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Regularization by noise for stochastic Hamilton–Jacobi equations

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

  1. See however the one-dimensional example in [29].

  2. Equations of this form arise as (simplified) models of fluctuating hydrodynamics of the zero range process about its hydrodynamic limit (cf. [14] and (1.10) below).

  3. In contrast, critical noise intensities regarding synchronization by noise have been observed before (cf. e.g. [1, 18, 40]).

  4. For a theory of pathwise entropy solutions to (6.9) we refer to [25].

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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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Correspondence to Paul Gassiat.

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

$$\begin{aligned} \begin{aligned} d u + \frac{1}{2} |D u|^2 \circ d{\xi }(t)&= F(t,x,u,Du,D^2u)dt \quad {\text {in }}\mathbb {R}^N\times (0,T] \\ u(0,\cdot )&= u_0\quad {\text {on }}\mathbb {R}^N\times \{0\}, \end{aligned} \end{aligned}$$
(A.1)

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,

$$\begin{aligned} \sup _{[0,T]\times \mathbb {R}^N} (u-v) \le \sup _{\mathbb {R}^N}(u_0-v_0)_+ + \Phi (\Vert \xi -\zeta \Vert _{C([0,T])}) , \end{aligned}$$
(A.2)

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

$$\begin{aligned} S: BUC(\mathbb {R}^N) \times C^1_0([0,T];\mathbb {R}^N) \rightarrow BUC([0,T]\times \mathbb {R}^N) \end{aligned}$$

admits a unique continuous extension to

$$\begin{aligned} S: BUC(\mathbb {R}^N) \times C^0_0([0,T];\mathbb {R}^N) \rightarrow BUC([0,T]\times \mathbb {R}^N). \end{aligned}$$

We then call \(u=S^\xi (u_0)\) the unique viscosity solution to (). One then has

$$\begin{aligned} \Vert S^{\xi }(u_0)-S^{\zeta }(v_0)\Vert _{C([0,T]\times \mathbb {R}^N)} \le \Vert u_0-v_0\Vert _{C(\mathbb {R}^N)}+\Phi \left( \Vert \xi -\zeta \Vert _{C([0,T])}\right) . \end{aligned}$$
(A.3)

In the case where \(F=F(p,X)\) only depends on its last two arguments, the estimate simplifies to

$$\begin{aligned} \sup _{[0,T]\times \mathbb {R}^N} (u-v) \le \sup _{x,y \in \mathbb {R}^N} \left( u_0(x)-v_0(y) - \frac{|x-y|^2}{\sup _{s \in [0,T]} (\xi (s)-\zeta (s))} \right) \end{aligned}$$
(A.4)

(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