Abstract
We study regularizing effects of nonlinear stochastic perturbations for fully nonlinear PDE. More precisely, path-by-path \(L^{\infty }\) bounds for the second derivative of solutions to such PDE are shown. These bounds are expressed as solutions to reflected SDE and are shown to be optimal.
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Notes
See however the one-dimensional example in [29].
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Acknowledgements
The work of PG was supported by the ANR, via the project ANR-16-CE40- 0020-01. The work of BG was supported by the DFG through CRC 1283.
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Appendix A: Stochastic viscosity solutions
Appendix A: Stochastic viscosity solutions
In this section we briefly recall the definition and main properties of stochastic viscosity solutions to fully nonlinear SPDE of the type
where \(u_0 \in BUC(\mathbb {R}^N)\), \(F\in C([0,T]\times \mathbb {R}^N\times \mathbb {R}\times \mathbb {R}^N\times S^N)\) and \(\xi \) is a continuous path.
We recall from [23, Theorem 1.2, Theorem 1.3]
Theorem A.1
Let \(u_0,v_0 \in BUC(\mathbb {R}^N)\), \(T>0\), \(\xi ,\zeta \in C^1_0([0,T];\mathbb {R})\) and assume that Assumption 2.1 holds. If \(u\in BUSC([0,T]\times \mathbb {R}^N)\), \(v\in BLSC([0,T]\times \mathbb {R}^N)\) are viscosity sub- and super-solutions to (A.1) driven by \(\xi ,\zeta \) respectively, then,
where \(\Phi \) depends only on T, the sup-norms and moduli of continuity of \(u_0, v_0\) and the quantities appearing in Assumption 2.1 (2)–(3)–(5), is non-decreasing and such that \(\Phi (0^+)=0\). In particular, the solution operator
admits a unique continuous extension to
We then call \(u=S^\xi (u_0)\) the unique viscosity solution to (). One then has
In the case where \(F=F(p,X)\) only depends on its last two arguments, the estimate simplifies to
(with convention \(0/0=0\), \(1/0=+\,\infty \)).
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Gassiat, P., Gess, B. Regularization by noise for stochastic Hamilton–Jacobi equations. Probab. Theory Relat. Fields 173, 1063–1098 (2019). https://doi.org/10.1007/s00440-018-0848-7
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DOI: https://doi.org/10.1007/s00440-018-0848-7
Keywords
- Stochastic Hamilton–Jacobi equations; regularization by noise
- Reflected SDE
- Stochastic p-Laplace equation
- Stochastic total variation flow
Mathematics Subject Classification
- 60H15
- 65M12
- 35L65