부산대학교 도서관

The authors consider the following system of multidimensional SDEs on the plane, with integral form: $$ X_{s,t}-X_{s,0}-X_{0,t}+X_{0,0}=\int_0^t \int_0^s b(\xi,\zeta,X_{\xi,\zeta}) {\rm d}\xi {\rm d}\zeta + W_{s,t}. $$ Here $(s,t)\in\Bbb{R}^2_+$, $b\:\Bbb{R}^2_+\times \Bbb{R}^d\to\Bbb{R}^d$ and $W$ is a $d$-dimensional Brownian sheet with ${\partial W=0}$,~where $\partial W$ stands for its restriction on the boundary $\partial \Bbb{R}^2_+$. Such an SDE can be formally written in differential form as $$ \frac{\partial^2 X_{s,t}}{\partial s\partial t} = b(s,t,X_{s,t}) + \frac{\partial^2 W_{s,t}}{\partial s\partial t}, $$ and can be connected, by a formal $\pi/4$ rotation transform, to the noisy nonlinear stochastic wave equation $$ \frac{\partial^2 Y}{\partial t^2} - \frac{\partial^2 Y}{\partial x^2} = b(t,x,Y) + \dot{W}, $$ where $\dot{W}$ now denotes space-time white noise. Alternatively, the SDE can also be regarded as a noisy version of the so-called Darboux problem. \par The main result of the paper shows that {\it path-by-path uniqueness} holds for the SDE under the mere assumptions that $b$ is of linear growth (i.e., $b(s,t,x)\leq M(1+|x|)$) and monotone in the space variable (i.e., $b_i(x)\leq b_i(y)$ whenever $x\preceq y$, $\preceq$ denoting the partial componentwise order on $\Bbb{R}^2_+$), in a {\it regularisation by noise} fashion. In particular, the authors strenghten a previous result by D. Nualart and S. Tindel [Potential Anal. {\bf 7} (1997), no.~3, 661--680; MR1473647] by relaxing a boundedness assumption on $b$ and improving the pathwise uniqueness statement to the stronger path-by-path one. As a byproduct, strong existence of solutions follows from the Yamada-Watanabe principle. The proof is based on revisiting the strategy originally proposed by A.~M. Davie [Int. Math. Res. Not. IMRN {\bf 2007}, no.~24, Art. ID rnm124; MR2377011] in the one-dimensional setting; it relies on the use of the Girsanov transform, suitable estimates for averaging operators along the Brownian sheet and a Gronwall-type lemma.