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On uniqueness of solutions to nonlinear Fokker-Planck-Kolmogorov equations. (English) Zbl 1336.35334
The authors study the uniqueness of the Cauchy problem for a nonlinear Fokker-Planck-Kolmogorov equation of the form $$\partial_t \mu = \partial_{x_i}\partial_{x_j}(a^{i,j}(\mu,x,t)\mu) - \partial_{x_i}(b^i(\mu,x,t),\mu)$$ with $$\mu|_{t=0}= \mu_0$$. A solution is a finite Borel $$\mu(dx\, dt) = \mu_t(dx)$$ given by a family of probability measures $$(\mu_t)_{t\in[0,T]}$$ on $$\mathbb{R}^d$$. Hence, the equations is to be understood in the sense of distributions. The diffusion matrix $$A=(a^{ij})$$ is assumed to be symmetric and non-negative definite. The main goal is to establish sufficient conditions for uniqueness allowing for nonsmooth and unbounded coefficients.
One way to establish well-posedness of the above equation is to use the connection to the martingale problem, where standard assumptions on the coefficients are a global Lipschitz bound. This restricts the nonlinear dependence of the coefficients $$A$$ and $$b$$ on $$\mu$$ to be a convolution with an at most polynomial growing kernel. Therefore the authors aim to establish uniqueness for non-Lipschitz coefficients with faster growing nonlinearities.
The proof is based on the Holmgren method: First, the adjoint problem $$\partial_s f + L_\mu f=0$$ with $$f|_{s=t} = \psi$$ and $$\psi \in C_0^\infty(\mathbb{R}^d)$$ is solved in the class of sufficiently smooth functions, where $$L_\mu = a^{i,j} \partial_{x_i} \partial_{x_j} + b^i \partial_{x_i}$$ is the formal adjoint. Then, for two solutions $$\mu,\sigma$$ with $$\mu|_{t=0}= \mu_0$$ and $$\sigma|_{t=0}=\sigma_0$$ one can formally deduce an a priori estimate of the form $$W_1(\mu_t,\sigma_t) \leq e^{Ct} W_1(\mu_0,\sigma_0)$$ with for instance $$W_1$$ the Kantorovich metric.
The main difficulty in this approach is solving the adjoint problem with nonregular and unbounded coefficients. Depending on the choice of the assumptions on the coefficients, there will be three steps necessary: Find a suitable approximation of the coefficients. Solve the adjoint problem. Find a suitable metric to which the Holmgren method can be applied. These steps are implemented for three classes of coefficients.
The first case assumes the diffusion matrix $$A$$ to be elliptic and independent of the solution. Under the assumption of the existence of a suitable Lyapunov function as well as additional regularity and growth assumptions on $$A$$ and $$b$$ the existence is established via Holmgren methods wrt. a weighted $$L^1$$-norm.
The second case considers degenerate but still independent diffusion matrix $$A$$, the extreme case being $$A\equiv 0$$ Since the solution $$\mu$$ has no reason to be absolute continuous wrt. Lebesgue, a family of metrics on the space of probability measures is introduced. For a continuous convex function $$W$$ on $$\mathbb{R}^d$$ with $$W\geq 1$$ and measures $$\mu,\sigma$$ such that $$|x| \sqrt{W(x)}\in L^1(\mu + \sigma)$$ the metric is defined by $$w_W(\mu,\sigma)=\sup\{ \int f d(\mu-\sigma) : f\in C_0^\infty(\mathbb{R}^d), |\nabla f(x)| \leq \sqrt{W(x)}\}$$. This metrics generalize the Kantorovich $$1$$-metric, which is obtained by the choice $$W\equiv 1$$. A particular choice leading to a metric on measures with $$p$$th moment is $$W(x) = (1+|x|^{p-1})^2$$, where the corresponding metric is denoted by $$w_p$$. This one can then be compared with the usual Kantorovich $$p$$-metric. The Holmgren method can then be implemented under suitable assumptions on the vectorfield $$b$$.
The last case deals with a non-degenerate but dependent diffusion matrix $$A$$. The ellipticity allows to still estimated the second derivatives of the solution to the adjoint problem in terms of first derivatives, which then connects back to the metric $$w_W$$. Hence, the essential assumption, besides regularity and growth control of $$A$$ and $$b$$ in space, is a modulus of continuity of the dependence of the coefficients on the solution wrt. the metric $$w_W$$. This allows to implement Holmgren method wrt. the metric $$w_W$$ also in this case. The authors give in the last section examples of non-uniqueness in the case of a degenerate and dependent diffusion matrix $$A$$. This also illustrates the necessity of certain assumptions made before.

##### MSC:
 35Q84 Fokker-Planck equations 35K55 Nonlinear parabolic equations 35Q83 Vlasov equations 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness 35B45 A priori estimates in context of PDEs
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##### References:
 [1] Ambrosio, L.; Gigli, N.; Savaré, G., (Gradient Flows in Metric Spaces and in the Space of Probability Measures, Lectures in Mathematics ETH Zürich, (2005), Birkhäuser Verlag Basel) [2] Benachour, S.; Roynette, B.; Talay, D.; Vallois, P., Nonlinear self-stabilizing processes. I: existence, invariant probability, propagation of chaos, Stochastic Process. Appl., 75, 2, 173-201, (1998) · Zbl 0932.60063 [3] Benachour, S.; Roynette, B.; Vallois, P., Nonlinear self-stabilizing processes. II: convergence to invariant probability, Stochastic Process. Appl., 75, 2, 203-224, (1998) · Zbl 0932.60064 [4] Bogachev, V. I., Measure theory, (2007), Springer-Verlag Berlin · Zbl 0576.28003 [5] Bogachev, V. I.; Da Prato, G.; Röckner, M., On parabolic equations for measures, Comm. Partial Differential Equations, 33, 397-418, (2008) · Zbl 1323.35058 [6] V.I. Bogachev, G. Da Prato, M. Rockner, S.V. Shaposhnikov, An analytic approach to infinite-dimensional continuity and Fokker-Planck-Kolmogorov equations, 2013. arXiv:1305.7348. · Zbl 1332.60099 [7] Bogachev, V. I.; Kolesnikov, A. V., The Monge-Kantorovich problem: achievements, connections, and perspectives, Russian Math. Surveys, 67, 5, 785-890, (2012) · Zbl 1276.28029 [8] Bogachev, V. I.; Krylov, N. V.; Roeckner, M., Elliptic and parabolic equations for measures, Russian Math. Surveys, 64, 6, 973-1078, (2009) · Zbl 1194.35481 [9] Bogachev, V. I.; Röckner, M., A generalization of khasminskii’s theorem on the existence of invariant measures for locally integrable drifts, Theory Probab. Appl., 45, 3, 363-378, (2001) · Zbl 1004.60061 [10] Bogachev, V. I.; Röckner, M.; Shaposhnikov, S. V., Nonlinear evolution and transport equations for measures, Dokl. Math., 80, 3, 785-789, (2009) · Zbl 1200.35047 [11] Bogachev, V. I.; Röckner, M.; Shaposhnikov, S. V., On uniqueness problems related to the Fokker-Planck-Kolmogorov equations for measures, J. Math. Sci. (N. Y.), 179, 1, 1-41, (2011) · Zbl 1291.35425 [12] Carrillo, J. A.; Difrancesco, M.; Figalli, A.; Laurent, T.; Slepcev, D., Global-in-time weak measure solutions and finite-time aggregation for non-local interaction equations, Duke Math. J., 156, 2, 229-271, (2011) · Zbl 1215.35045 [13] DiPerna, R. J.; Lions, P. L., On the Fokker-Planck-Boltzmann equation, Comm. Math. Phys., 120, 1, 1-23, (1988) · Zbl 0671.35068 [14] DiPerna, R. J.; Lions, P. L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., 98, 511-547, (1989) · Zbl 0696.34049 [15] Dobrushin, R. L., Vlasov equations, Funct. Anal. Appl., 13, 2, 115-123, (1979) · Zbl 0422.35068 [16] Frank, T. D., Nonlinear Fokker-Planck equations. fundamentals and applications, xii+404, (2005), Springer-Verlag Berlin · Zbl 1071.82001 [17] Friedman, A., Partial differential equations of parabolic type, (1964), Prentice-Hall NJ, Englewood Cliffs, MR 31 6062 · Zbl 0144.34903 [18] Funaki, T., A certain class of diffusion processes associated with nonlinear parabolic equations, Z. Wahrscheinlichkeitstheor. Verwandte Geb., 67, 331-348, (1984) · Zbl 0546.60081 [19] Hasminskii, R. Z., Ergodic properties of recurrent diffusion processes and stabilization of the solution of the Cauchy problem for parabolic equations, Theory Probab. Appl., 5, 179-196, (1960) · Zbl 0093.14902 [20] Jordan, R.; Kinderlehrer, D.; Otto, F., The variational formulation of the Fokker-Planck equation, SIAM J. Math. Anal., 29, 1, 1-17, (1998) · Zbl 0915.35120 [21] Knerr, B., Parabolic interior Schauder estimates by the maximum principle, Arch. Ration. Mech. Anal., 75, 51-58, (1980) · Zbl 0468.35014 [22] Kolmogorov, A. N., Über die analytischen methoden in der wahrscheinlichkeitsrechnung, Math. Ann., 104, 415-458, (1931) · Zbl 0001.14902 [23] Kozlov, V. V., The generalized Vlasov kinetic equation, Russian Math. Surveys, 63, 4, 691-726, (2008) · Zbl 1181.37006 [24] Kozlov, V. V., The Vlasov kinetic equation, dynamics of continuum and turbulence, Nelin. Dinam., 6, 3, 489-512, (2010) [25] Krylov, N. V.; Priola, E., Elliptic and parabolic second-order PDEs with growing coefficients, Comm. Partial Differential Equations, 35, 1, 1-22, (2009) · Zbl 1195.35160 [26] Le Bris, C.; Lions, P. L., Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients, Comm. Partial Differential Equations, 33, 1272-1317, (2008) · Zbl 1157.35301 [27] Lieberman, G. M., Second order parabolic differential equations, 439, (1996), World Sci. Singapore · Zbl 0884.35001 [28] Manita, O. A.; Shaposhnikov, S. V., Nonlinear parabolic equations for measures, Dokl. Math., 86, 3, 857-860, (2012) · Zbl 06149512 [29] Manita, O. A.; Shaposhnikov, S. V., Nonlinear parabolic equations for measures, St. Petersburg Math. J., 25, 1, 43-62, (2014) · Zbl 1286.35137 [30] McKean, H. P., A class of Markov processes associated with nonlinear parabolic equations, Proc. Natl. Acad. Sci. USA, 56, 1907-1911, (1966) · Zbl 0149.13501 [31] McKean, H. P., (Propagation of Chaos for a Class of Non-linear Parabolic Equations, Lecture Series in Differential Equations, Session 7, (1967), Catholic Univ.), 177-194 [32] Oleinik, O. A., On smoothness of solutions to degenerate elliptic and parabolic equations, Dokl. Acad. Sci. USSR, 163, 3, 577-580, (1965) · Zbl 0142.37201 [33] Rachev, S. T., Probability metrics and the stability of stochastic models, vol. 334, (1991), Wiley New York [34] Scheutzow, M., Uniqueness and non-uniqueness of solutions of Vlasov-Mckean equations, J. Aust. Math. Soc. Ser. A, 43, 2, 246-256, (1987) · Zbl 0625.60062 [35] Shaposhnikov, S. V., Fokker-Planck-Kolmogorov equations with potential terms and non-uniformly elliptic diffusion matrix, Tr. Mosk. Mat. Obs., MCCME, M, 74, 1, 17-34, (2013) · Zbl 1310.35229 [36] Stroock, D. W.; Varadhan, S. R.S., Multidimensional diffusion processes, (1979), Springer-Verlag Berlin, New York · Zbl 0426.60069 [37] A.Yu. Veretennikov, On Ergodic Measures for McKean-Vlasov Stochastic Equations, in: Monte Carlo and Quasi-Monte Carlo Methods 2004, 2006, 471-486. · Zbl 1098.60056 [38] Zolotarev, V. M., Approximation of the distribution of sums of independent variables with values in infinite-dimensional spaces, Theory Probab. Appl., 21, 4, 721-737, (1977) · Zbl 0378.60003 [39] Zolotarev, V. M., Probability metrics, Theory Probab. Appl., 28, 1, 278-302, (1984) · Zbl 0533.60025
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