Integral and probabilistic representations for systems of elliptic equations. (English) Zbl 0864.60064
Summary: We present probabilistic representations for some systems of elliptic equations constructed as expectations of functionals of some specific Markov chains, in particular, the walk on spheres processes. These representations are deduced from converse mean value theorems that we prove for the equations under analysis, especially the Lamé equation of elasticity theory. We construct Monte Carlo algorithms, and estimate the variances and the cost.
MSC:
| 60J60 | Diffusion processes |
| 35R60 | PDEs with randomness, stochastic partial differential equations |
| 35J99 | Elliptic equations and elliptic systems |
| 65C10 | Random number generation in numerical analysis |
Keywords:
direct and converse spherical mean value relations; local integral equations; walk on spheres process; Lamé equationReferences:
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