 it is proven that \((P_{t})\) remains strong Feller and topologically irreducible even under weaker hypotheses on \(b\) and \(\sigma\), consequently, the invariant probability measure \(\mu\) for (1) is unique and globally asymptotically stable in the total variation norm. Furthermore, dissipativity hypotheses are found under which \(P_{t}\varphi\) converges uniformly to \(\mu(\varphi)\) at a polynomial or exponential rate. In Chapter 3, the degenerate case is treated and a class of diffusion matrices \(\sigma\) is described such that \(\sigma(x)\) is non-invertible, nevertheless, the semigroup \((P_{t})\) on the space \(\text{BUC}(\mathbb R^{m})\) of bounded uniformly continuous functions is analytic.

Part II of the book is devoted to a thorough study of a stochastic reaction diffusion system \[ \partial_{t}u_{k} = {\mathcal A}_{k}(\xi,D)u_{k} + f_{k} (\xi,u_1,\ldots,u_{n}) + Q_{k}\partial^2_{t\xi}w_{k} \quad\text{in \(\mathbb R_{+}\times\mathcal O\)}, \;k=1,\ldots,n, \tag{5} \] with either Dirichlet or oblique boundary conditions. It is supposed that \(\mathcal O\subseteq\mathbb R^{m}\) is a bounded domain with a smooth boundary, \(m\leq 3\), \({\mathcal A}_{k}(\xi,D)\) are uniformly elliptic second-order differential operators with real regular coefficients, \(Q_{k}\) are nonnegative bounded linear operators in \(L^2(\mathcal O)\) (not necessarily boundedly invertible) and \(w_1,\ldots,w_{n}\) are independent standard cylindrical Wiener processes on \(L^2(\mathcal O)\). Again, smoothing properties of the transition semigroup defined by (5) and solvability of the associated Kolmogorov equation are investigated. Basically, the author’s approach remains the same as in Part I, however, many new technical complications arise (e.g., even if the functions \(f_{k}\) are very regular, the corresponding Nemytskij operator on \(L^2({\mathcal O};\mathbb R^{n})\) is not Fréchet differentiable with bounded derivatives in general), and methods used in the finite-dimensional case cannot be directly extended to equations like (5). First, in Chapters 4 and 5, the author studies smooth dependence of solutions to (5) on initial data and existence of solutions to the Kolmogorov equation under the additional hypothesis that \(f_{k}\)’s are globally Lipschitz, the results being generalized to nonlinear terms with a polynomial growth in the next two chapters. Existence and uniqueness of invariant measures for (5) are dealt with in the eighth chapter. In the last two chapters, applications to infinite-dimensional Hamilton-Jacobi-Bellman equations and to stochastic optimal control problems are given.

The book is concise and well organized; in particular we would like to mention the detailed introduction providing a useful survey of the theory developed in the book. It is a research monograph, based to some extent on the author’s numerous papers published recently. Now these interesting results are presented in a systematic manner for the first time and we believe that S. Cerrai’s treatise is a necessary reading for anyone interested in this area.

The main goal of the first part of the book under review is to extend this interplay between (1), (2) and (3) to problems with coefficients \(b\) and \(\sigma\) that are only locally Lipschitz, of a polynomial growth, and satisfy certain dissipativity assumptions. This task is highly nontrivial. For example, to show the mean square differentiability of solutions to (1) with respect to initial conditions, which is the first step in the proof of (4) in the “bounded” case, is no longer easy. This problem is studied in Chapter 1, the results obtained then being used to establish the Bismut-Elworthy formula for derivatives of the function \(P_{t}\varphi\). This in turn yields smoothing properties of the semigroup \((P_{t})\), and existence and uniqueness of classical solutions to (2) and (3) follow. Moreover, it is shown that the solutions satisfy Schauder type estimates in the space of Hölder continuous functions.

Smoothing properties of the semigroup \((P_{t})\) are closely related to its ergodic properties. In Chapter 2 it is proven that \((P_{t})\) remains strong Feller and topologically irreducible even under weaker hypotheses on \(b\) and \(\sigma\), consequently, the invariant probability measure \(\mu\) for (1) is unique and globally asymptotically stable in the total variation norm. Furthermore, dissipativity hypotheses are found under which \(P_{t}\varphi\) converges uniformly to \(\mu(\varphi)\) at a polynomial or exponential rate. In Chapter 3, the degenerate case is treated and a class of diffusion matrices \(\sigma\) is described such that \(\sigma(x)\) is non-invertible, nevertheless, the semigroup \((P_{t})\) on the space \(\text{BUC}(\mathbb R^{m})\) of bounded uniformly continuous functions is analytic.

Part II of the book is devoted to a thorough study of a stochastic reaction diffusion system \[ \partial_{t}u_{k} = {\mathcal A}_{k}(\xi,D)u_{k} + f_{k} (\xi,u_1,\ldots,u_{n}) + Q_{k}\partial^2_{t\xi}w_{k} \quad\text{in \(\mathbb R_{+}\times\mathcal O\)}, \;k=1,\ldots,n, \tag{5} \] with either Dirichlet or oblique boundary conditions. It is supposed that \(\mathcal O\subseteq\mathbb R^{m}\) is a bounded domain with a smooth boundary, \(m\leq 3\), \({\mathcal A}_{k}(\xi,D)\) are uniformly elliptic second-order differential operators with real regular coefficients, \(Q_{k}\) are nonnegative bounded linear operators in \(L^2(\mathcal O)\) (not necessarily boundedly invertible) and \(w_1,\ldots,w_{n}\) are independent standard cylindrical Wiener processes on \(L^2(\mathcal O)\). Again, smoothing properties of the transition semigroup defined by (5) and solvability of the associated Kolmogorov equation are investigated. Basically, the author’s approach remains the same as in Part I, however, many new technical complications arise (e.g., even if the functions \(f_{k}\) are very regular, the corresponding Nemytskij operator on \(L^2({\mathcal O};\mathbb R^{n})\) is not Fréchet differentiable with bounded derivatives in general), and methods used in the finite-dimensional case cannot be directly extended to equations like (5). First, in Chapters 4 and 5, the author studies smooth dependence of solutions to (5) on initial data and existence of solutions to the Kolmogorov equation under the additional hypothesis that \(f_{k}\)’s are globally Lipschitz, the results being generalized to nonlinear terms with a polynomial growth in the next two chapters. Existence and uniqueness of invariant measures for (5) are dealt with in the eighth chapter. In the last two chapters, applications to infinite-dimensional Hamilton-Jacobi-Bellman equations and to stochastic optimal control problems are given.

The book is concise and well organized; in particular we would like to mention the detailed introduction providing a useful survey of the theory developed in the book. It is a research monograph, based to some extent on the author’s numerous papers published recently. Now these interesting results are presented in a systematic manner for the first time and we believe that S. Cerrai’s treatise is a necessary reading for anyone interested in this area.

Reviewer: Jan Seidler (Praha)

##### MSC:

60-02 | Research exposition (monographs, survey articles) pertaining to probability theory |

35R15 | PDEs on infinite-dimensional (e.g., function) spaces (= PDEs in infinitely many variables) |

60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |

49L20 | Dynamic programming in optimal control and differential games |

60H10 | Stochastic ordinary differential equations (aspects of stochastic analysis) |

35K15 | Initial value problems for second-order parabolic equations |

35J15 | Second-order elliptic equations |