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I have learned that for the equation $\partial_tu+a(u)\partial_xu=0$, the entropy solution could be obtained as the limit of the equation $\partial_tu+a(u)\partial_xu=\epsilon u_{xx}$ with $\epsilon>0$ when $\epsilon \rightarrow 0$. When I study the stochastic equation, there is a connect of the stochastic equation with the classical ones.

Suppose $dX=b dt+G dW$ where $W$ is the Brownian motion. And $u(X,t)$ will follow $du=(\partial_t u+\frac{1}{2}G^2 u_{xx}+ b u_x)dt+G u_x dW$. If we take the parameter $G=\epsilon$, $du$ will have the classical solution as $\epsilon \rightarrow 0$. Here how could I prove the uniqueness and convergence of such a process? Here is the limit same with the definition of viscosity solution in the traditional case? Could one provide some references for the viscosity of stochastic differential equations?

I think it easy to generalize the one dimensional case to higher dimension and I want to define the entropy solution with the help of theory of SDEs and to study the shock solution in a similar method.

Thank you!

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  • $\begingroup$ Is the function $a$ dependent on $u$ or on $x$ only. In the former case it is not obvious to link it to stochastic differential equations easily as the PDE is not linear and your process $X_t$ will depend on $u$ which is unknown here. If $a$ depends only on $x$ you can represent it via Feyman-Kac and all what you need for the existence of the solution to the SDE equation on $X_t$ is that $a$ is Lipschitzian. $\endgroup$
    – Hicham
    Commented Nov 2, 2015 at 13:30

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Here are some references...

@article {MR1959710, AUTHOR = {Lions, P.-L. and Souganidis, P. E.}, TITLE = {Viscosity solutions of fully nonlinear stochastic partial differential equations}, NOTE = {Viscosity solutions of differential equations and related topics (Japanese) (Kyoto, 2001)}, JOURNAL = {S\=urikaisekikenky\=usho K\=oky\=uroku}, FJOURNAL = {S\=urikaisekikenky\=usho K\=oky\=uroku}, NUMBER = {1287}, YEAR = {2002}, PAGES = {58--65}, MRCLASS = {60H10 (35J60 35R60)}, MRNUMBER = {1959710}, }

@article {MR1807189, AUTHOR = {Lions, Pierre-Louis and Souganidis, Panagiotis E.}, TITLE = {Uniqueness of weak solutions of fully nonlinear stochastic partial differential equations}, JOURNAL = {C. R. Acad. Sci. Paris S\'er. I Math.}, FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique}, VOLUME = {331}, YEAR = {2000}, NUMBER = {10}, PAGES = {783--790}, ISSN = {0764-4442}, CODEN = {CASMEI}, MRCLASS = {35R60 (35A05)}, MRNUMBER = {1807189 (2002a:35214)}, MRREVIEWER = {Marek Capi{\'n}ski}, DOI = {10.1016/S0764-4442(00)01597-4}, URL = {htp://dx.doi.org/10.1016/S0764-4442(00)01597-4}, }

@article {MR1659958, AUTHOR = {Lions, Pierre-Louis and Souganidis, Panagiotis E.}, TITLE = {Fully nonlinear stochastic partial differential equations: non-smooth equations and applications}, JOURNAL = {C. R. Acad. Sci. Paris S\'er. I Math.}, FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique}, VOLUME = {327}, YEAR = {1998}, NUMBER = {8}, PAGES = {735--741}, ISSN = {0764-4442}, CODEN = {CASMEI}, MRCLASS = {60H15 (35R60)}, MRNUMBER = {1659958 (99j:60095)}, MRREVIEWER = {Sergey V. Lototsky}, DOI = {10.1016/S0764-4442(98)80161-4}, URL = {htp://dx.doi.org/10.1016/S0764-4442(98)80161-4}, }

@article {MR1647162, AUTHOR = {Lions, Pierre-Louis and Souganidis, Panagiotis E.}, TITLE = {Fully nonlinear stochastic partial differential equations}, JOURNAL = {C. R. Acad. Sci. Paris S\'er. I Math.}, FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences. S\'erie I. Math\'ematique}, VOLUME = {326}, YEAR = {1998}, NUMBER = {9}, PAGES = {1085--1092}, ISSN = {0764-4442}, CODEN = {CASMEI}, MRCLASS = {60H15 (35R60)}, MRNUMBER = {1647162 (99j:60094)}, MRREVIEWER = {Samy Tindel}, DOI = {10.1016/S0764-4442(98)80067-0}, URL = {http://dx.doi.org/10.1016/S0764-4442(98)80067-0}, }

etc...

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    $\begingroup$ Could you add some more explanation of how this article answers the question? $\endgroup$
    – Yemon Choi
    Commented Dec 27, 2014 at 23:06

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