\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2007(2007), No. 30, pp. 1--9.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu (login: ftp)} \thanks{\copyright 2007 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2007/30\hfil Subparabolic comparison principle] {A comparison principle for a class of subparabolic equations in Grushin-type spaces} \author[T. Bieske\hfil EJDE-2007/30\hfilneg] {Thomas Bieske} \address{Thomas Bieske \newline Department of Mathematics\\ University of South Florida\\ Tampa, FL 33620, USA} \email{tbieske@math.usf.edu} \thanks{Submitted November 27, 2006. Published February 14, 2007.} \subjclass[2000]{35K55, 49L25, 53C17} \keywords{Grushin-type spaces; parabolic equations; viscosity solutions} \begin{abstract} We define two notions of viscosity solutions to subparabolic equations in Grushin-type spaces, depending on whether the test functions concern only the past or both the past and the future. We then prove a comparison principle for a class of subparabolic equations and show the sufficiency of considering the test functions that concern only the past. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \section{Background and Motivation} In \cite{B:GS}, the author considered viscosity solutions to fully nonlinear subelliptic equations in Grushin-type spaces, which are sub-Riemannian metric spaces lacking a group structure. It is natural to consider viscosity solutions to subparabolic equations in this same environment. Our main theorem, found in Section 4, is a comparison principle for a class of subparabolic equations in Grushin-type spaces. We begin with a short review of the key geometric properties of Grushin-type spaces in Section 2 and in Section 3, we define two notions of viscosity solutions to subparabolic equations. Section 4 contains a parabolic comparison principle and the corollary showing the sufficiency of using test functions that concern only the past. \section{Grushin-type Spaces} We begin with $\mathbb{R}^n$, possessing coordinates $p = (x_1,x_2,\dots,x_n)$ and vector fields $$X_i = \rho_i (x_1,x_2,\dots,x_{i-1}) \frac{\partial}{\partial x_i}$$ for $i=2,3,\dots, n$ where $\rho_i (x_1,x_2,\dots,x_{i-1})$ is a (possibly constant) polynomial. We decree that $\rho_1 \equiv 1$ so that $$X_1=\frac{\partial}{\partial x_1}.$$ A quick calculation shows that when $i < j$, the Lie bracket is given by \begin{equation*} X_{ij} \equiv [X_i,X_j]= \rho_i (x_1,x_2,\dots,x_{i-1}) \frac{\partial \rho_j (x_1,x_2,\dots,x_{j-1})}{\partial x_i} \frac{\partial }{\partial x_j}. \end{equation*} Because the $\rho_i$'s are polynomials, at each point there is a finite number of iterations of the Lie bracket so that $\frac{\partial}{\partial x_i}$ has a non-zero coefficient. It follows that H\"{o}rmander's condition \cite{H:H} is satisfied by these vector fields. We may further endow $\mathbb{R}^N$ with an inner product (singular where the polynomials vanish) so that the span of the $\{X_i\}$ forms an orthonormal basis. This produces a sub-Riemannian manifold that we shall call $g_n$, which is also the tangent space to a generalized Grushin-type space $G_n$. Points in $G_n$ will also be denoted by $p=(x_1,x_2,\dots, x_n)$. We observe that if $\rho_i\equiv 1$ for all $i$, then $g_n=G_n=\mathbb{R}^n$. Given a smooth function $f$ on $G_n$, we define the horizontal gradient of $f$ as $$\nabla_0f(p) = (X_1f(p),X_2f(p),\dots, X_nf(p))$$ and the symmetrized second order (horizontal) derivative matrix by $$((D^2f(p))^{\star})_{ij} = \frac{1}{2} (X_iX_jf(p)+X_jX_if(p))$$ for $i,j=1,2,\dots n$. \begin{definition} \label{def1} \rm The function $f: G_n \to \mathbb{R}$ is said to be $C^1_{\rm sub}$ if $X_if$ is continuous for all $i=1,2,\dots,n$. Similarly, the function $f$ is $C^2_{\rm sub}$ if $X_iX_jf(p)$ is continuous for all $i,j=1,2,\dots,n$. \end{definition} Though $G_n$ is not a Lie group, it is a metric space with the natural metric being the Carnot-Carath\'{e}odory distance, which is defined for points $p$ and $q$ as follows: \begin{equation*} d_C(p,q)= \inf_{\Gamma} \int_{0}^{1} \| \gamma '(t) \| dt. \end{equation*} Here $\Gamma$ is the set of all curves $\gamma$ such that $\gamma (0) = p$, $\gamma (1) = q$ and $$\gamma '(t) \in \mathop{\rm span} \{\{X_i(\gamma(t))\}_{i=1}^n\} .$$ By Chow's theorem (see, for example, \cite{BR:SRG}) any two points can be joined by such a curve, which means $d_C(p,q)$ is an honest metric. Using this metric, we can define Carnot-Carath\'{e}odory balls and bounded domains in the usual way. The Carnot-Carath\'{e}odory metric behaves differently at points where the polynomials $\rho_i$ vanish. Fixing a point $p_0$, consider the $n$-tuple $r_{p_0}=(r^1_{p_0},r^2_{p_0},\dots,r^n_{p_0})$ where $r^i_{p_0}$ is the minimal number of Lie bracket iterations required to produce $$[X_{j_1},[X_{j_2},[\cdots[X_{j_{r^i_{p_0}}},X_i]\cdots](p_0) \neq 0.$$ Note that though the minimal length is unique, the iteration used to obtain that minimum is not. Note also that $$\rho_i(p_0) \neq 0 \leftrightarrow r^i_{p_0}=0.$$ Setting $R^i(p_0)=1+r^i_{p_0}$ we obtain the local estimate at $p_0$ $$\label{distest} d_C(p_0,p) \sim \sum _{i=1}^n |x_i-x_i^0|^\frac{1}{R^i(p_0)}$$ as a consequence of \cite[Theorem 7.34]{BR:SRG}. Using this local estimate, we can construct a local smooth Grushin gauge at the point $p_0$, denoted $\mathcal{N}(p_0,p)$, that is comparable to the Carnot-Carath\'{e}odory metric. Namely, $$\label{gauge} (\mathcal{N}(p_0,p))^{2\mathcal{R}} =\sum_{i=1}^n (x_i-x_i^0)^{\frac{2\mathcal{R}}{R^i(p_0)}}$$ with $$\mathcal{R}(p_0)=\prod_{i=1}^n R^i(p_0).$$ \section{Subparabolic Jets and Solutions to Subparabolic Equations} In this section, we define and compare various notions of solutions to parabolic equations in Grushin-type spaces, in the spirit of \cite[Section 8]{CIL:UGTVS}. We begin by letting $u(p,t)$ be a function in $G_n \times [0,T]$ for some $T>0$ and by denoting the set of $n \times n$ symmetric matrices by $S^{n}$. We consider parabolic equations of the form $$\label{main} u_t+F(t,p,u,\nabla_0 u,(D^2u)^{\star})=0$$ for continuous and proper $F:[0,T]\times G_n \times \mathbb{R} \times g_n \times S^{n} \to \mathbb{R}$. Recall that $F$ is proper means $$F(t,p,r,\eta,X)\leq F(t,p,s,\eta,Y)$$ when $r\leq s$ and $Y\leq X$ in the usual ordering of symmetric matrices. \cite{CIL:UGTVS} We note that the derivatives $\nabla_0 u$ and $(D^2u)^{\star}$ are taken in the space variable $p$. We call such equations \emph{subparabolic}. Examples of subparabolic equations include the subparabolic $P$-Laplace equation for $2 \leq P < \infty$ given by $$u_t+\Delta_Pu = u_t - \textmd{div}(\|\nabla_0u\|^{P-2}\nabla_0u)=0$$ and the subparabolic infinite Laplace equation $$u_t+\Delta_{\infty}u = u_t - \langle(D^2u)^\star\nabla_0u, \nabla_0u \rangle =0.$$ Let $\mathcal{O}\subset G_n$ be an open set containing the point $p_0$. We define the parabolic set $\mathcal{O}_T \equiv \mathcal{O} \times (0,T)$. Following the definition of Grushin jets in \cite{B:GS}, we can define the subparabolic superjet of $u(p,t)$ at the point $(p_0,t_0) \in \mathcal{O}_T$, denoted $P^{2,+}u(p_0,t_0)$, by using triples $(a,\eta,X) \in \mathbb{R} \times g_n \times S^{n}$ with $\eta=\sum_{i=1}^n \eta_jX_j$ and the $ij$-th entry of $X$ denoted $X_{ij}$. We then have that $(a,\eta,X) \in P^{2,+}u(p_0,t_0)$ if \begin{align*} u(p,t) &\leq u(p_0,t_0)+ a(t-t_0)+\sum_{j \notin \mathcal{N}} \frac{1}{\rho_j(p_0)}(x_j-x_j^0)\eta_j\\ &\quad +\frac{1}{2}\sum_{j \notin \mathcal{N}} \frac{1}{(\rho_j(p_0))^2}(x_j-x_j^0)^2X_{jj} \\ & \quad + \sum_{\stackrel{i,j \notin \mathcal{N}}{i < j}}(x_i-x_i^0)(x_j-x_j^0)\big(\frac{1}{\rho_j(p_0)\rho_i(p_0)}X_{ij}- \frac{1}{2}\frac{1}{(\rho_j(p_0))^2}\frac{\partial \rho_j}{\partial x_i}(p_0)\eta_j\big) \\ &\quad + \sum_{k \in \mathcal{N}} \frac{1}{\beta}\sum_{j=1}^n(x_{k}-x_{k}^0)\frac{2}{\rho_j(p_0)} (\frac{\partial \rho_{k}}{\partial x_j}(p_0))^{-1}X_{jk} +o(|t-t_0|+d_C(p_0,p)^2). \end{align*} Here, as in \cite{B:GS}, $\beta$ is the number of non-zero terms in the final sum and we understand that if $\rho_j(p_0)=0$ or $\frac{\partial \rho_{i_m}}{\partial x_j}(p_0)=0$ then that term in the final sum is zero. We define the subjet $P^{2,-}u(p_0,t_0)$ by $$P^{2,-}u(p_0,t_0)=-P^{2,+}(-u)(p_0,t_0).$$ We also define the set theoretic closure of the superjet, denoted $\overline{P}^{2,+}u(p_0,t_0)$, by requiring $(a,\eta,X) \in \overline{P}^{2,+}u(p_0,t_0)$ exactly when there is a sequence $$(a_n,p_n,t_n,u(p_n,t_n),\eta_n,X_n)\to (a,p_0,t_0,u(p_0,t_0),\eta,X)$$ with the triple $(a_n,\eta_n,X_n)\in P^{2,+}u(p_n,t_n)$. A similar definition holds for the closure of the subjet. As in the subelliptic case, we may also define jets using the appropriate test functions. Namely, we consider the set $\mathcal{A}u(p_0,t_0)$ by $\mathcal{A}u(p_0,t_0)=\{\phi \in C^2_{\rm sub}(\mathcal{O}_T): u(p,t) -\phi(p,t) \leq u(p_0,t_0)-\phi(p_0,t_0)=0\}$ consisting of all test functions that touch from above. We define the set of all test functions that touch from below, denoted $\mathcal{B}u(p_0,t_0)$, by $\mathcal{B}u(p_0,t_0)=\{\phi \in C^2_{\rm sub}(\mathcal{O}_T): u(p,t)-\phi(p,t) \geq u(p_0,t_0)-\phi(p_0,t_0)=0\}.$ The following lemma is proved in the same way as the Euclidean version (\cite{C:VS} and \cite{I:I}) except we replace the Euclidean distance $|p-p_0|$ with the local Grushin gauge $\mathcal{N}(p_0,p)$. \begin{lemma} With the above notation, we have $P^{2,+}u(p_0,t_0)=\{(\phi_t(p_0,t_0),\nabla_0 \phi(p_0,t_0), (D^2\phi(p_0,t_0))^\star): \phi \in \mathcal{A}u(p_0,t_0)\}$ and $P^{2,-}u(p_0,t_0)=\{(\phi_t(p_0,t_0),\nabla_0 \phi(p_0,t_0), (D^2\phi(p_0,t_0))^\star): \phi \in \mathcal{B}u(p_0,t_0)\}.$ \end{lemma} We may now relate the traditional Euclidean parabolic jets found in \cite{CIL:UGTVS} to the Grushin subparabolic jets via the following lemma. \begin{lemma} \label{jets} Let the coordinates of the points $p,p_0 \in \mathbb{R}^n$ be $p=(x_1,x_2,\dots,x_n)$ and $p_0=(x_1^0,x_2^0,\dots, x_n^0)$. Let $P_{\rm eucl}^{2,+}u(p_0,t_0)$ be the traditional Euclidean parabolic superjet of $u$ at the point $(p_0,t_0)$ and let $(a,\eta, X) \in \mathbb{R}\times \mathbb{R}^{n} \times S^{n}$ with $\eta=(\eta_1,\eta_2,\dots,\eta_n)$. Then $$(a,\eta, X) \in \overline{P}_{\rm eucl}^{2,+}u(p_0,t_0)$$ gives the element $$(a,\tilde{\eta},\mathcal{X}) \in \overline{P}^{2,+}u(p_0,t_0)$$ where the vector $\tilde{\eta}$ is defined by $$\tilde{\eta} = \sum_{i=1}^n \rho_i(p_0) \eta_i X_i$$ and the symmetric matrix $\mathcal{X}$ is defined by $\mathcal{X}_{ij}=\begin{cases} \rho_i(p_0)\rho_j(p_0)X_{ij}+\frac{1}{2}\frac{\partial \rho_j}{\partial x_i}(p_0)\rho_i(p_0)\eta_j &\text{if } i \leq j \\ \mathcal{X}_{ji} &\text{if } i > j. \end{cases}$ \end{lemma} The proof matches the subelliptic case in Grushin-type spaces as found in \cite{B:GS}. We then use these jets to define subsolutions and supersolutions to Equation \eqref{main}. \begin{definition} \rm Let $(p_0,t_0)\in \mathcal{O}_T$ be as above. The upper semicontinuous function $u$ is a \emph{viscosity subsolution} in $\mathcal{O}_T$ if for all $(p_0,t_0) \in \mathcal{O}_T$ we have $(a,\eta,X) \in P^{2,+}u(p_0,t_0)$ produces $$\label{sub} a+F(t_0,p_0,u(p_0,t_0),\eta,X)\leq 0.$$ A lower semicontinuous function $u$ is a \emph{viscosity supersolution} in $\mathcal{O}_T$ if for all $(p_0,t_0) \in \mathcal{O}_T$ we have $(b,\nu,Y) \in P^{2,-}u(p_0,t_0)$ produces $$\label{super} b+F(t_0,p_0,u(p_0,t_0),\nu,Y)\geq 0.$$ A continuous function $u$ is a \emph{viscosity solution} in $\mathcal{O}_T$ if it is both a viscosity subsolution and viscosity supersolution. \end{definition} We observe that the continuity of the function $F$ allows Equations \eqref{sub} and \eqref{super} to hold when $(a,\eta,X) \in \overline{P}^{2,+}u(p_0,t_0)$ and $(b,\nu,Y) \in \overline{P}^{2,-}u(p_0,t_0)$, respectively. We also wish to define what \cite{Ju:P} refers to as parabolic viscosity solutions. We first need to consider the sets $$\mathcal{A}^-u(p_0,t_0)=\{\phi \in C^2_{\rm sub}(\mathcal{O}_T): u(p,t)-\phi(p,t) \leq u(p_0,t_0)-\phi(p_0,t_0)=0 \text{ for } t < t_0\}$$ consisting of all functions that touch from above only when $t0$, there are elements $(a,\tau \Upsilon_{p_{\tau}},\mathcal{X}^{\tau}) \in \overline{P}^{2,+}u(p_\tau,t_\tau)$ and $(a,\tau \Upsilon{q_{\tau}},\mathcal{Y}^{\tau}) \in \overline{P}^{2,-}v(q_\tau,t_\tau)$ where \begin{gather*} (\Upsilon_{p_\tau})_i \equiv \rho_i(p_\tau)\frac{\partial \varphi(p_\tau, q_\tau)}{\partial x_i} = \rho_i(p_\tau)(x^{\tau}_i-y^{\tau}_i)^{2^i-1}, \\ (\Upsilon_{q_\tau})_i \equiv -\rho_i(q_\tau)\frac{\partial \varphi(p_\tau, q_\tau)}{\partial y_i} = \rho_i(q_\tau)(x^{\tau}_i-y^{\tau}_i)^{2^i-1} \end{gather*} so that if $$\lim_{\tau \to \infty}\tau\varphi(p_\tau,q_\tau)=0,$$ then we have \begin{gather}\label {vectordiff} |\,\|\Upsilon_{q_\tau}\|^2- \|\Upsilon_{p_\tau}\|^2\,| = O(\varphi(p_\tau,q_\tau)^2),\\ \label{matrixest} \mathcal{X}^{\tau}\leq \mathcal{Y}^{\tau} +\mathcal{R}^{\tau} \textmd{\; where\ }\lim_{\tau\to \infty}\mathcal{R}^{\tau}=0. \end{gather} We note that Equation \eqref{matrixest} uses the usual ordering of symmetric matrices. \end{theorem} \begin{proof} We first need to check that condition 8.5 of \cite{CIL:UGTVS} is satisfied, namely that there exists an $r>0$ so that for each $M$, there exists a $C$ so that $b \leq C$ when $(b,\eta,X) \in P_{\rm eucl}^{2,+}u(p,t), |p-p_\tau|+|t-t_\tau|0$, we have an $M$ so that for all $C$, $b>C$ when $(b,\eta,X) \in P_{\rm eucl}^{2,+}u(p,t)$. By Lemma \ref{jets} we would have $$(b,\tilde{\eta},\mathcal{X})\in P^{2,+}u(p,t)$$ contradicting the fact that $u$ is a subsolution. A similar conclusion is reached for $-v$ and so we conclude that this condition holds. We may then apply Theorem 8.3 of \cite{CIL:UGTVS} and obtain, by our choice of $\varphi$, \begin{gather*} (a,\tau D_p\varphi(p_\tau,q_\tau),X^\tau) \in \overline{P}^{2,+}_{\rm eucl}u(p_\tau,t_\tau), \\ (a,-\tau D_q\varphi(p_\tau,q_\tau),Y^\tau) \in \overline{P}^{2,-}_{\rm eucl}v(q_\tau,t_\tau). \end{gather*} Using Lemma \ref{jets} we define the vectors $\Upsilon_{p_{\tau}}(p_\tau,q_\tau)$ and $\Upsilon_{q_{\tau}}(p_\tau,q_\tau)$ by \begin{gather*} \Upsilon_{p_{\tau}}(p_\tau,q_\tau) = \widetilde{D_p\varphi}(p_\tau,q_\tau),\\ \Upsilon_{q_{\tau}}(p_\tau,q_\tau) = -\widetilde{D_q\varphi}(p_\tau,q_\tau) \end{gather*} and we also define the matrices $\mathcal{X}$ and $\mathcal{Y}$ as in Lemma \ref{jets}. Then by Lemma \ref{jets}, \begin{gather*} (a,\tau \Upsilon_{p_{\tau}}(p_\tau,q_\tau),\mathcal{X}^\tau) \in \overline{P}^{2,+}u(p_\tau,t_\tau), \\ (a,\tau \Upsilon_{q_{\tau}}(p_\tau,q_\tau),\mathcal{Y}^\tau) \in \overline{P}^{2,-}v(q_\tau,t_\tau). \end{gather*} Equations \eqref{vectordiff} and \eqref{matrixest} are in \cite[Lemma 4.2]{B:GS}. \end{proof} Using this theorem, we now define a class of parabolic equations to which we shall prove a comparison principle. \begin{definition} \rm We say the continuous, proper function $$F:[0,T]\times \overline{\Omega}\times \mathbb{R}\times g_n \times S^{n} \to \mathbb{R}$$ is \emph{admissible} if for each $t \in [0,T]$, there is the same function $\omega:[0,\infty] \to [0,\infty]$ with $\omega(0+)=0$ so that $F$ satisfies $$\label{cond} F(t,q,r,\nu,\mathcal{Y})-F(t,p,r,\eta,\mathcal{X}) \leq\omega\big(d_C(p,q)+\big|\;\|\nu\|^2-\|\eta\|^2\big|+\|\mathcal{Y}-\mathcal{X}\|\big).$$ \end{definition} We now formulate the comparison principle for the following problem. \begin{gather} %\label{problem} u_t+F(t,p,u,\nabla_0 u, (D^2u)^\star) = 0 \quad \textmd{in } (0,T)\times \Omega \label{E}\\ u(p,t)=h(p,t) \quad p \in \partial \Omega,\ t \in [0,T) \label{BC}\\ u(p,0) = \psi(p) \quad p \in \overline{\Omega} \label{IC} \end{gather} Here, $\psi \in C(\overline{\Omega})$ and $h \in C(\overline{\Omega} \times [0,T))$. We also adopt the convention in \cite{CIL:UGTVS} that a subsolution $u(p,t)$ to Problem \eqref{E}--\eqref{IC} is a viscosity subsolution to \eqref{E}, $u(p,t) \leq h(p,t)$ on $\partial \Omega$ with $0 \leq t < T$ and $u(p,0) \leq \psi(p)$ on $\overline{\Omega}$. Supersolutions and solutions are defined in an analogous matter. \begin{theorem}\label{comp} Let $\Omega$ be a bounded domain in $G_n$. Let $F$ be admissible. If $u$ is a viscosity subsolution and $v$ a viscosity supersolution to Problem \eqref{E}--\eqref{IC} then $u \leq v$ on $[0,T) \times \Omega$. \end{theorem} \begin{proof} Our proof follows that of \cite[Thm. 8.2]{CIL:UGTVS} and so we discuss only the main parts. For $\epsilon > 0$, we substitute $\tilde{u}=u-\frac{\varepsilon}{T-t}$ for $u$ and prove the theorem for \begin{gather*} u_t+F(t,p,u,\nabla_0 u,(D^2u)^\star) \leq -\frac{\varepsilon}{T^2} < 0, \\ \lim_{t \uparrow T}u(p,t) = -\infty \quad \text{uniformly on } \overline{\Omega} \end{gather*} and take limits to obtain the desired result. Assume the maximum occurs at $(p_0,t_0)\in \Omega \times (0,T)$ with $$u(p_0,t_0)-v(p_0,t_0)= \delta >0.$$ Let $$M_\tau=u(p_\tau,t_\tau)-v(q_\tau,t_\tau)-\tau\varphi(p_\tau,q_\tau)$$ with $(p_\tau,q_\tau,t_\tau)$ the maximum point in $\overline{\Omega} \times \overline{\Omega} \times [0,T)$ of $u(p,t)-v(q,t)-\tau \varphi(p,q)$. Using the same proof as \cite[Lemma 5.2 ]{B:HG} we conclude that $$\lim_{\tau\to \infty}\tau\varphi(p_\tau,q_\tau) =0.$$ If $t_\tau=0$, we have $$0 < \delta \leq M_\tau \leq \sup_{\overline{\Omega}\times\overline{\Omega}}(\psi(p)-\psi(q) -\tau \varphi(p,q))$$ leading to a contradiction for large $\tau$. We therefore conclude $t_\tau >0$ for large $\tau$. Since $u \leq v$ on $\partial \Omega \times [0,T)$ by Equation \eqref{BC}, we conclude that for large $\tau$, we have $(p_\tau,q_\tau,t_\tau)$ is an interior point. That is, $(p_\tau,q_\tau,t_\tau) \in \Omega \times \Omega \times (0,T)$. Using Lemma \ref{jets}, we obtain \begin{gather*} (a,\tau \Upsilon_{p_{\tau}}(p_\tau,q_\tau), \mathcal{X}^\tau) \in \overline{P}^{2,+}u(p_\tau,t_\tau), \\ (a,\tau \Upsilon_{q_{\tau}}(p_\tau,q_\tau), \mathcal{Y}^\tau) \in \overline{P}^{2,-}v(q_\tau,t_\tau) \end{gather*} satisfying the equations \begin{gather*} a+F(t_\tau,p_\tau,u(p_\tau,t_\tau),\tau\Upsilon(p_\tau,q_\tau), \mathcal{X}^\tau) \leq -\frac{\varepsilon}{T^2}, \\ a+F(t_\tau,q_\tau,v(q_\tau,t_\tau),\tau\Upsilon(p_\tau,q_\tau), \mathcal{Y}^\tau) \geq 0. \end{gather*} Using the fact that $F$ is proper, the fact that $u(p_\tau,t_\tau)\geq v(q_\tau,t_\tau)$ (otherwise $M_\tau < 0$), and Equations \eqref{vectordiff} and \eqref{matrixest}, we have \begin{align*} 0 <\frac{\varepsilon}{T^2} & \leq F(t_\tau,q_\tau,v(q_\tau,t_\tau),\tau\Upsilon_{q_{\tau}} (p_\tau,q_\tau), \mathcal{Y}^\tau)\\ &\quad -F(t_\tau,p_\tau,u(p_\tau,t_\tau),\tau\Upsilon_{p_{\tau}}(p_\tau,q_\tau), \mathcal{X}^\tau)\\ & \leq \omega(d_C(p_\tau,q_\tau)+\tau |\; \|\Upsilon_q(p,q)\|^2 -\|\Upsilon_p(p,q)\|^2 |+\|\mathcal{Y}^\tau-\mathcal{X}^\tau\|)\\ & = \omega (d_C(p_\tau,q_\tau)+C\tau\varphi(p_\tau,q_\tau) +\|\mathcal{R}_\tau\|). \end{align*} We arrive at a contradiction as $\tau \to \infty$. \end{proof} We then have the following corollary, showing the equivalence of parabolic viscosity solutions and viscosity solutions. \begin{corollary} For admissible $F$, we have the parabolic viscosity solutions are exactly the viscosity solutions. \end{corollary} \begin{proof} We showed above that parabolic viscosity sub(super-)solutions are viscosity sub(super-)solutions. To prove the converse, we will follow the proof of the subsolution case found in \cite{Ju:P}, highlighting the main details. Assume that $u$ is not a parabolic viscosity subsolution. Let $\phi \in \mathcal{A}^-u(p_0,t_0)$ have the property that $$\phi_t(p_0,t_0)+F(t_0,p_0,\phi(p_0,t_0),\nabla_0 \phi(p_0,t_0), (D^2\phi(p_0,t_0))^\star) \geq \epsilon > 0$$ for a small parameter $\epsilon$. Let $r > 0$ be sufficiently small so that the gauge $\mathcal{N}(p_0,p)$ is comparable to the distance $d_C(p_0,p)$. Define the gauge ball $B_{\mathcal{N}(p_0)}(r)$ by  B_{\mathcal{N}(p_0)}(r)=\{p\in G_n:\mathcal{N}(p_0,p) \tilde{\phi}(p_0,t_0)$. Thus, the comparison principle, Theorem \ref{comp}, does not hold. Thus,$u\$ is not a viscosity subsolution. The supersolution case is identical and omitted. \end{proof} \begin{thebibliography}{CIL} \bibitem{BR:SRG} Bella\" {\i}che, Andr\' {e}. \emph{The Tangent Space in Sub-Riemannian Geometry}. 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