Maximum Principle for Spdes and Its Applications

The maximum principle for SPDEs is established in multidimensional C domains. An application is given to proving the Hölder continuity up to the boundary of solutions of one-dimensional SPDEs. The maximum principle is one of the most powerful tools in the theory of second-order elliptic and parabolic partial differential equations. However, until now it did not play any significant role in the theory of SPDEs. In this paper we show how to apply it to one-dimensional SPDEs on the half line R+ = (0,∞) and prove the Hölder continuity of solutions on [0,∞). This result was previously known when the coefficients of the first order derivatives of solution appearing in the stochastic term in the equation obeys a quite unpleasant condition. On the other hand, if they just vanish, then the Hölder continuity was well known before (see, for instance, [6] and the references therein). To the best of our knowledge the maximum principle was first proved in [12] (see also [14] for the case of random coefficients) for SPDEs in the whole space by the method of random characteristics introduced there and also in [15]. Later the method of random characteristics was used in many papers for various purposes, for instance, to prove smoothness of solutions (see, for instance, [1], [2], [3], [17] and the references therein). It was very tempting to try to use this method for proving the maximum principle for SPDEs in domains. However, the implementation of the method turns out to become extremely cumbersome and inconvenient if the coefficients of the equation are random processes. Also, it requires more regularity of solutions than actually needed. Here in Section 1 we state the maximum principle in domains under minimal assumptions. We prove it in Section 3 by using methods taken from PDEs after we prepare some auxiliary results in Section 2. Section 5 contains an application of the maximum principle to investigating the Hölder continuity up to the boundary of solutions of one-dimensional SPDEs. Note that, for instance, in [1], [2] and in many other papers that can be found from our list of references the regularity properties are proved 1991 Mathematics Subject Classification. 60H15, 35R60.