Abstract
We study the asymptotic behavior of solutions of semilinear PDEs. Neither periodicity nor ergodicity will be assumed. On the other hand, we assume that the coecients have averages in the Cesaro sense. In such a case, the averaged coecients could be discontinuous. We use a probabilistic approach based on weak convergence of the associated backward stochastic dierential equation (BSDE) in the Jakubowski $S$-topology to derive the averaged PDE. However, since the averaged coecients are discontinuous, the classical viscosity is not dened for the averaged PDE. We then use the notion of solution introduced in [7]. The existence of $L_p$-viscosity to the averaged PDE is proved here by using BSDEs techniques.
Highlights
In this paper, we study the limit of the solution of the semi-linear PDEs of the form∂vε ∂s (s, x1, x2) = Lε(x1, x2)vε(s, x1, x2) + f (x1 ε x2, vε(s, x2))s ∈ (0, t) vε(0, x1, x2) = H(x1, x2) (1.1)The infinitesimal generator Lε is associated to the IR × IRd-diffusion process (x1t, ε, x2t, ε) x1t, ε = x11 ε t 0 φ(x1s, ε, x2s, ε)dWs x2t, ε = x2 +
We use probabilistic approach based on weak convergence techniques for the associated backward stochastic differential equation in the S-topology
Is the infinitesimal generator associated to the IR × IRd-diffusion process (ε x1t, ε, x2t, ε) defined by
Summary
We study the limit of the solution of the semi-linear PDEs of the form. The system (1.2) have been considered by Krylov and Khasminskii [4] studying weak convergence without ergodicity and periodicity assumptions. They defined averaged coefficients as a limit in Cesaro sense. The classical probabilistic representation of viscosity solution for PDE fails due, to the discontinuity of the coefficients. We use a probabilistic representation of Lp-viscosity solution of nonlinear PDE to make sense the connection to BSDE.
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