Infinite dimensional stochastic differential equations of Ornstein-Uhlenbeck type (Q2490067)

Let \(\lambda_i\) be a sequence of positive reals tending to infinity, let \(\sigma_{ij}\) and \(b_i\) be functions defined on a suitable Hilbert space, which satisfy certain continuity and non-degeneracy conditions, and let \(W_t^i\) be a sequence of independent one-dimensional Brownian motions. The authors consider the countable system of stochastic differential equations \[ dX_t^i=\sum_{j=1}^\infty \sigma_{ij}(X_t)\,dW_t^i-\lambda_i b_i(X_t)X_t^i\, dt,\quad i=1,2,\dots, \tag{1} \] and investigate sufficient conditions for weak existence and weak uniqueness to hold. Note that when the \(\sigma_{ij}\) and \(b_i\) are constant, we have the stochastic differential equations characterizing the infinite-dimensional Ornstein-Uhlenbeck process. The authors approach the weak existence and uniqueness of (1) by means of the martingale problem for the corresponding operator \[ \mathcal Lf(x)=\frac{1}{2}\sum_{i,j=1}^\infty a_{ij}(x)\frac{\partial^2f}{\partial x_i\partial x_j}(x)- \sum_{i,j=1}^\infty \lambda_ix_ib_i(x)\frac{\partial f}{\partial x_i}(x) \] operating on a suitable class of functions, where \(a_{ij}(x) = \sum_{j=1}^\infty \sigma_{ik}(x)\sigma_{jk}(x)\). The main theorem says that if the \(a_{ij}\) are nondegenerate and bounded, the \(b_i\) are bounded above and below, and the \(a_{ij}\) and \(b_i\) satisfy appropriate Hölder continuity conditions, then existence and uniqueness hold for the martingale problem for \(\mathcal L\). There has been considerable interest in infinite-dimensional operators whose coefficients are only Hölder continuous. Consider the one-dimensional SPDE \[ \frac{\partial u}{\partial t}(t,x)= \frac{1}{2} \frac{\partial^2u}{\partial x^2}(x,t)+A(u)\,d\dot W\tag{2} \] where \(\dot W\) is space-time white noise. If one sets \[ X_t^j=\int_0^{2\pi}e^{\text{i}jx}u(x,t)\,dx,\quad j=0,\pm1,\pm2,\dots, \] then the collection \(\{X^i\}_{i=-\infty}^\infty\) can be shown to solve system (1) with \(\lambda_i = i^2\), the \(b_i\) constant, and the \(a_{ij}\) defined in an explicit way in terms of \(A\). The original interest in the problem solved in this paper is to understand (2) when the coefficients \(A\) are bounded above and below but were only Hölder continuous as a function of \(u\). The main novelties of the paper are the following: (1) \(C^\alpha\) estimates (i.e., Schauder estimates) for the infinite-dimensional Ornstein-Uhlenbeck process. These were already known but the authors point out that in contrast to using interpolation theory, the derivation in this paper is quite elementary and relies on a simple real variable lemma together with some semigroup manipulations. (2) Localization. The authors use perturbation theory along the lines of Stroock-Varadhan to establish uniqueness of the martingale problem when the coefficients are sufficiently close to constant. The authors then perform a localization procedure to establish the main result. In infinite dimensions, localization is much more involved, and this argument represents an important feature of this work. (3) A larger class of perturbations. Unlike much of the previous work on the subject, the authors do not require that the perturbation of the second order term be bounded by an operator that is nonnegative. The price the authors pay is that they require additional conditions on the off-diagonal \(a_{ij}\)'s. The paper also contains some specific examples where the main result applies. This includes coefficients \(a_{ij}\) which depend on a finite number of local coordinates near \((i,j)\) in a Hölder manner.