PATHWISE DEFINITION OF SECOND ORDER SDES LLUÍS QUER-SARDANYONS AND SAMY TINDEL Abstract. In this article, a class of second order differential equations on [0, 1], driven by a ?-Hölder continuous function for any value of ? ? (0, 1) and with multiplicative noise, is considered. We first show how to solve this equation in a pathwise manner, thanks to Young integration techniques. We then study the differentiability of the solution with respect to the driving process and consider the case where the equation is driven by a fractional Brownian motion, with two aims in mind: show that the solution we have produced coincides with the one which would be obtained with Malliavin calculus tools, and prove that the law of the solution is absolutely continuous with respect to the Lebesgue measure. 1. Introduction During the last past years, a growing activity has emerged, aiming at solving stochastic PDEs beyond the Brownian case. In some special situations, namely in linear (additive noise) or bilinear (noisy term of the form u B˙) cases, stochastic analysis techniques can be applied [14, 30]. When the driving process of the equation exhibits a Hölder continuity exponent greater than 1/2, Young integration or fractional calculus tools also allow to solve those equations in a satisfying way [10, 17, 25]. Eventually, when one wishes to tackle non-linear problems in which the driving noise is only Hölder continuous with Hölder regularity exponent ≤ 1/2, rough paths analysis must come into the picture.
- malliavin calculus
- solution could
- linear multiplicative
- young integration
- hölder continuity
- technique can
- additive noise
- noise x˙