Rough paths and 1d SDE with a time dependent distributional drift: application to polymers

Abstract

Motivated by the recent advances in the theory of stochastic partial differential equations involving nonlinear functions of distributions, like the Kardar–Parisi–Zhang (KPZ) equation, we reconsider the unique solvability of one-dimensional stochastic differential equations, the drift of which is a distribution, by means of rough paths theory. Existence and uniqueness are established in the weak sense when the drift reads as the derivative of a \(\alpha \)-Hölder continuous function, \(\alpha >1/3\). Regularity of the drift part is investigated carefully and a related stochastic calculus is also proposed, which makes the structure of the solutions more explicit than within the earlier framework of Dirichlet processes.

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Correspondence to François Delarue.

Appendix

Appendix

Lemma 33

Given a sequence of smooth paths \((Y^n)_{n \ge 1}\) such that, for some \(T_{0}>0\) and any \(T \in [0,T_{0}]\), the sequence \((W^{n,T}=(Y^n,Z^{n,T}))_{n \ge 1}\) satisfies the assumption of Proposition 6, with \(\kappa =\sup _{0 \le T \le T_{0}} \sup _{n \ge 1} \kappa _{\alpha ,\chi }((W_{t}^{n,T},{\fancyscript{W}}_{t}^{n,T})_{0 \le t <T}) < \infty \), then, we can assume that, for any \(n \ge 1\), \(Y^n\) has bounded derivatives on the whole space.

Proof

For \(N \in {\mathbb N} {\setminus } \{0\}\), we consider a smooth function \(\varphi ^N : {\mathbb {R}}\rightarrow [0,1]\), symmetric, equal to \(1\) on \([0,N]\) and to \(0\) on \([2N,+\infty )\), non-increasing on \([N,2N]\), satisfying \(\Vert \mathrm{d}^p \varphi ^N/\mathrm{d}x^p \Vert _{\infty } \le c_{p}/N^p\) for some \(c_{p} \ge 1\), independent of \(N\), for any integer \(p \ge 1\). Then, we let \(Y^{n,N}_{t}(x)=Y^n_{t}(0)+\int _{0}^x \varphi ^N(y) \partial _{x} Y^n_{t}(y) \mathrm{d}y\) and, for a given \(T>0\), we define \(Z^{n,N,T}\), \(W^{n,N,T}\) and \({\fancyscript{W}}^{n,N,T}\) accordingly.

For a given \(n\), \((Y^{n,N})_{N \ge 1}\) (resp. \(\partial _{x}^p Y^{n,N}\) for an integer \(p \ge 1\)) converges towards \(Y^n\) (resp. \(\partial _{x}^p Y^n\)) as \(N\) tends to \(\infty \), uniformly in \(x\) in compact sets and in \(t \in [0,T)\). Using the representations of \(Z_{t}^{n,N,T}\) and \(Z^{n,T}_{t}\), see (13), the same holds true for the sequence \((Z^{n,N,T})_{N \ge 1}\) (resp. \(({W}^{n,N,T})_{N \ge 1}\)) with \(Z^{n,T}\) (resp. \(W^{n,T}\)) as limit path. Hence, \(({{\fancyscript{W}}}_{t}^{n,N,T})_{N \ge 1}\) converges towards \({\fancyscript{W}}_{t}^{n,T}\) in norm \(\Vert \cdot \Vert _{\alpha }\), uniformly in \(t \in [0,T)\). Using the same notation as in Proposition 6, \((\Vert (W^{n,N,T}-W^{n,T},{\fancyscript{W}}^{n,N,T} - {\fancyscript{W}}^{n,T})\Vert _{0,\alpha }^{[0,T) \times {\mathbb {I}}})_{N \ge 1}\) tends to \(0\) as \(N\) tends to \(\infty \). Therefore, we can find a sequence \((N_{n})_{n \ge 1}\) such that \(\Vert (W^{n,N_{n},T}-W^{n,T},{\fancyscript{W}}^{n,N_{n},T} - {\fancyscript{W}}^{n,T})\Vert _{0,\alpha }^{[0,T) \times {\mathbb {I}}}\), and thus \(\Vert (W^{n,N_{n},T}-W^{T},{\fancyscript{W}}^{n,N_{n},T} - {\fancyscript{W}}^{T})\Vert _{0,\alpha }^{[0,T) \times {\mathbb {I}}}\), tend to \(0\) as \(n\) tends to \(\infty \), which fits (1) in Proposition 6.

We now discuss (2) in Proposition 6. We start with the Hölder estimate of \(Y^{n,N}_{t}\). For \(0\le x \le y \le a\), with \(a \ge 1\), the second mean-value theorem yields \({Y}^{n,N}_{t}(y) - {Y}^{n,N}_{t}(x) = \varphi _{N}(x) [Y^{n}_{t}(y')-Y_{t}^{n}(x)]\), for \(y' \in [x,y]\). We deduce that \(\vert Y^{n,N}_{t}(y) - Y^{n,N}_{t}(x) \vert \le \kappa a^{\chi } \vert y - x \vert ^{\alpha }\). The same holds true when \(-a \le y \le x \le 0\). Changing \(\kappa \) into \(2\kappa \), we get the same result for any \(x,y \in [-a,a]\). By Lemma 19, the bound \(\vert {Z}^{n,N}_{t}(x) - {Z}^{n,N}_{t}(y) \vert \le \kappa a^{\chi } \vert x-y \vert ^{\alpha }\) follows.

We finally discuss the regularity of the second-order integrals. As discussed in Sect. 5, it suffices to focus on the cross-integral \(\int _{x}^y [ {Z}^{n,N}_{t}(z) - {Z}^{n,N}_{t}(x)] \mathrm{d}{Y}^{n,N}_{t}(z)\).

By (13), \(\partial _{t} Z_{t}^{n,N,T}(x) + (1/2) \partial ^2_{x} Z_{t}^{n,N,T}(x) = - \partial ^2_{x} [Y_{t}^{n,N}](x) = - \partial _{x} [ \varphi ^N \partial _{x} Y^{n}_{t}](x)\). Similarly, \(\partial _{t} Z_{t}^{n,T}(x) + (1/2) \partial ^2_{x} Z_{t}^{n,T}(x) = - \partial ^2_{x} [Y_{t}^{n}](x)\). Therefore,

$$\begin{aligned} \partial _{t} \bigl [ Z_{t}^{n,N,T} - \varphi ^N Z_{t}^{n,T} \bigr ] + \tfrac{1}{2} \partial ^2_{x} \bigl [ Z_{t}^{n,N,T} -\varphi ^N Z_{t}^{n,T} \bigr ] = - \varphi _{N}' \partial _{x} \bigl [Y^{n}_{t} + Z^{n,T}_{t} \bigr ] - \tfrac{1}{2} \varphi _{N}'' Z^{n,T}_{t}, \end{aligned}$$

with \(Z_{T}^{n,N,T} - Z_{T}^{n,T}=0\). Therefore, integrating against \(p_{s-t}\) and then integrating by parts,

$$\begin{aligned}&Z_{t}^{n,N,T}(x) - \varphi ^N(x) Z_{t}^{n,T}(x) \nonumber \\&\quad = \int _{t}^T \int _{{\mathbb {R}}} \partial _{x} p_{s-t}(x-y) \varphi _{N}'(y) \bigl [Y^{n}_{t} + Z^{n,T}_{t} \bigr ](y) \mathrm{d}y \mathrm{d}s\nonumber \\&\quad \quad - \int _{t}^T \int _{{\mathbb {R}}} p_{s-t}(x-y) \varphi _{N}''(y) \Bigl ( \bigl [Y^{n}_{t} + Z^{n,T}_{t} \bigr ](y) +\frac{1}{2} Z^{n,T}_{t}(y) \Bigr ) \mathrm{d}y \mathrm{d}s. \end{aligned}$$
(96)

The aim is to differentiate both sides of the equality in order to estimate the derivative of the left-hand side. In order to bound the derivative of the right-hand side, we discuss the Hölder constant of the integrands right above. We have \(\vert \varphi _{N}'(y) Y^{n}_{t}(y) - \varphi _{N}'(x) Y^{n}_{t}(x) \vert \le c_{2} \vert Y^n_{t}(x) \vert \vert y - x \vert /N^2 + (c_{1}\kappa /N) a^{\chi } \vert y -x \vert ^{\alpha }\), for \(x,y \in [-a,a]\), \(a \ge 1\). Modifying \(\kappa \) if necessary \(\vert Y^n_{t}(x) \vert \le \kappa a^{1+\chi }\). Therefore, we can find a constant \(C \ge 0\) such that

$$\begin{aligned} \vert \varphi _{N}'(y) Y^{n}_{t}(y) - \varphi _{N}'(x) Y^{n}_{t}(x) \vert \le C a^{1+\chi } \vert y - x \vert /N^2 + C a^{\chi } \vert y -x \vert ^{\alpha }/N. \end{aligned}$$

Since \(\varphi _{N}' =0\) outside \([-2N,2N]\), we can always assume that \(x,y \in [-2N,2N]\) (by projecting \(x\) and \(y\) onto \([-2N,2N])\) and thus that \(a \le 2N\). Then, the left-hand side is less than \(C a^{\chi } \vert y -x \vert ^{\alpha }/N\). Using a similar argument for all the other terms of the same type in the right-hand side of (96), we deduce that the left-hand side in (96) is differentiable and that \(\vert \partial _{x} [ Z_{t}^{n,N,T} - \varphi ^N Z_{t}^{n,T}](x) \vert \le C a^{\chi }/N\), when \(x \in [-a,a]\), \(a \ge 1\). By integration by parts,

$$\begin{aligned}&\biggl \vert \int _{x}^y \Bigl ( \bigl [ Z_{t}^{n,N,T} - \varphi ^N Z_{t}^{n,T}\bigr ](z) - \bigl [ Z_{t}^{n,N,T} - \varphi ^N Z_{t}^{n,T}\bigr ](x) \Bigr ) \mathrm{d}Y^{n,N}_{t}(z) \biggr \vert \\&\quad = \biggl \vert \int _{x}^y \bigl [Y^{n,N}(y) - Y^{n,N}(z) \bigr ] \partial _{x} [ Z_{t}^{n,N,T} - \varphi ^N Z_{t}^{n,T}](z) \mathrm{d}z \biggr \vert \le C \frac{a^{2\chi }\vert x-y \vert ^{1+\alpha }}{N}. \end{aligned}$$

Since \(\partial _{x}Y^{n,N}_{t}(z)=0\) when \(\vert z \vert \ge 2N\), we can always assume that \(x,y \in [-2N,2N]\) and \(a \le 2N\). We deduce that the term in the first line is less than \(C a^{2\chi } \vert x-y \vert ^{2\alpha }/N^{\alpha }\). To end up the analysis, it thus suffices to prove that

$$\begin{aligned} \biggl \vert \int _{x}^y \bigl ( \varphi ^N(z) Z_{t}^{n,T}(z) - \varphi ^N(x) Z_{t}^{n,T}(x) \bigr ) \mathrm{d}Y^{n,N}_{t}(z) \biggr \vert \le C a^{2\chi }\vert x-y \vert ^{2\alpha }. \end{aligned}$$

Since \(\partial _{x} Y^{n,N}_{t}(z) = \varphi ^N(z) \partial _{x} Y^n_{t}(z)\), we can use again the second mean-value theorem to handle \(\int _{x}^y \varphi ^N(z)[ Z_{t}^{n,T}(z) - Z_{t}^{n,T}(x)] \mathrm{d}Y^{n,N}_{t}(z) = \int _{x}^y (\varphi ^N(z))^2[ Z_{t}^{n,T}(z) - Z_{t}^{n,T}(x)] \mathrm{d}Y^{n}_{t}(z)\). Therefore, it suffices to focus on \(Z_{t}^{n,T}(x) \int _{x}^y [ \varphi ^N(z) - \varphi ^N(x)] \mathrm{d}Y^{n,N}_{t}(z)\). By integration by parts,

$$\begin{aligned}&\biggl \vert Z_{t}^{n,T}(x) \int _{x}^y [ \varphi ^N(z) - \varphi ^N(x)] \mathrm{d}Y^{n,N}_{t}(z) \biggr \vert \\&\quad = \biggl \vert Z_{t}^{n,T}(x) \int _{x}^y \bigl [Y^{n,N}_{t}(y)-Y^{n,N}_{t}(z) \bigr ] ( \varphi ^N)'(z) \mathrm{d}z \biggr \vert , \end{aligned}$$

which is less than \(C a^{2\chi } \vert y -x \vert ^{1+\alpha }/N\) (following Lemma 19, \(Z^{n,T}_{t}\) satisfies \(\vert Z^{n,T}_{t}(x) \vert \le C a^{\chi }\) –better than the elementary but rough bound \(\vert Z^{n,T}_{t}(x) \vert \le C a^{1+\chi }\)–). Limiting the analysis to the case \(a \le 2N\), we conclude as above. \(\square \)

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Delarue, F., Diel, R. Rough paths and 1d SDE with a time dependent distributional drift: application to polymers. Probab. Theory Relat. Fields 165, 1–63 (2016). https://doi.org/10.1007/s00440-015-0626-8

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Mathematics Subject Classification

  • Primary 60H10
  • Secondary 60H05
  • 82D60