Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness

Rehmeier M (2021)
Journal of Evolution Equations 21(1): 17-31.

Zeitschriftenaufsatz | Veröffentlicht | Englisch
 
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Abstract / Bemerkung
Let the coefficients a(ij) and b(i), i, j <= d, of the linear Fokker-Planck-Kolmogorov equation (FPK-eq.) partial derivative(t)mu(t) = partial derivative(i)partial derivative(j)(a(ij)mu(t)) - partial derivative(i)(b(i)mu(t)) be Borel measurable, bounded and continuous in space. Assume that for every s. [0, T] and every Borel probability measure. on Rd there is at least one solution mu = (mu(t))t is an element of([s,T]) to the FPK-eq. such that mu(s) = v and t bar right arrow mu(t) is continuous w.r.t. the topology of weak convergence of measures. We prove that in this situation, one can always select one solution mu(s,v) for each pair (s,v) such that this family of solutions fulfills mu(s,v)(t) = mu(r,mu rs,v)(t) for all 0 <= s <= r <= t <= T, which one interprets as a flow property of this solution family. Moreover, we prove that such a flow of solutions is unique if and only if the FPK-eq. is well-posed.
Stichworte
Fokker-Planck-Kolmogorov equation; Flow property; Martingale problem; Superposition principle
Erscheinungsjahr
2021
Zeitschriftentitel
Journal of Evolution Equations
Band
21
Ausgabe
1
Seite(n)
17-31
ISSN
1424-3199
eISSN
1424-3202
Page URI
https://pub.uni-bielefeld.de/record/2942725

Zitieren

Rehmeier M. Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness. Journal of Evolution Equations. 2021;21(1):17-31.
Rehmeier, M. (2021). Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness. Journal of Evolution Equations, 21(1), 17-31. https://doi.org/10.1007/s00028-020-00569-y
Rehmeier, Marco. 2021. “Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness”. Journal of Evolution Equations 21 (1): 17-31.
Rehmeier, M. (2021). Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness. Journal of Evolution Equations 21, 17-31.
Rehmeier, M., 2021. Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness. Journal of Evolution Equations, 21(1), p 17-31.
M. Rehmeier, “Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness”, Journal of Evolution Equations, vol. 21, 2021, pp. 17-31.
Rehmeier, M.: Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness. Journal of Evolution Equations. 21, 17-31 (2021).
Rehmeier, Marco. “Existence of flows for linear Fokker-Planck-Kolmogorov equations and its connection to well-posedness”. Journal of Evolution Equations 21.1 (2021): 17-31.
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2022-11-14T12:40:05Z
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