Stochastic Lagrangian transport and generalized relative entropies. (English) Zbl 1120.35046

Let us consider a linear operator \[ D\rho=\nu\partial_{i}(a_{ij}\partial_{j}\rho)-\text{ div}_{x}(U\rho)+V\rho \] in \(\mathbb{R}^{n}\), where \(U(x,t)=(U_{j}(x,t))_{j=1,\ldots,n}\) is a smooth \((C^2)\) function, \(V=V(x,t)\) is a continuous and bounded scalar potential and \(a_{ij}(x,t)=\sigma_{ip}(x,t)\sigma_{jp}(x,t)\), \(\sigma(x,t)\) is smooth \((C^2)\), bounded matrix. \(U\) and \(\nabla_{x}\sigma\) decay at infinity. Let \(X(a,t)\) be the strong solution of the stochastic differential system \(dX_{j}(t)=v_{j}(X,t)dt+\sqrt{2\nu}\sigma_{jp}(X,t)dW_{p}\) with initial data \(X(a,0)=a\), where \[ v_{j}(x,t)=U_{j}-\nu(\partial_{k}\sigma_{kp})\sigma_{jp} +\nu(\partial_{k}\sigma_{jp})\sigma_{kp}. \] Let the stochastic map \(x\mapsto A(x,t)\) be the inverse of the flow map \(a\mapsto X(a,t)\). Then the process \[ \psi(x,t)=f_0(A(x,t))\exp \left\{\int_{0}^{t}P(X(a,s),s)\,ds |_{a=A(x,t)}\right\}, \] where \(P=V-\text{ div}_{x}(U)\), solves \(d\psi-(D\psi)dt+\sqrt{2\nu}\nabla_{x}\psi\sigma \,dW=0\) with initial data \(\psi(x,0)=f_0(x)\). Also the stochastic integrals of motion and generalized relative entropies are discussed.


35K45 Initial value problems for second-order parabolic systems
60H30 Applications of stochastic analysis (to PDEs, etc.)
35R60 PDEs with randomness, stochastic partial differential equations
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