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Inverse problems for stochastic transport equations. (English) Zbl 1322.35172

The authors consider the equations \(\partial_tu=\partial_xu+V(x)u+\dot W\) and \(\partial_tu=\partial_xu+V(x)u+u\dot W\) with an initial condition \(u(0)=u_0\) on \(\mathbb R\), where \(\dot W\) denotes either a temporal or a spatial Wiener process on \(\mathbb R\) and \(V\) is a bounded deterministic function. The objective of the paper is to reconstruct (approximatively) the potential \(V\) from the pair of \(u_0\) and the random path of the solution \(\{u(t,0):t>0\}\). It is proved that a random process \(\hat V_n\) can be constructed from \((u_0,u(\cdot,0))\) such that \(\hat V_n(t)\) converges to \(V(t)\) almost surely at a precise rate. In case of the equation with the additive temporal noise, the potential must have the form \(V(x)=aq(x)\), where \(q\) is either a periodic bounded function or an indicator of an interval \([0,b]\) and \(a\) is an unknown parameter.

MSC:

35R30 Inverse problems for PDEs
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
35R60 PDEs with randomness, stochastic partial differential equations
65C30 Numerical solutions to stochastic differential and integral equations
82C70 Transport processes in time-dependent statistical mechanics
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