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.

MSC:

 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
Full Text: