Singular HJB equations with applications to KPZ on the real line

Abstract

This paper is devoted to studying Hamilton-Jacobi-Bellman equations with distribution-valued coefficients, which are not well-defined in the classical sense and are understood by using the paracontrolled distribution method introduced in (Gubinelli et al. in Forum Math Pi 3(6):1, 2015). By a new characterization of weighted Hölder spaces and Zvonkin’s transformation we prove some new a priori estimates, and therefore establish the global well-posedness for singular HJB equations. As applications, we obtain global well-posedness in polynomial weighted Hölder spaces for KPZ type equations on the real line, as well as modified KPZ equations for which the Cole–Hopf transformation is not applicable.

Appendices

Appendix A: Uniqueness of paracontrolled solutions

In this subsection we use Hairer and Labbé’s argument [30] to show the uniqueness of paracontrolled solutions. For this aim, we use the following time-dependent exponential weight: for $\ell \in (0,1)$,

$$\begin{aligned} \mathbf{e}^\ell _t(x):=\exp (-(1+t)\langle x\rangle ^\ell ),\ t\geqslant 0,\ x\in {{\mathbb {R}}}^d. \end{aligned}$$

We can similarly define the Hölder space with weight $\mathbf{e}^\ell$ (see [45]). For instance,

$$\begin{aligned} \Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^\alpha (\mathbf{e}^\ell )}:=\sup _{t\in [0,T]}\Vert f(t,\cdot )\Vert _{{{\mathbf {C}}}^\alpha (\mathbf{e}^\ell _t)}, \end{aligned}$$

and for $\alpha \in (0,1)$,

$$\begin{aligned} \Vert f\Vert _{C^\alpha _TL^\infty (\mathbf{e}^\ell )}&:= \sup _{0\leqslant t\leqslant T} \Vert f(t)\mathbf{e}^\ell _t\Vert _{L^\infty } +\sup _{0\leqslant s\ne t\leqslant T} \frac{\Vert f(t)-f(s)\Vert _{L^\infty (\mathbf{e}^\ell _{t\vee s})}}{|t-s|^{\alpha }}. \end{aligned}$$

In particular, for $\alpha \in (0,2)$, we also set

$$\begin{aligned} {{\mathbb {S}}}^\alpha _T(\mathbf{e}^\ell ):=\Vert f\Vert _{L^\infty _T{{\mathbf {C}}}^\alpha (\mathbf{e}^\ell )}+\Vert f\Vert _{C^{\alpha /2}_TL^\infty (\mathbf{e}^\ell )}. \end{aligned}$$

By [43, Lemma 2.10], for any $T>0$, there is a $C=C(T,\ell ,d)>0$ such that for all $s,t\in [0,T]$ and $j\geqslant -1$,

$$\begin{aligned} \Vert \Delta _jP_tf\Vert _{L^\infty (\mathbf{e}^\ell _s)}\lesssim \mathrm {e}^{-2^{2j}t}\Vert \Delta _jf\Vert _{L^\infty (\mathbf{e}^\ell _s)}. \end{aligned}$$

(A.1)

Moreover, Lemmas 2.8, 2.10, 2.11 and 2.12 still hold for exponential weight $\mathbf{e}^\ell _t$ (see [45]). The following result corresponds to Lemma 2.9.

Lemma A.1

Let $\alpha ,\ell \in (0,1)$, $\kappa \in (0,(1-\frac{\alpha }{2})\ell )$. For any $q\in (\frac{1}{1-\alpha /2-\kappa /\ell },\infty ]$ and $T>0$, there is a constant $C=C(T, d,\alpha ,\ell ,\theta ,\kappa ,q)>0$ such that

$$\begin{aligned} \Vert {{\mathscr {I}}}f\Vert _{{{\mathbb {S}}}_T^{2-\frac{2}{q}-\frac{2\kappa }{\ell }-\alpha }(\mathbf{e}^\ell )}\lesssim _C \Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa }\mathbf{e}^\ell )}. \end{aligned}$$

Proof

First of all we have the following simple observation:

$$\begin{aligned} \mathbf{e}^\ell _t(x)\lesssim \langle x\rangle ^{-\kappa } \mathbf{e}^\ell _s(x)/|t-s|^{\kappa /\ell },\ \ 0\leqslant s<t<\infty . \end{aligned}$$

(A.2)

Let $\frac{1}{p}+\frac{1}{q}=1$ and $t\in (0,T]$. By (A.1) and Hölder’s inequality, we have for $j\geqslant -1$,

$$\begin{aligned} \Vert \Delta _j{{\mathscr {I}}}f(t)\Vert _{L^\infty (\mathbf{e}^\ell _t)}&\lesssim \int ^t_0\mathrm {e}^{-2^{2j}(t-s)}\Vert \Delta _jf(s)\Vert _{L^\infty (\mathbf{e}^\ell _t)}{\mathord {\mathrm{d}}}s \\ {}&\lesssim \int ^t_0\frac{\mathrm {e}^{-2^{2j}(t-s)}}{|t-s|^{\kappa /\ell }}\Vert \Delta _jf(s)\Vert _{L^\infty (\rho _\kappa \mathbf{e}^\ell _s)}{\mathord {\mathrm{d}}}s \\ {}&\lesssim 2^{\alpha j}\left( \int ^t_0\frac{\mathrm {e}^{-p2^{2j}(t-s)}}{|t-s|^{p\kappa /\ell }}{\mathord {\mathrm{d}}}s\right) ^{1/p} \Vert f\Vert _{L^q_t{{\mathbf {C}}}^{-\alpha }(\rho _\kappa \mathbf{e}^\ell )}\\&\lesssim 2^{-(\frac{2}{p}-\frac{2\kappa }{\ell }-\alpha ) j}\Vert f\Vert _{L^q_t{{\mathbf {C}}}^{-\alpha }(\rho _\kappa \mathbf{e}^\ell )}, \end{aligned}$$

which in turn gives that

$$\begin{aligned} \Vert {{\mathscr {I}}}f\Vert _{L^\infty _T{{\mathbf {C}}}^{2-\frac{2}{q}-\frac{2\kappa }{\ell }-\alpha }(\mathbf{e}^\ell )} \lesssim \Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa }\mathbf{e}^\ell )}. \end{aligned}$$

(A.3)

On the other hand, for $0\leqslant t_1<t_2\leqslant T$, we have

$$\begin{aligned} \Vert {{\mathscr {I}}}f(t_2)- {{\mathscr {I}}}f(t_1)\Vert _{L^\infty (\mathbf{e}^\ell _{t_2})}&\leqslant \Vert (P_{t_2-t_1}-I){{\mathscr {I}}}f(t_1)\Vert _{L^\infty (\mathbf{e}^\ell _{t_2})}\\&\quad +\left\| \int _{t_1}^{t_2}P_{t_2-s}f(s) {\mathord {\mathrm{d}}}s\right\| _{L^\infty (\mathbf{e}^\ell _{t_2})}=:I_1+I_2. \end{aligned}$$

For $I_1$, by (2.10) and (A.3) we have

$$\begin{aligned} I_1&\lesssim (t_2-t_1)^{1-\frac{\alpha }{2}-\frac{1}{q}- \frac{\kappa }{\ell }} \Vert {{\mathscr {I}}}f(t_1)\Vert _{{{\mathbf {C}}}^{2-\alpha -\frac{2}{q}- \frac{2\kappa }{\ell }}(\mathbf{e}^\ell _{t_2})} \\ {}&\lesssim (t_2-t_1)^{1-\frac{\alpha }{2}-\frac{1}{q}- \frac{\kappa }{\ell }}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa }{} \mathbf{e}^\ell )}. \end{aligned}$$

For $I_2$, by (2.8), (A.2) and Hölder’s inequality, we have

$$\begin{aligned} I_2&\lesssim \int _{t_1}^{t_2}(t_2-s)^{-\frac{\alpha }{2}}\Vert f(s)\Vert _{{{\mathbf {C}}}^{-\alpha }(\mathbf{e}^\ell _{t_2})}{\mathord {\mathrm{d}}}s \\ {}&\lesssim \int _{t_1}^{t_2}(t_2-s)^{-\frac{\alpha }{2}-\frac{\kappa }{\ell }}\Vert f(s)\Vert _{{{\mathbf {C}}}^{-\alpha }(\rho _\kappa \mathbf{e}^\ell _s)}{\mathord {\mathrm{d}}}s \\ {}&\lesssim (t_2-t_1)^{1-\frac{\alpha }{2}-\frac{1}{q}- \frac{\kappa }{\ell }}\Vert f\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{\kappa }{} \mathbf{e}^\ell )}. \end{aligned}$$

Combining the above estimates, we obtain the desired estimate. $\square$

Now we consider the following linear equation:

$$\begin{aligned} {{\mathscr {L}}}u=(b+{\bar{b}})\cdot \nabla u+hu,\ \ u(0)\equiv 0, \end{aligned}$$

(A.4)

where $b\in \cap _{T>0}{{\mathbb {B}}}^\alpha _T(\rho _\kappa )$ and ${\bar{b}},h\in \cap _{T>0}L^\infty _T(\rho _\eta )$. Let

$$\begin{aligned} (u,u^\sharp )\in \cap _{T>0}{{\mathbb {S}}}^{2-\alpha }_T(\rho _\eta )\times {{\mathbb {S}}}^{3-2\alpha }_T(\rho _{2\eta }) \end{aligned}$$

be the paracontrolled solution of PDE (A.4). That is,

$$\begin{aligned} u=\nabla u\prec \!\!\!\prec {{\mathscr {I}}}b+u^\sharp , \end{aligned}$$

(A.5)

with $u^\sharp$ solving the following PDE in weak sense

$$\begin{aligned} {{\mathscr {L}}}u^\sharp&=\nabla u\prec b-\nabla u\prec \!\!\!\prec b+\nabla u\succ b+b\circ \nabla u\nonumber \\ {}&\quad +{\bar{b}}\cdot \nabla u+hu-[{{\mathscr {L}}}, \nabla u\prec \!\!\!\prec ]{{\mathscr {I}}}b, \end{aligned}$$

(A.6)

where

$$\begin{aligned} b\circ \nabla u&=b\circ (\nabla ^2 u\prec {{\mathscr {I}}}b)+(b\circ \nabla {{\mathscr {I}}}b)\cdot \nabla u+\text {com}\nonumber \\&\quad +\text {com}_1+b\circ \nabla u^\sharp , \end{aligned}$$

(A.7)

and

$$\begin{aligned} \text {com}_1:=b\circ \nabla [\nabla u\prec \!\!\!\prec {{\mathscr {I}}}b-\nabla u\prec {{\mathscr {I}}}b] \end{aligned}$$

and

$$\begin{aligned} \text {com}:=\mathrm {com}(\nabla u, \nabla {{\mathscr {I}}}b,b). \end{aligned}$$

Theorem A.2

Let $\ell \in (0,1)$ and $\kappa \in (0,\tfrac{(2-3\alpha )\ell }{6})$, $\eta \in (0,\frac{(1-\alpha )\ell }{2})$. Suppose that

$$\begin{aligned}&b\in \cap _{T>0}{{\mathbb {B}}}^\alpha _T(\rho _\kappa ),\ \ {\bar{b}},h\in \cap _{T>0}{{\mathbb {L}}}^\infty _T(\rho _\eta ),\\&\beta \in (\alpha ,(2-2\alpha -\tfrac{6\kappa }{\ell })\wedge (1-\tfrac{2\eta }{\ell })),\ \ \gamma \in (\alpha ,2-2\alpha -\tfrac{4\kappa }{\ell }). \end{aligned}$$

The unique paracontrolled solution to PDE (A.4) in the sense of Definition 3.1 with

$$\begin{aligned} (u,u^\sharp )\in {{\mathbb {S}}}_T^{\gamma +\alpha }(\mathbf{e}^\ell )\times L_T^\infty {{\mathbf {C}}}^{\beta +1}(\mathbf{e}^\ell ) \end{aligned}$$

is zero.

Proof

Let $T>0$. Choose q large enough such that

$$\begin{aligned}\alpha<\gamma \leqslant 2-2\alpha -\tfrac{2}{q}-\tfrac{4\kappa }{\ell },\ \ \alpha <\beta \leqslant (2-2\alpha -\tfrac{2}{q}-\tfrac{6\kappa }{\ell })\wedge (1-\tfrac{2\eta }{\ell }). \end{aligned}$$

First of all, by Lemmas A.1 and 2.10, we have

$$\begin{aligned} \begin{aligned}&\Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha -\frac{2}{q}-\frac{4\kappa }{\ell }}(\mathbf{e}^\ell )} \\ {}&\lesssim \Vert b\prec \nabla u+b\succ \nabla u+b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{2\kappa }{} \mathbf{e}^\ell )}+\Vert {\bar{b}}\cdot \nabla u+hu\Vert _{L_T^qL^\infty (\rho _{\eta }{} \mathbf{e}^\ell )} \\ {}&\lesssim \Vert b\Vert _{L^\infty _T{{\mathbf {C}}}^{-\alpha }(\rho _\kappa )}\Vert \nabla u\Vert _{L^q_TL^\infty (\mathbf{e}^\ell )} +\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{-\alpha }(\rho _{2\kappa }{} \mathbf{e}^\ell )} \\ {}&+\Vert {\bar{b}}\Vert _{{{\mathbb {L}}}_T^\infty (\rho _\eta )}\Vert \nabla u\Vert _{L_T^qL^\infty (\mathbf{e}^\ell )}+\Vert h\Vert _{{{\mathbb {L}}}_T^\infty (\rho _\eta )}\Vert u\Vert _{L_T^qL^\infty (\mathbf{e}^\ell )}, \end{aligned} \end{aligned}$$

and by the corresponding version of Lemma 2.12 for exponential weight $\mathbf{e}^\ell$ (see [45, Lemma 2.10]),

$$\begin{aligned} \begin{aligned} \Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\mathbf{e}^\ell )}&\lesssim \Vert \nabla u\prec b-\nabla u\prec \!\!\!\prec b+\nabla u\succ b-[{{\mathscr {L}}}, \nabla u\prec \!\!\!\prec ]{{\mathscr {I}}}b\Vert _{L^\infty _T{{\mathbf {C}}}^{1-2\alpha -\frac{2}{q}-\frac{4\kappa }{\ell }}(\rho _\kappa \mathbf{e}^\ell )}\\&\quad +\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }\mathbf{e}^\ell )}+\Vert {\bar{b}}\cdot \nabla u+hu\Vert _{{{\mathbb {L}}}_T^\infty (\rho _{\eta }\mathbf{e}^\ell )} \\&\lesssim \Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }\mathbf{e}^\ell )}+\Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha -\frac{2}{q}-\frac{4\kappa }{\ell }}(\mathbf{e}^\ell )} \\ {}&\quad +\Vert {\bar{b}}\Vert _{{{\mathbb {L}}}_T^\infty (\rho _\eta )}\Vert \nabla u\Vert _{{{\mathbb {L}}}_T^\infty (\mathbf{e}^\ell )}+\Vert h\Vert _{{{\mathbb {L}}}_T^\infty (\rho _\eta )}\Vert u\Vert _{{{\mathbb {L}}}_T^\infty (\mathbf{e}^\ell )} \\ {}&\lesssim \Vert u\Vert _{{{\mathbb {S}}}_T^{2-\alpha -\frac{2}{q}-\frac{4\kappa }{\ell }}(\mathbf{e}^\ell )}+\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }{} \mathbf{e}^\ell )}. \end{aligned}\end{aligned}$$

Moreover, by Lemma 3.3 with $(\rho ,{\bar{\rho }})=(\rho _\kappa ,\mathbf{e}^\ell _t)$,

$$\begin{aligned} \Vert (b\circ \nabla u)(t)\Vert _{{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }\mathbf{e}^\ell _t)} \lesssim \Vert u\Vert _{{{\mathbb {S}}}^{\gamma +\alpha }_t(\mathbf{e}^\ell )}+\Vert u^\sharp (t)\Vert _{{{\mathbf {C}}}^{\beta +1}(\rho _\kappa \mathbf{e}^\ell _t)}. \end{aligned}$$

Combining the above three estimates, we obtain

$$\begin{aligned}&\Vert u\Vert _{{{\mathbb {S}}}_T^{\gamma +\alpha }(\mathbf{e}^\ell )}+\Vert u^\sharp \Vert _{L_T^\infty {{\mathbf {C}}}^{\beta +1}(\mathbf{e}^\ell )} \\ {}&\lesssim \Vert \nabla u\Vert _{L^q_TL^\infty (\mathbf{e}^\ell )} +\Vert u\Vert _{L_T^qL^\infty (\mathbf{e}^\ell )}+\Vert b\circ \nabla u\Vert _{L^q_T{{\mathbf {C}}}^{1-2\alpha }(\rho _{2\kappa }{} \mathbf{e}^\ell )}\\&\lesssim \left( \int ^T_0\Big (\Vert u\Vert ^q_{{{\mathbb {S}}}^{\gamma +\alpha }_t(\mathbf{e}^\ell )}+\Vert u^\sharp (t)\Vert ^q_{{{\mathbf {C}}}^{\beta +1}(\rho _\kappa \mathbf{e}^\ell _t)}\Big ){\mathord {\mathrm{d}}}t\right) ^{1/q}, \end{aligned}$$

which implies $u\equiv 0$ by Gronwall’s inequality. $\square$

Appendix B: Exponential moment estimates for SDEs

In this section we consider the following SDE:

$$\begin{aligned} {\mathord {\mathrm{d}}}X_t=b(t,X_t){\mathord {\mathrm{d}}}t+\sigma (t,X_t){\mathord {\mathrm{d}}}W_t, \ X_0=x. \end{aligned}$$

We have the following exponential moment estimates for $X_t$.

Lemma B.1

Suppose that $\sigma$ is bounded and b is linear growth. Then for any $\alpha \in [0,2)$ and $T, \gamma >0$, there is a constant $C>0$ such that for all $x\in {{\mathbb {R}}}^d$,

$$\begin{aligned} {{\mathbb {E}}}\mathrm {e}^{\gamma \sup _{t\in [0,T]} \langle X_t\rangle ^\alpha }\leqslant C\mathrm {e}^{\langle x\rangle ^\alpha }. \end{aligned}$$

Proof

Let $\beta \in (\alpha ,2)$. Recall $\langle x\rangle ^\beta =(1+|x|^2)^{\beta /2}$. By Itô’s formula, we have

$$\begin{aligned} M_t:=\mathrm {e}^{-\lambda t}\langle X_t\rangle ^\beta&=\langle x\rangle ^\beta +\int ^t_0 \eta _s{\mathord {\mathrm{d}}}s+\int ^t_0\xi _s{\mathord {\mathrm{d}}}W_s, \end{aligned}$$

where

$$\begin{aligned} \eta _s&:=\mathrm {e}^{-\lambda s}\beta \Big [X_s\cdot b(s,X_s)+ \mathrm {tr}(\sigma \sigma ^*)(s,X_s)/2\Big ]\langle X_s\rangle ^{\beta -2}\\&\quad +\beta (\tfrac{\beta }{2}-1)\mathrm {e}^{-\lambda s}|\sigma ^*(s,X_s) X_s|^2 \langle X_s\rangle ^{\beta -4}-\lambda \mathrm {e}^{-\lambda s} \langle X_s\rangle ^\beta , \end{aligned}$$

and

$$\begin{aligned} \xi _s:=\beta \mathrm {e}^{-\lambda s}\sigma ^*(s,X_s)X_s\langle X_s\rangle ^{\beta -2}. \end{aligned}$$

By the linear growth of b and the boundedness of $\sigma$, there is a $\lambda$ large enough so that

$$\begin{aligned} \eta _s\leqslant 0 \end{aligned}$$

and

$$\begin{aligned} |\xi _s|^2\leqslant C\mathrm {e}^{-\lambda s}\langle X_s\rangle ^{2(\beta -1)}\leqslant C M^{2-\frac{2}{\beta }}_s. \end{aligned}$$

Now by [33, Theorem 1.1], we obtain the desired estimate. $\square$