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Distances between transition probabilities of diffusions and applications to nonlinear Fokker-Planck-Kolmogorov equations. (English) Zbl 1345.35051
The authors consider the Cauchy problem for measures \begin{aligned} \partial_t\mu &=\mathcal{L}_{A,b}^*\mu, \\ \mu|_{t=0}&=\nu,\end{aligned}\tag{1} in $$\mathbb{R}^d\times (0,T)$$, where $$\nu$$ is a Borel probability measure on $$\mathbb{R}^d$$ and the coefficients of the operator $\mathcal{L}_{A,b}=\sum_{i,j=1}^d a^{ij}\partial_{x_i}\partial_{x_j}+\sum_{i=1}^db^i\partial_{x_i}$ are Borel-measurable. The diffusion matrix $$A(x,t)=(a^{ij}(x,t))_{ij}$$ is positive symmetric, assumed to be locally Lipschitz in $$x$$ and locally strictly positive, and the drift $$b(x,t)=(b^i(x,t))_i$$ is locally bounded.
The main result of the paper are estimates for the entropy of two solutions of the Cauchy problem $$(1)$$ with different diffusion matrices and drifts (but the same initial data $$\nu$$). More precisely, let $$A^{(i)}$$ and $$b^{(i)}$$ be diffusion matrices and drift functions satisfying the above conditions, and $$\mu^{(i)}$$ be the corresponding solutions of $$(1)$$ with densities $$\rho^{(i)}$$, $$i=1,2.$$ If either the coefficients of elliptic operator satisfy some mild integrability conditions, or a certain Lyapunov function exists, estimates for the entropy $H(\mu_t^{(1)}|\mu_t^{(2)})= \int_{\mathbb{R}^d}\rho^{(1)}(x, t)\log\Big(\frac{\rho^{(1)}(x, t)}{\rho^{(2)}(x, t)}\Big)dx$ are established, and as a consequence, bounds for the total variation and Kantorovich distance between $$\rho^{(1)}$$ and $$\rho^{(2)}$$ deduced. The article is based on a series of previous works by the authors, the main novelty is to consider equations with different diffusion matrices (and drifts).
Finally, the results are applied to prove the local existence and uniqueness of solutions of nonlinear Fokker-Planck-Kolmogorov equations and to formulate sufficient conditions for their differentiability. Further applications are an optimal control problem and mean field game models.

##### MSC:
 35K15 Initial value problems for second-order parabolic equations 60J60 Diffusion processes 35Q84 Fokker-Planck equations 35K55 Nonlinear parabolic equations
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