Stochastic approach for a multivalued Dirichlet-Neumann problem

We prove the existence and uniqueness of a viscosity solution of the parabolic variational inequality with a nonlinear multivalued Neumann-Dirichlet boundary condition:% {equation*} \{{array}{r} \dfrac{\partial u(t,x)}{\partial t}-\mathcal{L}_{t}u(t,x) {+}{% \partial ϕ}\big(u(t,x)\big)\ni f\big(t,x,u(t,x),(\nabla uσ)(t,x)\big), t>0, x\in \mathcal{D},\medskip \multicolumn{1}{l}{\dfrac{\partial u(t,x)}{\partial n}+{\partial ψ}\big(% u(t,x)\big)\ni g\big(t,x,u(t,x)\big), t>0, x\in Bd(\mathcal{D}%),\multicolumn{1}{l}{u(0,x)=h(x), x\in \bar{\mathcal{D}},}% {array}%. {equation*}% where $\partial ϕ$ and $\partial ψ$ are subdifferentials operators and $\mathcal{L}_{t}$ is a second differential operator. The result is obtained by a Feynman-Kaç representation formula starting from the backward stochastic variational inequality:% {equation*} \{{array}{l} dY_{t}{+}F(t,Y_{t},Z_{t}) dt{+}G(t,Y_{t}) dA_{t}\in \partial ϕ(Y_{t}) dt{+}\partial ψ(Y_{t}) dA_{t}{+}Z_{t}dW_{t}, 0\leq t\leq T,\medskip \ Y_{T}=ξ.% {array}%. {equation*}

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