You are an expert at summarizing long articles. Proceed to summarize the following text: 
this paper focuses on a random partial differential equation consisting of a parabolic pde with irregular noise in the drift . formulation , existence ( with uniqueness in a certain sense ) and double probabilistic representation are discussed . the equation itself is motivated by _ random irregular media models_.    let @xmath0 , @xmath1 be a continuous function and @xmath2 a generalized random field playing the role of a noise . let @xmath3 , @xmath4 \times{\mathbb r}\rightarrow{\mathbb r}$ ] be continuous . consider the problem @xmath5 \\[-8pt ] \nonumber v(0,x ) & = & u^0 ( x),\end{aligned}\ ] ] where @xmath6 is the derivative in the sense of distributions of a continuous process . among examples of possible @xmath7 , we have in mind not only different possibilities of continuous processes as classical wiener process and ( multi ) fractional brownian motion , but also non - gaussian processes . the derivative in the sense of distributions @xmath2 will be the associated noise . ( [ ei-0 ] ) is a new type of spde , not yet studied in any real depth even when @xmath7 is a classical brownian motion . for the situation where @xmath2 is replaced by a space - time white noise @xmath8 , some relevant work was done by nualart and viens ( see , e.g. , @xcite ) . in this article , time dependence is useful for the corresponding stochastic integration . equation  ( [ ei-0 ] ) is equivalent to the following _ dual _ problem : @xmath9 \\[-8pt ] \nonumber u(t , x ) & = & u^0 ( x).\end{aligned}\ ] ] formally speaking , setting @xmath10 , @xmath11 solves ( [ ei-0 ] ) if and only if @xmath12 solves ( [ ei-0dual ] ) . this is rigorously confirmed in section  [ s9 ] so that at this stage , the choice of whether to work with equation ( [ ei-0 ] ) or ( [ ei-0dual ] ) is arbitrary . we have decided to concentrate on equation ( [ ei-0dual ] ) because it corresponds to the standard form for probabilistic representation . the idea of this paper is to first freeze the realization @xmath13 , to set @xmath14 and then to consider the deterministic cauchy problem associated with  ( [ ei-0dual ] ) , @xmath15 \\[-8pt ] \nonumber u(t , x ) & = & u^0 ( x),\end{aligned}\ ] ] where @xmath16 is the derivative of the continuous function @xmath17 .    since the product of a distribution and a continuous function is not defined in the theory of schwarz distributions , we must develop some substitution tools . ideally , we would like to represent the parabolic pde probabilistically through a diffusion which is the solution of the stochastic differential equation ( sde ) @xmath18 with generalized drift . we will give a meaning to ( [ ei-2 ] ) at three different levels :    * the level of a martingale problem ; * the level of a stochastic differential equation in the sense of probability laws ; * the level of a stochastic differential equation in the strong sense . for each of these levels , we shall provide conditions for equation ( [ ei-2 ] ) , with given initial data , to be well posed . later , the notion of a @xmath19-solution to the generalized parabolic pde ( [ ei-0det ] ) will be defined ; related to this , existence , uniqueness and probabilistic representation will be shown .    when @xmath7 is a strong finite cubic variation process and @xmath20 , the solutions to  ( [ ei-0det ] ) obtained for @xmath21 provide solutions to the spde ( [ ei-0 ] ) . this is shown in the last part of the paper . a typical example of a strong zero cubic variation process is the fractional brownian motion with hurst index @xmath22 . equation ( [ ei-0det ] ) will be understood in some _ weak distributional _ sense that we can formally reconstruct as follows . we freeze @xmath21 as a realization and formally integrate equation ( [ ei-0 ] ) from @xmath23 to @xmath24 in time against a smooth test function @xmath25 with compact support in space . the result is @xmath26 the integral @xmath27 needs interpretation since @xmath17 is not generally of bounded variation and it involves the product of the distribution @xmath16 and the function @xmath28 ; in general , this function is , unfortunately , only continuous . as expected this operation is deterministically undefined , unless one uses a generalized functions theory . however , since @xmath17 is a frozen realization of a stochastic process @xmath7 , we can hope to justify the integral in a stochastic sense . note that it can not be of it type , even if @xmath7 were a semimartingale , since @xmath29 is not necessarily adapted to some corresponding filtration . we will , in fact , interpret the stochastic integral element @xmath30 or @xmath31 as a symmetric ( stratonovich ) integral @xmath32 of regularization type ; see section  [ s3 ] . [ dei-1rig ] a continuous random field @xmath33 , x \in{\mathbb r } ) $ ] , a.s . in @xmath340,t [ \times{\mathbb r})$ ] , is said to be a ( _ weak _ ) _ solution _ to the spde ( [ ei-0 ] ) if @xmath35 for every smooth function with compact support @xmath25 . if we integrate equation ( [ ei-0dual ] ) from @xmath24 to @xmath36 in time against a smooth test function  @xmath25 with compact support in space , we are naturally led to the following . [ dei-1dual ] a continuous random field @xmath37 , x \in{\mathbb r } ) $ ] , a.s . in @xmath340,t [ \times{\mathbb r})$ ] , is said to be a ( _ weak _ ) _ solution _ to the spde ( [ ei-0dual ] ) if @xmath38 for every smooth function with compact support @xmath25 . we will show that the _ probabilistic _ solutions that we construct through stochastic equation ( [ ei-2 ] ) will , in fact , solve ( [ ei-1 ] ) . diffusions in the generalized sense were studied by several authors beginning with ( at least to our knowledge ) @xcite . later , many authors considered special cases of stochastic differential equations with generalized coefficients . it is difficult to quote them all . in particular , we refer to the case when @xmath17 is a measure @xcite . in all of these cases , solutions were semimartingales . more recently , @xcite considered special cases of nonsemimartingales solving stochastic differential equations with generalized drift ; those cases include examples coming from bessel processes . @xcite and @xcite treated well - posedness of the martingale problem , it s formula under weak conditions , semimartingale characterization and the lyons  zheng decomposition . the only assumption was the strict positivity of @xmath39 and the existence of the function @xmath40 with appropriate regularizations . bass and chen  @xcite were also interested in ( [ ei-2 ] ) and provided a well - stated framework when @xmath39 is @xmath41-hlder continuous and @xmath17 is @xmath42-hlder continuous , @xmath43 .    beside the martingale problem , in the present paper , we shall emphasize the formulation of ( [ ei-2 ] ) as a stochastic differential equation which can be solved by introducing more assumptions on the coefficients . several examples are provided for the case of weak and strong solutions of ( [ ei-2 ] ) . the paper is organized as follows . section  [ s2 ] is devoted to basic preliminaries , including definitions and properties related to young integrals . section  [ s3 ] is devoted to some useful remainder in stochastic calculus via regularization . in section  [ s4 ] , we introduce the formal _ elliptic _ operator @xmath44 and recall the concept of a @xmath45-generalized solution of @xmath46 for continuous real functions @xmath47 . we further introduce a fundamental hypothesis on @xmath44 for the sequel , called technical assumption  , and we illustrate several examples where it is verified . in section  [ s5 ] , we discuss different notions of martingale problems . section  [ s6 ] provides notions of solutions to _ stochastic differential equations with distributional drift _ and their connections with martingale problems . the notion of solution is coupled with a property of _ extended local time regularity_. this concept of solution is new , even when the drift is an ordinary function . section  [ s7 ] presents the notion of a @xmath19-solution for a parabolic equation @xmath48 , where @xmath49 is bounded and continuous with @xmath50 . we also provide existence , uniqueness and probabilistic representations of @xmath19-solutions to @xmath51 . section  [ s8 ] discusses _ mild _ solutions to the previous parabolic pde and useful integrability properties for its solutions . in section  [ s9 ] , we finally show that the @xmath19-solutions provide , in fact , true _ weak _ solutions to the spde ( [ ei-0 ] ) if @xmath20 . in this paper , @xmath36 will be a fixed horizon time , unless otherwise specified . a function @xmath52 defined on @xmath53 $ ] ( resp . , @xmath54 ) will be extended , without mention , by setting @xmath55 for @xmath56 and @xmath57 for @xmath58 [ resp . , @xmath59 for @xmath60 . @xmath61 will indicate the set of continuous functions defined on @xmath62 @xmath63 the space of real functions with differentiability class @xmath64 . we denote by @xmath65 [ resp . , @xmath66 the space of continuous ( continuous differentiable ) functions vanishing at zero . when there is no confusion , we will also simply use the symbols @xmath67 . we denote by @xmath68 \times{\mathbb r})$ ] the space of real continuous bounded functions defined on @xmath53 \times { \mathbb r}$ ] . @xmath69 , or simply @xmath70 , indicates the space of continuous bounded functions defined on @xmath71 . the vector spaces @xmath72 and @xmath73 are topological frchet spaces , or spaces , according to the terminology of @xcite , chapter  1.2 . they are equipped with the following natural topology . a sequence @xmath74 belonging to @xmath75 [ resp . , @xmath73 ] is said to converge to @xmath52 in the @xmath77 [ resp . , @xmath78 sense if @xmath74 ( resp . , @xmath74 and all derivatives up to order @xmath79 ) converges ( resp . , converge ) to @xmath52 ( resp . , to @xmath52 and all its derivatives ) uniformly on each compact of @xmath71 .    we will consider functions @xmath80 \times{\mathbb r } \rightarrow{\mathbb r}$ ] which are bounded and continuous . a sequence @xmath81 in @xmath82 \times{\mathbb r})$ ] will be said to converge _ in a bounded way _ to @xmath12 if :    * @xmath83 \times{\mathbb r}$ ] ; * there exists a constant @xmath84 , independent of the sequence , such that @xmath85    if the sequence @xmath81 does not depend on @xmath24 , we similarly define the convergence of @xmath86 to @xmath87 in a bounded way .    given two functions @xmath88 \times{\mathbb r } \rightarrow { \mathbb r}$ ] , the composition notation @xmath89 means @xmath90 .    for positive integers @xmath91 , @xmath92 will indicate functions in the corresponding differentiability class . for instance , @xmath93 will be the space of @xmath94 functions which are @xmath45 on @xmath95 ( i.e. , once continuously differentiable ) and such that @xmath96 exists and is continuous .    @xmath97 will indicate the set of functions @xmath92 such that the partial derivatives of all orders are bounded . if @xmath98 is a real compact interval and @xmath990,1[$ ] , we denote by @xmath100 the vector space of real functions defined on @xmath98 which are hlder with parameter @xmath42 . we denote by @xmath101 , or simply @xmath102 , the space of locally hlder functions , that is , hlder on each real compact interval . suppose @xmath103 $ ] , @xmath104 being two real numbers such that @xmath105 . here , @xmath36 does not necessarily need to be positive . recall that @xmath106 belongs to @xmath107 if @xmath108 clearly , @xmath109 defines a norm on @xmath107 which makes it a banach space . @xmath110 is an f - space if equipped with the topology of convergence related to @xmath100 for each compact interval @xmath98 . a sequence @xmath111 in @xmath112 converges to @xmath52 if it converges according to @xmath107 for every compact interval @xmath98 .    we will also provide some reminders about the so - called _ young integrals _ ( see  @xcite ) but will remain , however , in a simplified framework , as in @xcite or @xcite . we recall the essential inequality , stated , for instance , in @xcite :    let @xmath113 be such that @xmath114 . if @xmath115 , then @xmath116 for any @xmath117\subset i$ ] and @xmath1180,\gamma+\beta-1[$ ] , where @xmath119 is a constant not depending on @xmath120 . the bilinear map sending @xmath121 to @xmath122 can be continuously extended to @xmath123 with values in @xmath124 . by definition , that object will be called the _ young integral _ of @xmath52 with respect to @xmath125 on @xmath98 . we also denote it @xmath126 .    by additivity , we set , for @xmath127 $ ] , @xmath128    moreover , the bilinear map defined on @xmath129 by @xmath130 extends continuously to @xmath131 onto @xmath77 . again , that object , defined on the whole real line , will be called _ young integral _ of @xmath52 with respect to @xmath125 and will again be denoted by @xmath132 . [ rexvq7 ] inequality ( [ fyoung ] ) remains true for @xmath133 . in particular , @xmath134 belongs to @xmath135 . in fact , @xmath136    through the extension of the bilinear operator sending @xmath121 to @xmath137 , it is possible to get the following chain rule for young integrals . [ pcruley ] let @xmath138 , @xmath103 $ ] . we suppose that @xmath139 , @xmath140 , @xmath141 with @xmath114 , @xmath142 we define @xmath143 . then @xmath144    if @xmath145 , then the result is obvious . we remark that @xmath146 . repeatedly using inequality ( [ fyoung ] ) , one can show that the two linear maps @xmath147 and @xmath148 are continuous from @xmath149 to @xmath150 . this concludes the proof of the proposition .    by a _ mollifier _ , we mean a function @xmath151 ( i.e. , a @xmath152-function such that itself and all its derivatives decrease to zero faster than any power of @xmath153 as @xmath154 ) with @xmath155 . we set @xmath156 . the result below shows that mollifications of a hlder function @xmath52 converge to @xmath52 with respect to the hlder topology . [ pyoungreg ] let @xmath157 be a mollifier and let @xmath158 . we write @xmath159 . then @xmath160 in the @xmath107 topology for any @xmath161 . we need to show that @xmath162 converges to zero . we set @xmath163 . let @xmath164 . we will establish that @xmath165 without loss of generality , we can suppose that @xmath166 . we distinguish between two cases . _ case _ @xmath167 . we have @xmath168    _ case _ @xmath169 .    in this case , we have @xmath170 ) is verified with @xmath171 . this implies that @xmath172 which allows us to conclude .    for convenience , we introduce the topological vector space defined by @xmath173 it is also a _ vector algebra _ , that is , @xmath174 is a vector space and an algebra with respect to the _ sum and product of functions_.    the next corollary is a consequence of the definition of the young integral and remark  [ rexvq7 ] . [ cyoung ] let @xmath175 , @xmath176 with @xmath177 . then @xmath178 is well defined and belongs to @xmath179 . @xmath174 is not a metric space , but an inductive limit of the f - spaces @xmath102 ; the weak version of the banach  steinhaus theorem for f - spaces can be adapted . in fact , a direct consequence of the banach  steinhaus theorem of @xcite , section  2.1 , is the following . [ tbs ] let @xmath180 be an inductive limit of f - spaces @xmath181 and @xmath182 another f - space . let @xmath183 be a sequence of continuous linear operators . suppose that @xmath184 exists for any @xmath185 . then is again a continuous ( linear ) operator . we recall here a few notions related to stochastic calculus via regularization , a theory which began with  @xcite . we refer to a recent survey paper @xcite . the stochastic processes considered may be defined on @xmath53 , { \mathbb r}_+$ ] or @xmath71 . let @xmath186 be a continuous process and @xmath187 be a process with paths in @xmath188 . for the paths of process @xmath189 with parameter on @xmath53 $ ] ( resp . , @xmath54 ) , we apply the same convention as was applied at the beginning of previous section for functions . so we extend them without further mention , setting @xmath190 for @xmath191 and @xmath192 for @xmath58 ( resp . , @xmath190 for @xmath191 ) . @xmath193 will denote the vector algebra of continuous processes . it is an f - space if equipped with the topology of u.c.p . ( uniform convergence in probability ) convergence .    in the sequel , we recall the most useful rules of calculus ; see , for instance , @xcite or @xcite . the forward symmetric integrals and the covariation process are defined by the following limits in the u.c.p . sense , whenever they exist : @xmath194_t & : = & \lim_{\varepsilon\to0 + } c^{\varepsilon}(x , y)_t,\end{aligned}\ ] ] where @xmath195 all stochastic integrals and covariation processes will of course be elements of  @xmath196 . if @xmath197 $ ] , @xmath198 $ ] and @xmath199 $ ] exist , we say that @xmath200 _ has all of its mutual covariations_.    [ r1.0 ] if @xmath201 is ( locally ) of bounded variation , we have :    * @xmath202 , where the third integral is meant in the lebesgue stieltjes sense ; * @xmath203 \equiv0 $ ] . [ r1.1 ] ( a ) @xmath204 $ ] provided that two of the three integrals or covariations exist . @xmath205_t $ ] provided that two of the three integrals or covariations exist . @xmath206 provided that one of the two integrals exists . [ r1.2 ] ( a ) if @xmath198 $ ] exists , then it is always an increasing process and @xmath201 is called a _ finite quadratic variation process_. if @xmath198 = 0 $ ] , then @xmath201 is said to be a _ zero quadratic variation process_.    let @xmath201 , @xmath189 be continuous processes such that @xmath200 has all of its mutual covariations . then @xmath197 $ ] has locally bounded variation . if @xmath207 , then @xmath208_t = \int_0^t f'(x)g'(y)\,d[x , y].\ ] ]    if @xmath209 is a zero quadratic variation process and @xmath201 is a finite quadratic variation process , then @xmath210 \equiv0 $ ] . a bounded variation process is a zero quadratic variation process . ( _ classical it formula_. ) if @xmath211 , then @xmath212 exists and is equal to @xmath213.\ ] ]    if @xmath214 and @xmath215 , then the forward integral @xmath216 is well defined .    in this paper , all filtrations are supposed to fulfill the usual conditions . if @xmath217}$ ] is a filtration , @xmath201 an @xmath218-semimartingale and @xmath189 is an @xmath218-adapted cadlag process , then @xmath219 is the usual it integral . if @xmath189 is an @xmath218-semimartingale , then @xmath220 is the classical fisk  stratonovich integral and @xmath197 $ ] is the usual covariation process @xmath221 . we now introduce the notion of dirichlet process , which was essentially introduced by fllmer @xcite and has been considered by many authors ; see , for instance , @xcite for classical properties .    in the present section , @xmath222 will denote a classical @xmath223-brownian motion . [ d81 ] an @xmath224-adapted ( continuous ) process is said to be a _ dirichlet process _ if it is the sum of an local martingale @xmath225 and a zero quadratic variation process @xmath209 . for simplicity , we will suppose that @xmath226 a.s . [ r81 ] ( i ) process @xmath227 in the previous decomposition is anadapted process . an @xmath224-semimartingale is an @xmath223-dirichlet process . the decomposition @xmath228 is unique . let @xmath229 be of class  @xmath45 and let @xmath201 be an @xmath224-dirichlet process . then @xmath230 is again an @xmath231-dirichlet process with local martingale part @xmath232 .    the class of semimartingales with respect to a given filtration is known to be stable with respect to @xmath233 transformations . remark  [ r1.2](b ) says that finite quadratic variation processes are stable through @xmath45 transformations . the last point of the previous remark states that @xmath45 stability also holds for dirichlet processes . young integrals introduced in section  [ s2 ] can be connected with the forward and symmetric integrals via the regularization appearing before remark  [ r1.0 ] . the next proposition was proven in  @xcite . let @xmath234 be processes whose paths are respectively in @xmath235 and @xmath236 , with @xmath237 , @xmath238 and @xmath239 . for any symbol @xmath240 , the integral @xmath241 coincides with the young integral @xmath242 . [ rexvq7a ] suppose that @xmath201 and @xmath189 satisfy the conditions of proposition  [ pexvq7 ] . then remark  [ r1.1](a ) implies that @xmath197=0 $ ] . we need an extension of stochastic calculus via regularization in the direction of higher @xmath243-variation . the properties concerning variation higher than @xmath244 can be found , for instance , in @xcite . we set @xmath245^\varepsilon_t = \frac { 1}{\varepsilon } \int_0^t ( x_{s+\varepsilon}-x_s)^3\ , ds.\ ] ] we also define @xmath246^\varepsilon \|_t = \frac{1}{\varepsilon } \int_0^t    @xmath247^\varepsilon_t$ ] when @xmath248 exists for any @xmath24 , we denote it by @xmath247_t$ ] . if the limiting process @xmath247 $ ] has a continuous version , we say that @xmath201 is a _ finite cubic variation process_.    if , moreover , there is a positive sequence @xmath249 converging to zero such that @xmath250^{\varepsilon_n }   \|_t < + \infty,\ ] ] then we say that @xmath201 is a ( _ strong _ ) _ finite cubic variation process_. if @xmath201 is a ( strong ) finite cubic variation process such that @xmath247 = 0 $ ] , then @xmath201 will be said to be a ( strong ) zero _ finite cubic variation process_.    for instance , if @xmath251 , a fractional brownian motion with hurst index @xmath252 , then @xmath201 is a finite quadratic variation process if and only if @xmath253 ; see @xcite . it is a strong zero cubic variation process if and only if @xmath254 ; see @xcite . on the other hand , @xmath255  is a zero cubic variation process if and only if @xmath256 ; see @xcite . it is clear that a finite quadratic variation process is a strong zero cubic variation process . on the other hand , processes whose paths are h " older continuous with parameter greater than @xmath257 are strong zero cubic variation processes .    as for finite quadratic variation and dirichlet processes , the @xmath45-stability also holds for finite cubic variation processes . the next proposition is a particular case of a result contained in @xcite . [ pstacub ] let @xmath201 be a strong finite cubic variation process , @xmath258 a locally bounded variation process and @xmath259 of class @xmath260 . then @xmath261 is again a strong finite cubic variation process and @xmath262_t = \int_0^t \partial_x f(v_s , x_s)^3\ , d[x , x , x]_s.\ ] ]    moreover , an it chain rule property holds , as follows .    [ crule ] let @xmath201 be a strong finite cubic variation process , @xmath263 a bounded variation process and @xmath189 a cadlag process . let @xmath259 be of class  @xmath264 . then @xmath265_s.\end{aligned}\ ] ]    we deduce , in particular , that a @xmath45 transformation of a strong zero cubic variation process is again a strong zero cubic variation process . we conclude the section by introducing a concept of _ definite integral _ via regularization . if processes @xmath266 are indexed by the whole real line , a.s . with compact support , we define @xmath267 where the limit is understood in probability . integration by parts , proposition  [ pexvq7 ] and the chain rule property ( proposition  [ crule ] ) can all be immediately adapted to these definite integrals . let @xmath39 , @xmath268 be such that @xmath269 . without loss of generality , we will suppose that @xmath270 . we consider a formal pde operator of the following type : @xmath271 if @xmath17 is of class @xmath45 , so that @xmath16 is continuous , we will say that @xmath44 is a _ classical _ pde operator .    for a given mollifier @xmath157 , we denote @xmath272 we then consider @xmath273 \\[-8pt ] \nonumber { \mathcal   l}_n u & = & \partial_t u + l_n u\qquad\mbox{for } u \in c^{1,2}([0,t[\times{\mathbb r}),\end{aligned}\ ] ] where @xmath274 acts on @xmath275 . a priori , @xmath276 , @xmath277 and the operator @xmath274 depend on the mollifier  @xmath157 . previous definitions are slightly different from those in papers @xcite , but a considerable part of the analysis of @xmath44 and the study of the martingale problem can be adapted . in those papers , there was only regularization but no truncation ; here , truncation is used to study the associated parabolic equations . [ dc1gen ] a function @xmath278 is said to be a _ @xmath45-generalized solution _ to @xmath279 where @xmath280 if for any mollifier @xmath157 , there are sequences @xmath111 in @xmath233 and @xmath281 in @xmath282 such that @xmath283    [ t24 ] there is a solution @xmath284 to @xmath285 such that @xmath286 for every @xmath287 if and only if @xmath288 exists in @xmath282 , independently of the mollifier . moreover , in this case , any solution @xmath52 to @xmath289 fulfills @xmath290    this result follows in a very similar way to the proof of proposition  2.3 in  @xcite  first at the level of regularization and then passing to the limit .    for the remainder of this paper , we will suppose the existence of this function  @xmath291 . we will consider @xmath292 such that @xmath293 in particular , @xmath294 holds . even though we discuss the general case with related nonexplosion conditions in @xcite , here , in order to ensure conservativeness , we suppose that @xmath295 \\[-8pt ] \nonumber \int_{- \infty}^0 \frac{e^ { \sigma(x)}}{\sigma^2}\ , dx & = & \int^ { \infty}_0 \frac{e^ { \sigma(x)}}{\sigma^2}\ , dx = + \infty.\end{aligned}\ ] ] previous assumptions are of course satisfied if @xmath39 is lower bounded by a positive constant and @xmath17 is constant outside a compact interval .    condition ( [ nonexplo ] ) implies that the image set of @xmath296 is @xmath71 . [ r2.5 ] proposition  [ t24 ] implies uniqueness of the problem @xmath297 for every @xmath298 , @xmath299 . [ r2.6 ] we present four important examples where @xmath291 exists :    if @xmath300 for some @xmath3010,1]$ ] , then @xmath302 and @xmath303 if @xmath304 , the operator @xmath44 can be formally expressed in divergence form as @xmath305    suppose that @xmath17 is locally of bounded variation . we then get @xmath306 since @xmath307 in the weak-@xmath308 topology and @xmath309 is continuous . if @xmath39 has bounded variation , then we have @xmath310 in particular , this example contains the case where @xmath20 for any  @xmath17 . suppose that @xmath39 is locally hlder continuous with parameter @xmath42 and that @xmath17 is locally hlder continuous with parameter @xmath311 such that @xmath312 . since @xmath39 is locally bounded , @xmath313 is also locally h " older continuous with parameter @xmath42 . proposition  [ pyoungreg ] implies that @xmath314 in @xmath315 and @xmath316 in @xmath317 for every @xmath318 and @xmath319 . since @xmath39 is strictly positive on each compact , @xmath320 in @xmath315 . by remark  [ rexvq7 ] , @xmath291 is well defined and locally h " older continuous with parameter @xmath321 . again , the following lemma can be proven at the level of regularizations ; see also lemma 2.6 in @xcite . [ t27 ] the unique solution to problem ( [ e2.9 ] ) is given by @xmath322    [ r2.3 ] if @xmath323 and @xmath324 is a classical solution to @xmath325 , then @xmath52 is clearly also a @xmath45-generalized solution . [ r2.9bis ] given @xmath326 , we denote by @xmath327 the unique @xmath45-generalized solution @xmath52 to problem ( [ e2.9 ] ) with @xmath328 , @xmath329 . the unique solution to the general problem ( [ e2.9 ] ) is given by @xmath330 we write @xmath331 , that is , the solution with @xmath332    [ r2.9 ] let @xmath333 . there is at most one @xmath334 such that @xmath325 . in fact , to see this , it is enough to suppose that @xmath335 . lemma [ t27 ] implies that @xmath336 consequently , @xmath337 is forced to be zero . this consideration allows us to define without ambiguity @xmath338 @xmath339 , where @xmath340 is the set of all @xmath278 which are @xmath45-generalized solution to @xmath341 for some @xmath280 . in particular , @xmath342 . a direct consequence of lemma [ t27 ] is the following useful result . [ l28bis ] @xmath343 is the set of @xmath333 such that there exists @xmath344 with @xmath345    in particular , it gives us the following density proposition . [ t29 ] @xmath340 is dense in @xmath45 . it is enough to show that every @xmath233-function is the @xmath45-limit of a sequence of functions in @xmath340 . let @xmath346 be a sequence in @xmath45 converging to @xmath347 in  @xmath282 . it follows that @xmath348 converges to @xmath333 and @xmath349 . we must now discuss technical aspects of the way @xmath44 and its domain @xmath340 are transformed by @xmath296 . we recall that @xmath285 and that @xmath350 is strictly positive . condition  ( [ nonexplo ] ) implies that the image set of @xmath296 is @xmath71 . let @xmath351 be the classical pde operator @xmath352 where @xmath353 @xmath351 is a classical pde map ; however , we can also consider it at the formal level and introduce @xmath354 . [ t212 ] @xmath355 , @xmath356 . @xmath357 . @xmath358 holds if and only if @xmath359 . moreover , we have @xmath360 for every @xmath361 . this follows similarly as for proposition  2.13 of @xcite . we will now discuss another operator related to @xmath44 . given a function @xmath52 , we need to provide a suitable definition of @xmath362 , that is , some primitive of @xmath363 . * one possibility is to define that map , through previous expression , for @xmath364 . * otherwise , we try to define it as linear map on @xmath233 . for this , first suppose that @xmath16 is continuous . then integrating by parts , we obtain @xmath365 we remark that the right - hand side of this expression makes sense for any @xmath215 and continuous @xmath17 . we will thus define @xmath366 as follows : @xmath367    one may ask if , in the general case , the two definitions @xmath368 on @xmath369 and @xmath370 on @xmath233 are compatible . we will later see that under assumption  [ a0 ] , this will be the case . however , in general , @xmath371 may be empty . thus far , we have learned how to eliminate the first - order term in a formal pde operator through the transformation @xmath296 introduced at ( [ efh ] ) ; when @xmath44 is classical , this was performed by zvonkin ( see @xcite ) . we would now like to introduce a transformation which puts the pde operator in a divergence form . let @xmath44 be a pde operator which is formally of type ( [ e2.1 ] ) : @xmath372    we consider a function of class @xmath45 , namely @xmath373 such that @xmath374 according to assumptions ( [ nonexplo ] ) , @xmath375 is bijective on @xmath71 . if there is no drift term , that is , @xmath376 , then we have @xmath377 . [ t216 ] we consider the formal pde operator given by @xmath378 where @xmath379 then :    @xmath380 if and only if @xmath381 ;    for every @xmath380 , we have @xmath382 . it is practically the same as in lemma 2.16 of @xcite . we now give a lemma whose proof can be easily established by investigation . suppose that @xmath44 is a classical pde operator . then @xmath383 is well defined for @xmath384 functions where @xmath44 acts on the second variable . given a function @xmath385 \times{\mathbb r})$ ] , we will hereafter set @xmath386 \times{\mathbb r}\longrightarrow{\mathbb r}$ ] by @xmath387 . [ l28 ] let us suppose that @xmath388 . we set @xmath389 . we define the pde operator @xmath390 by @xmath391 , where @xmath351 is a classical operator acting on the space variable @xmath275 and @xmath392 if @xmath393 and @xmath394 in the classical sense , then @xmath395 . we will now formulate a supplementary assumption which will be useful when we study singular stochastic differential equations in the proper sense and not only in the form of a martingale problem . [ a0 ] let @xmath396 be a topological f - space which is a linear topological subspace of @xmath72 ( or , eventually , an inductive limit of sub - f - spaces ) . the @xmath396-convergence implies convergence in @xmath282 and , therefore , pointwise convergence . we say that @xmath44 fulfills assumption if the following conditions hold :    @xmath397 , which is dense .    for every @xmath398 , the multiplicative operator @xmath399 maps @xmath396 into itself . let @xmath400 as defined in lemma [ t27 ] , that is , @xmath401 is such that @xmath402 we recall that @xmath401 solves problem @xmath403 with @xmath404 . we suppose that @xmath36 admits a continuous extension to @xmath396 . let @xmath405 . for every @xmath215 with @xmath406 and @xmath407 so that ] , we have @xmath408 and @xmath409 , where @xmath410 denotes the continuous extension of @xmath411 ( see remark  [ r2.9bis ] ) to @xmath396 , which exists by  . the set @xmath412 is dense in @xmath413 . [ r2.21a ] let @xmath405 . remark  [ r2.9bis ] and point ( iii ) above together imply that @xmath414 extends continuously to @xmath396 . moreover , @xmath415    point ( iv ) above shows that @xmath416 and @xmath417 , where @xmath418 ; in fact , @xmath419 , @xmath420 and ( [ e213a ] ) implies that @xmath421 . point ( i ) above is satisfied if , for instance , the map @xmath36 is closable as a map from @xmath282 to @xmath45 . in that case , @xmath396 may be defined as the domain of the closure of @xmath45 , equipped with the graph topology related to @xmath422 . below , we give some sufficient conditions for points ( iv ) and ( v ) of the technical assumption to be satisfied . we define by @xmath423 the vector space of functions @xmath333 such that @xmath424 . this will be an f - space if equipped with the following topology . a sequence @xmath111 will be said to converge to @xmath52 in @xmath423 if @xmath425 and @xmath426 converges to @xmath427 in @xmath396 . in particular , a sequence converging according to @xmath428 also converges with respect to  @xmath45 . on the other hand , @xmath429 and a sequence converging in @xmath430 also converges with respect to @xmath431 . moreover , @xmath233 is dense in @xmath431 because @xmath45 is dense in  @xmath396 . [ l2.21 ] suppose that points to of the technical assumption are fulfilled . we suppose , moreover , that :    @xmath432 .    for every @xmath215 , @xmath433 , @xmath434 , @xmath435 , we have @xmath408 and . @xmath436 is well defined and has a continuous extension to @xmath423 , still denoted by @xmath370 , such that @xmath437 . @xmath438 . @xmath439 is the identity map on @xmath440 for every @xmath405 are injective and points and of the technical assumption are satisfied . the injectivity of @xmath36 follows from point ( e ) . the injectivity of @xmath411 is a consequence of remark  [ r2.9bis ] . we prove point ( iv ) . point ( c ) says that @xmath437 . we set @xmath441 , @xmath211 , where @xmath442 , @xmath443 . clearly , @xmath444 and @xmath445 . point ( b ) implies that @xmath446 . hence , @xmath447 and ( iv ) is satisfied . concerning point ( v ) , let @xmath408 with @xmath448 and set @xmath449 . since @xmath52 belongs to @xmath423 by ( c ) , @xmath427 belongs to @xmath396 . point ( i ) of the technical assumption implies that there exists a sequence @xmath426 of @xmath45 functions converging to @xmath427 in the @xmath396 sense and thus also in @xmath282 . let @xmath111 be the sequence of primitives of @xmath426 ( which are of class  @xmath233 ) such that @xmath450 . in particular , we have that @xmath111 converges to @xmath52 in the @xmath431-sense . by ( c ) , there exists @xmath49 in @xmath396 which is the limit of @xmath451 in the @xmath396-sense . observe that because of  ( b ) , @xmath452 . on the other hand , @xmath453 in @xmath45 . applying @xmath36 and using ( iii ) of the technical assumption , we obtain @xmath454 the injectivity of @xmath36 allows us to conclude that @xmath455 . [ r2.21aa ] under the assumptions of lemma [ l2.21 ] , we have :    * @xmath456 ; * @xmath457 , @xmath458    in fact , let @xmath459 . without loss of generality , we can suppose that @xmath433 . let @xmath460 and set @xmath461 so that @xmath462 . setting @xmath463 lemma [ t27 ] implies that @xmath464 , where @xmath465 . so @xmath466 . since @xmath467 , it follows that @xmath468 , by additivity . on the other hand , @xmath469 by point ( e ) of lemma [ l2.21 ] . [ 4ex ] we provide here a series of four significant examples when technical assumption is verified . we only comment on the points which are not easy to verify . the first example is simple . it concerns the case when the drift @xmath16 is continuous . this problem , to be studied later , corresponds to an ordinary sde where @xmath470    @xmath44 is _ close to divergence type _ , that is , @xmath471 and where @xmath311 is a locally bounded variation function vanishing at zero . the operator is of divergence type with an additional radon measure term , that is , we have @xmath472 . in this case , we have @xmath473 points ( i ) and ( ii ) of the technical assumption are trivial . we have , in fact , @xmath474 @xmath36 defined at point ( iii ) of the technical assumption is such that @xmath475 , where @xmath433 and @xmath476 consequently , the extension of @xmath36 to @xmath477 , always still denoted by the same letter @xmath36 , is given by @xmath401 with @xmath433 and @xmath478   \\[-8pt ]   \nonumber \qquad & & \hspace*{11.2 mm } { } \times \biggl ( \ell(0 ) +   \int_0^x \ell(y ) \exp \biggl(2\int_0^y\frac{d\beta}{\sigma ^2 } \biggr ) \frac{1}{\sigma^2(y ) } \,d\beta(y ) \biggr ) \biggr\}.\end{aligned}\ ] ] points ( iv ) and ( v ) are seen to be satisfied via lemma [ l2.21 ] . we have @xmath479 . point ( a ) is obvious since @xmath480 and so @xmath481 . let @xmath215 . using lebesgue stieltjes calculus , we can easily show that @xmath482 this shows that @xmath483 and therefore the first part of ( b ) . we remark that we can , in fact , consider @xmath484 because @xmath485 the expression of @xmath486 extends continuously to @xmath333 , which yields the first part of point  ( c ) . moreover , inserting the expression for @xmath350 into @xmath427 in ( [ e2.27a ] ) , one shows that @xmath437 . suppose , now , that in expression ( [ e2.27a ] ) , @xmath215 , @xmath487 , @xmath434 . a simple investigation shows that @xmath488 , so the second part of point ( b ) is fulfilled ; point  ( d ) is also clear because of ( [ e2.27 ] ) . finally point ( d ) holds because one can prove by inspection that @xmath439 is the identity on @xmath489 . we recall the notation @xmath490 which indicates the topological vector space of locally hlder continuous functions defined on @xmath71 with parameter @xmath491 . we recall that @xmath490 is a vector algebra . suppose that @xmath492 and @xmath493 ( or @xmath494 and @xmath495 ) . remark  [ r2.6](d ) implies that @xmath291 also belongs to @xmath496 . we set @xmath497 . technical assumption [ a0 ] is verified for the following reasons .    since @xmath498 , @xmath499 belongs to the same space . point ( i ) follows because of proposition  [ pyoungreg ] and point ( ii ) follows because @xmath496 is an algebra . corollary [ cyoung ] yields that for every @xmath500 , the function @xmath501 is well defined and belongs to @xmath496 . this shows that @xmath36 can be continuously extended to @xmath396 and point ( iii ) is established . concerning points ( iv ) and ( v ) , we again use lemma [ l2.21 ] . we observe that @xmath502 point ( a ) is obvious since @xmath503 . let @xmath504 . considering @xmath17 as a deterministic process and recalling the definition of @xmath370 as in ( [ e213a ] ) , integration by parts in remark  [ r1.1](c ) and proposition  [ pexvq7 ] together imply that @xmath505 the first part of point ( b ) follows because of proposition  [ pcruley ] . of course , the previous expression can be extended to @xmath468 and this shows the first part of point  ( c ) . showing that the second part of point ( c ) of lemma [ l2.21 ] holds consists of verifying that @xmath437 . substituting @xmath506 into the previous expression , through proposition  [ pcruley ] , we obtain @xmath507 concerning the second part of point ( b ) , let @xmath215 so that @xmath508 . we want to show that @xmath509 coincides with @xmath52 .    since @xmath510 , it remains to check that @xmath511 . we recall that @xmath512    twice applying the chain rule of proposition  [ pcruley ] and using ( [ e1.1b ] ) , the fact that @xmath513 and integration by parts , we obtain @xmath514 point ( b ) is therefore completely established . point ( d ) follows because in ( [ eimt ] ) , when @xmath408 , it follows that @xmath424 .    clearly , as for the previous example , @xmath438 . it remains to show that @xmath515 is the identity map @xmath516 .    for this , we first remark that @xmath517 in fact , by proposition  [ pexvq7 ] and integration by parts contained in remark  [ r1.1](c ) , we obtain @xmath518 by the chain rule of proposition  [ pcruley ] , we obtain the right - hand side of ( [ eyverif1 ] ) .    at this point , by definition , if @xmath519 , we have @xmath520 therefore , ( [ eyverif1 ] ) and proposition  [ pcruley ] allow us to conclude that @xmath521    suppose @xmath17 is locally with bounded variation . then the technical assumption is satisfied for @xmath522 , where @xmath523 is the space of continuous real functions , locally with bounded variation @xmath11 , equipped with the following topology . a sequence @xmath524 in @xmath523 converges to @xmath11 if @xmath525 the arguments for proving that the technical assumption is satisfied are similar , but easier , than those for the previous point . young - type calculus is replaced by classical lebesgue  stieltjes calculus . in this section , we consider a pde operator satisfying the same properties as in previous section , that is , @xmath526 where @xmath269 and @xmath17 are continuous . in particular , we assume that @xmath527 exists in @xmath282 , independently of the chosen mollifier . then @xmath296 defined by @xmath528 and @xmath529 is a solution to @xmath285 with @xmath530 .    here , we aim to introduce different notions of martingale problem , trying , when possible , to also clarify the classical notion . for the next two definitions , we consider the following convention . let @xmath531 equipped with a filtration @xmath532 fulfill the _ usual conditions _ ; see , for instance , @xcite , definition 2.25 , chapter 1 .    [ dmp ] a process @xmath201 is said to solve _ the martingale problem _ related to @xmath44 ( with respect to the aforementioned filtered probability space ) with initial condition @xmath533 , @xmath534 , if @xmath535 is an @xmath536-local martingale for @xmath537 and @xmath533 .    more generally , for @xmath538 , @xmath539 , we say that @xmath540 solves the martingale problem related to @xmath44 with initial value @xmath275 at time @xmath541 if for every @xmath542 , @xmath543 is an @xmath544-local martingale . we remark that @xmath545 solves the martingale problem at time @xmath541 if and only if @xmath546 solves the martingale problem at time @xmath23 . [ dsmp ] let @xmath222 be an @xmath224-classical wiener process . anprogressively measurable process @xmath547 is said to solve _ the sharp martingale problem _ related to @xmath44 ( on the given filtered probability space ) with initial condition @xmath533 , @xmath534 , if @xmath548 for every @xmath542 .    more generally , for @xmath538 , @xmath539 , we say that @xmath549 solves the sharp martingale problem related to @xmath44 with initial value @xmath275 at time @xmath541 if for every @xmath542 , @xmath550    [ r3.1a ] let @xmath222 be an @xmath224-wiener process . if @xmath16 is continuous , then a process @xmath201 solves the ( corresponding ) sharp martingale problem with respect to @xmath44 if and only if it is a classical solution of the sde @xmath551 for this , a simple application of the classical it formula gives the result . [ r3.2 ] ( i ) in general , @xmath552 does not belong to @xmath340 , otherwise a solution to the martingale problem with respect to @xmath44 would be a semimartingale . according to remark  [ rsem ] , this is generally not the case . in @xcite , we gave necessary and sufficient conditions on @xmath17 so that @xmath201 is a semimartingale . \(ii ) given a solution @xmath201 to the martingale problem related to @xmath44 , we are interested in the operators @xmath553 and @xmath554 where @xmath193 is the vector algebra of continuous processes .    we may ask whether @xmath555 and @xmath209 are closable in @xmath45 and @xmath282 , respectively . we will see that @xmath555 admits a continuous extension to @xmath45 . however , @xmath209 can be extended continuously to some topological vector subspace @xmath396 of @xmath282 , where @xmath396 includes the drift , only when assumption  [ a0 ] is satisfied .    similarly , as in the case of classical stochastic differential equations , it is possible to distinguish two types of existence and uniqueness for the martingale problem . even if we could treat initial conditions which are random @xmath556-measurable solutions , here we will only discuss deterministic ones . we will denote by @xmath557 [ resp . @xmath558 the martingale problem ( resp . sharp martingale problem ) related to @xmath44 with initial condition @xmath559 the notions will only be formulated with respect to the initial condition at time  0 . [ d112 ] we will say that @xmath560 admits _ strong existence _ if the following holds . given any probability space @xmath561 , a filtration @xmath562 and an @xmath563-brownian motion @xmath564 , there is a process @xmath565 which solves the sharp martingale problem with respect to @xmath44 and initial condition @xmath566 . [ d113 ] we will say that @xmath560 admits _ pathwise uniqueness _ if the following property is fulfilled . let @xmath531 be a probability space with filtration @xmath562 andbrownian motion @xmath567 . if two processes @xmath568 are two solutions of the sharp martingale problem with respect to @xmath44 and @xmath566 , such that @xmath569 a.s . , then @xmath201 and @xmath570 coincide . [ d114 ] we will say that @xmath571 _ admits weak existence _ if there is a probability space @xmath531 , a  filtration @xmath572 and a process @xmath565 which is a solution of the corresponding martingale problem . we say that @xmath573 admits weak existence if @xmath571 admits weak existence for every @xmath566 . [ ( _ uniqueness in law_)][d115 ] we say that @xmath571 has a _ unique solution in law _ if the following holds . we consider an arbitrary probability space @xmath574 with a filtration @xmath562 and a solution @xmath201 of the corresponding martingale problem . we also consider another probability space @xmath575 equipped with another filtration @xmath576 and a solution @xmath570 . we suppose that @xmath577 , and @xmath578 , @xmath579-a.s . then @xmath201 and @xmath570 must have the same law as a r.v.s with values in @xmath580 ( or @xmath581 $ ] ) . [ rsclass ] let us suppose @xmath16 to be a continuous function . we do not suppose @xmath39 to be strictly positive ( only continuous ) .    the @xmath560 then admits strong existence and pathwise uniqueness if the corresponding classical sde @xmath582 admits strong existence and pathwise uniqueness . in this case , @xmath583 and to establish this , it is enough to use the classical it formula . it is well known ( see @xcite ) that weak existence ( resp . , uniqueness in law ) of the martingale problem is equivalent to weak existence ( resp . , uniqueness in law ) of the corresponding sde .    for the rest of the section let @xmath584 $ ] , @xmath585 moreover , let @xmath586 be a fixed filtered probability space fulfilling the usual conditions .    the first result concerning solutions to the martingale problem related to @xmath44 is the following . [ p3.3 ] let @xmath587    a process @xmath201 solves the martingale problem related to @xmath44 with initial condition @xmath275 at time @xmath541 if and only if @xmath588 is a local martingale which solves , on the same probability space , @xmath589 where @xmath590 and where @xmath222 is an @xmath224-classical brownian motion . let @xmath222 be an @xmath224-classical brownian motion . if @xmath189 is a solution to equation  ( [ e3.3 ] ) , then @xmath591 is a solution to the sharp martingale problem with respect to @xmath44 with initial condition @xmath275 at time @xmath541 . [ r3.4 ] let @xmath201 be a solution to the martingale problem with respect to @xmath44 and set @xmath592 as in point  ( i ) above . since @xmath189 is a local martingale , we know from remark  [ r81](iv ) that @xmath593 is an @xmath223-dirichlet process with martingale part @xmath594 in particular , @xmath201 is a finite quadratic variation process with @xmath595 = [ m^x , m^x]_t = \int_0^t \sigma^2(x_s)\,ds.\ ] ]    proof of proposition  [ p3.3 ] for simplicity , we will set @xmath596 . first , let @xmath201 be a solution to the martingale problem related to @xmath44 . since @xmath597 and @xmath285 , we know that @xmath588 is an @xmath224-local martingale . in order to calculate its bracket , we recall that @xmath355 and @xmath598 hold by proposition  [ t212](a ) . thus , @xmath599 is an @xmath224-local martingale . this implies that @xmath600_t = \int_0^t ( \sigma h')^2 ( h^{-1}(y_s))\,ds = \int_0^t \tilde\sigma_h^2(y_s)\,ds.\ ] ] finally , @xmath189 is a solution to the sde ( [ e3.3 ] ) with respect to the standard @xmath601-brownian motion @xmath602 given by @xmath603 where @xmath604 is the canonical filtration generated by @xmath189 . now , let @xmath588 be a solution to ( [ e3.3 ] ) and let @xmath537 . proposition  [ t212](c ) says that @xmath605 , where @xmath606 we can therefore apply it s formula to evaluate @xmath607 , which coincides with @xmath230 . this gives @xmath608_s.\ ] ] using @xmath609_s = \tilde\sigma_h^2(y_s)\,ds$ ] and taking into account ( [ e3.4 ] ) , we conclude that @xmath610 this establishes the proposition . [ r3.5 ] from proposition  [ p3.3 ] in particular , we have the following . let @xmath586 be a filtered probability space fulfilling the usual conditions . let @xmath611 and @xmath201 be a solution to the martingale problem related to @xmath44 with initial condition @xmath566 . then there exists a classical brownian motion @xmath222 such that @xmath201 is a solution to the sharp martingale problem related to @xmath44 with initial condition @xmath566 . [ c3.5 ] let @xmath201 be a solution to the martingale problem related to @xmath44 with initial condition @xmath566 . then map @xmath555 admits a continuous extension from @xmath340 to @xmath45 with values in @xmath612 which we will again denote by @xmath555 . moreover , @xmath613 is a zero quadratic variation process for every @xmath614 . @xmath555 has a continuous extension because of ( [ e3.5 ] ) . @xmath615 is a zero quadratic variation process because @xmath201 is a dirichlet process with martingale part @xmath616 and because of remark  [ r81 ] . [ r3.6 ] the extension of ( [ e3.5 ] ) to @xmath45 gives @xmath617 choosing @xmath618 in ( [ e3.6 ] ) , we get @xmath619    we will see that if there is a subspace @xmath396 of @xmath282 such that technical assumption  [ a0 ] is verified , then the operator @xmath209 will be extended to @xmath396 . if @xmath17 is an element of that space , then it will be possible to write @xmath421 and @xmath620 . in that case , we will be able to indicate that @xmath201 is a solution of the generalized sde with diffusion coefficient @xmath39 and distributional drift @xmath16 . a similar result to proposition  [ p3.3 ] can be deduced for the case of a transformation through function @xmath375 and the divergence - type operator introduced at ( [ efk ] ) . [ pmardiv ] we consider the transformation @xmath375 and the pde operator @xmath621 introduced at and in lemma , respectively . a process @xmath201 solves the martingale problem related to @xmath44 with initial condition @xmath566 at time @xmath541 if and only if @xmath622 solves the martingale problem related to @xmath621 with initial condition @xmath623 at time  @xmath541 . this is an easy consequence of lemma [ t216 ] . let @xmath611 , @xmath624 . let @xmath625 be as in section  [ s4 ] . we set @xmath626 .    from proposition  [ p3.3 ] , we have the following .    [ cmp ] strong existence ( resp . , pathwise uniqueness ) holds for @xmath627 if and only if strong existence ( resp . , pathwise uniqueness ) holds for the sde @xmath628 with initial condition @xmath629 . an analogous equivalence holds for weak existence ( resp . , uniqueness in law ) . from proposition  [ p3.3 ] , we can deduce two other corollaries concerning the well - posedness of our martingale problem . [ cwe ] under the same assumptions as the previous corollary , @xmath630 admits weak existence and uniqueness in law . the statement follows from point ( i ) of corollary [ cmp ] and from the fact that the sde ( [ e3.3 ] ) admits weak existence and uniqueness in law because see theorem  5.7 , chapter  5 of  @xcite , or @xcite . [ rsem ] by corollary 5.11 of @xcite , it is immediate to see that the solution is a semimartingale for each initial condition if and only if @xmath291 is locally of bounded variation . if @xmath44 is in divergence form [ see remark  [ r2.6](a ) with @xmath304 ] , then the solution corresponds to the process constructed and studied by , for instance , stroock  @xcite . [ c3.3ss ] suppose that either @xmath632 or @xmath633 and , moreover , that is satisfied . then @xmath630 admits strong existence and pathwise uniqueness .    in this case , @xmath291 is well defined [ see remark  [ r2.6](d ) ] and @xmath39 belongs to @xmath634 . since @xmath635 is of class @xmath45 , @xmath636 is hlder continuous with parameter @xmath41 . the sde ( [ e3.3 ] ) admits pathwise uniqueness because of theorem 3.5(ii ) of @xcite and weak existence , again through theorem  5.7 of @xcite . the yamada watanabe theorem ( see @xcite , corollary 3.23 , chapter  5 ) also implies strong existence for  ( [ e3.3 ] ) . the result follows from point ( i ) of corollary [ cmp ] . in this section , we will discuss the case where the martingale problem is equivalent to a stochastic differential equation to be specified . first , one would need to give a precise sense to the generalized drift @xmath637 , @xmath17 being a continuous function . we will introduce a property related to a general process @xmath201 . first , we consider the linear map @xmath638 defined on @xmath639 with values in  @xmath640 . [ deltr ] let @xmath641 be a topological f - space ( or , eventually , an inductive limit of @xmath182-spaces ) which is a topological linear subspace of @xmath642 and such that @xmath643 . we will say that @xmath201 has _ extended local time regularity with respect to @xmath641 _ if :    * @xmath644 admits a continuous extension to @xmath641 , which will still be denoted by the same symbol ; * @xmath645 exists for every @xmath646 and every @xmath647    [ rby ] the terminology related to _ local time _ is natural in this context . to illustrate this , we consider a general continuous process @xmath201 having a local time @xmath648 , a \in{\mathbb r})$ ] with respect to lebesgue measure , that is , fulfilling the density occupation identity @xmath649,\ ] ] for every positive borel function @xmath650 . @xmath201 trivially has extended local time regularity , at least with respect to @xmath651 . let @xmath326 . suppose for a moment that @xmath652 is a semimartingale in @xmath653 , as is the case , for instance , if @xmath201 is a classical brownian motion . in that case , one would have @xmath654 clearly , the rightmost integral can be extended continuously in probability to any @xmath655 , which implies that @xmath201 also has extended local time regularity related to @xmath656 . we remark that @xcite gives general conditions on semimartingales @xmath201 under which @xmath657 is a good integrator , even if @xmath652 is not necessarily a semimartingale in @xmath653 . [ dsol ] let @xmath658 a filtered probability space , @xmath222 a classical @xmath223-brownian motion and @xmath659 an @xmath660-measurable random variable . a process @xmath201 will be called a @xmath641-_solution _ of the sde @xmath661 if :    * @xmath201 has the extended local time regularity with respect to @xmath641 ; * @xmath662 ; * @xmath201 is a finite quadratic variation process .    [ rsol ] suppose that @xmath663 . if @xmath664 , then a @xmath665-solution is also a @xmath641-solution . the previous definition is also new in the classical case , that is , when @xmath16 is a continuous function . a @xmath641-solution with @xmath666 corresponds to a solution to the sde in the classical sense . on the other hand , a @xmath641-solution with @xmath641 strictly including @xmath45 is a solution whose local time has a certain additional regularity .    even in this generalized framework , it is possible to introduce the notions of _ strong @xmath641-existence _ , _ weak @xmath641-existence _ , _ pathwise @xmath641-uniqueness _ and _ @xmath641-uniqueness in law_. this can be done similarly as in definition [ d115 ] according to whether or not the filtered probability space with the classical brownian motion is fixed a priori . [ l3.9 ] we suppose that technical assumption is satisfied . if @xmath201 is a solution to a martingale problem related to a pde operator @xmath44 , then it has extended local time regularity with respect to @xmath667 . let @xmath668 . since @xmath201 solves the martingale problem with respect to @xmath44 , setting @xmath519 , it follows that @xmath669 continuity of @xmath36 on @xmath396 implies that @xmath644 can be extended to @xmath396 . now , let @xmath670 and @xmath671 . since @xmath230 equals a local martingale plus @xmath672 , it remains to show that @xmath673 exists for any @xmath674 . integrating by parts , the previous integral ( [ e3.8 ] ) equals @xmath675.\ ] ] remark  [ r1.2](b ) , ( f ) shows that the rightmost term member is well defined . [ l3.10 ] let @xmath201 be a process having extended local time regularity with respect to some @xmath182-space ( or inductive limit ) @xmath641 . suppose that for fixed @xmath214 , the application @xmath676 is continuous from @xmath641 to @xmath641 . then for every @xmath674 and every @xmath677 , we have @xmath678 where @xmath679    the banach  steinhaus - type theorem [ tbs ] implies that for every , @xmath680 is continuous from @xmath641 to @xmath196 . in fact , expression ( [ e3.11 ] ) is the u.c.p . limit of @xmath681 note that @xmath157 is a continuous bilinear map from @xmath682 to @xmath641 . since @xmath683 is continuous , the mapping @xmath684 is also continuous from @xmath641 to @xmath193 . in order to conclude the proof , we need to check identity ( [ e3.9 ] ) for @xmath326 . in that case , since @xmath685 both sides of ( [ e3.9 ] ) equal @xmath686    we will now explore the relation between the martingale problem associated with @xmath44 and the stochastic differential equations with distributional drift .    [ p3.11 ] let @xmath611 . suppose that @xmath44 fulfills technical assumption  . let @xmath658 be a filtered probability space fulfilling the usual conditions and let @xmath222 be a classical @xmath231-brownian motion . if @xmath201 solves the sharp martingale problem with respect to @xmath44 with initial condition  @xmath566 , then @xmath201 is a @xmath396-solution to the stochastic differential equation @xmath687 \\[-8pt ] \nonumber x_0 & = & x_0.\end{aligned}\ ] ]    [ r2.21 ] in particular , if @xmath44 is close to divergence type , as in example  [ 4ex](ii ) , then @xmath201 is a @xmath282-solution to the previous equation with @xmath688 . let @xmath201 be a solution to the martingale problem related to @xmath44 . we know , by lemma [ l3.9 ] , that @xmath201 has extended local time regularity with respect to @xmath641 . on the other hand , by remark  [ r3.4 ] , @xmath201 is a finite quadratic variation process . it remains to show that @xmath689 let @xmath326 and set @xmath690 . by definition of a sharp martingale problem , we have    @xmath691    according to remark  [ r2.21a](i ) concerning the continuity of the map @xmath692 , previous expression can be extended to any @xmath408 .    by remark  [ r2.21a](ii ) , @xmath693 and @xmath694 . replacing this in ( [ e3.13 ] ) , we obtain @xmath695 since @xmath696 , the proof is complete . [ c3.11 ] let @xmath611 . suppose that @xmath44 fulfills technical assumption  . if @xmath630 [ resp . @xmath560 ] admits weak ( resp . , strong ) existence , then the sde also admits weak ( resp . , strong ) existence . the statement concerning strong solutions is obvious . concerning weak solutions , let us admit the existence of a filtered probability space , where there is a solution to the martingale problem with respect to @xmath44 with initial condition @xmath566 . then according to remark  [ r3.5 ] , this solution is also a solution to a sharp martingale problem and the result follows .    if @xmath201 is some @xmath641-solution to ( [ e3.12 ] ) , is it a solution to the ( sharp ) martingale problem related to some operator @xmath44 ? this is a delicate question . in the following proposition , we only provide the converse of proposition  [ p3.11 ] as a partial answer . [ c3.12 ] suppose that the pde operator @xmath44 fulfills technical assumption . let @xmath658 be a filtered probability space fulfilling the usual conditions and let @xmath222 be a classical @xmath231-brownian motion . let @xmath201 be a progressively measurable process . @xmath201 solves the sharp martingale problem related to @xmath44 with respect to some initial condition @xmath566 if and only if it is a @xmath396-solution to the stochastic differential equation @xmath698 \\[-8pt ] \nonumber   x_0 & = & x_0 .\end{aligned}\ ] ]    [ cswequ ] let @xmath611 . suppose that @xmath44 fulfills technical assumption . then weak existence and uniqueness in law ( resp . , strong existence and pathwise uniqueness ) hold for equation if and only if the same holds for  @xmath630 [ resp . @xmath560 ] . proof of proposition  [ c3.12 ] suppose that @xmath201 is a @xmath396-solution to  ( [ e3.15 ] ) . then it is a finite quadratic variation process . let @xmath699 . since @xmath201 solves  ( [ e3.12 ] ) and @xmath700 always exists by the classical it formula [ see remark  [ r1.2](e ) of section  [ s1 ] ] , we know that @xmath701 also exists and is equal to @xmath702 therefore , this it formula says that @xmath703 holds .    by lemma [ l3.10 ] , the linearity of mapping @xmath644 and ( [ e213a ] ) , we obtain @xmath704    this shows that @xmath705 for every @xmath706 . in reality , it is possible to show the previous equality for any @xmath215 . in fact , the left - hand side extends continuously to @xmath233 and even to @xmath45 . the right - hand side is also allowed to be extended to @xmath233 for the following reason . for @xmath707 , let @xmath111 be a sequence of functions in @xmath708 converging to @xmath52 when @xmath709 , according to the @xmath233 topology . in particular , the convergence also holds in @xmath431 . since @xmath370 is continuous with respect to the @xmath431 topology with values in @xmath396 , we have @xmath710 in @xmath396 . finally , @xmath711 u.c.p . because of the extended local time regularity with respect to @xmath396 . we will , in fact , use the validity of ( [ e3.14 ] ) for @xmath215 with @xmath433 and @xmath460 and @xmath712 according to technical assumption [ a0](iv ) , we have @xmath713 . therefore , ( [ e3.14 ] ) gives @xmath714 again using extended local time regularity with respect to @xmath715 and the continuity of @xmath411 , we can state the validity of the previous expression for each @xmath408 with @xmath716 , in particular , for @xmath326 with @xmath716 . but in this case , for any @xmath717 with @xmath433 and @xmath718 , we obtain @xmath719 this shows the validity of the identity in definition [ dsmp ] for @xmath717 and that @xmath720 and @xmath721 . if @xmath722 , we replace @xmath52 by @xmath723 in the previous identity and use the fact that @xmath724 for any @xmath725 . it follows that @xmath201 fulfills a sharp martingale problem with respect to  @xmath44 . this shows the reversed sense of the statement . the direct implication was proven in proposition  [ p3.11 ] . [ ceqstrong ] we suppose that @xmath492 and @xmath493 , or @xmath494 and @xmath495 , with conditions  . we set @xmath726 .    then equation ( [ e3.15 ] ) admits @xmath396-strong existence and pathwise uniqueness . the result follows from corollaries [ cswequ ] and [ c3.3ss ] . in this section , we want to discuss the related parabolic cauchy problem with final condition , which is associated with our stochastic differential equations with distributional drift . we will adopt the same assumptions and conventions as in section  [ s4 ] . we consider the formal operator @xmath50 , where @xmath44 will hereafter act on the second variable . [ d21 ] let @xmath49 be an element of @xmath727 \times{\mathbb r } ) $ ] and let @xmath728 . a function @xmath729 \times{\mathbb r})$ ] will be said to be a _ @xmath19-generalized solution _ to @xmath730 \\[-8pt ] \nonumber u(t,\cdot ) & = & u^0,\end{aligned}\ ] ] if the following are satisfied :    for any sequence @xmath731 in @xmath732 \times{\mathbb r})$ ] converging to @xmath49 in a bounded way,=1    for any sequence @xmath733 in @xmath734 converging in a bounded way to @xmath735 ,    such there are classical solutions @xmath81 in @xmath736\times{\mathbb r})$ ] of class@xmath737 to @xmath738 , @xmath739 ,    then @xmath81 converges in a bounded way to @xmath12 . [ r21 ] ( a ) @xmath12 is said to solve @xmath740 if there exists @xmath741 such that ( [ f25 ] ) holds . the previous definition depends in principle on the mollifier , but it could be easily adapted so as not to depend on it . the regularized problem admits a solution : if @xmath742 and @xmath743 \times{\mathbb r})$ ] , then there is a classical solution @xmath744 in @xmath745\times{\mathbb r})$ ] of @xmath746 for this , it suffices to apply theorem of  . we now state a result concerning the case when the operator @xmath44 is classical . even if the next proposition could be stated when the drift @xmath16 is a continuous function , we will suppose it to be zero . in fact , it will later be applied to @xmath747 . [ p39 ] we suppose that @xmath376 . let @xmath748\times{\mathbb r } ) , n \in{\mathbb n},$ ] such that @xmath749 , @xmath750 in a bounded way on @xmath71 and @xmath53\times{\mathbb r}$ ] . let @xmath39 be a strictly positive real continuous function . suppose that there exist @xmath751\times { \mathbb r } ) $ ] such that @xmath752 then @xmath81 will converge to @xmath753\times{\mathbb r})$ ] in a bounded way , where the function @xmath12 is defined by @xmath754 where @xmath755 is the unique solution ( in law ) to @xmath756 and where @xmath222 is a classical brownian motion on some suitable filtered probability space . [ r310 ] usual it calculus implies that @xmath757 where @xmath758 is the unique solution in law to the problem @xmath759    theorem ( chapter   of ) affirms that it is possible to construct a solution ( unique in law ) @xmath760 to the sde ( [ f310quater ] ) resp . , @xmath761 to . suppose that @xmath44 is a classical pde operator . let @xmath762 be bounded and continuous on @xmath53\times{\mathbb r}$ ] . again , it calculus shows that @xmath12 can be represented by ( [ f311ter ] ) and ( [ f310quater ] ) . in particular , a classical solution @xmath12 to @xmath763 is also a generalized solution . proof of proposition  [ p39 ] we fix @xmath584 $ ] , @xmath764 . using the engelbert schmidt construction ( see , e.g. , the proof of theorem 5.4 , chapter  5 and 5.7 of  @xcite ) , it is possible to construct a solution @xmath765 of the sde on some fixed probability space which solves ( [ f310quater ] ) with respect to some classical wiener process  @xmath222 . we set @xmath596 for simplicity . the procedure is as follows . we fix a standard brownian motion @xmath766 on some fixed probability space one set @xmath767 @xmath768 is a.s . a homeomorphism on @xmath54 and we define @xmath209 as the inverse of @xmath768 . a solution @xmath189 will be then given by @xmath769 ; in fact , it is possible to show that the quadratic variation of the local martingale @xmath189 is @xmath770 the brownian motion @xmath602 is constructed a posteriori and is adapted to the natural filtration of @xmath189 by setting @xmath771    so , on the same probability space , we can set @xmath772 , @xmath773 being the inverse of @xmath774 , where @xmath775 . consequently , on the same probability space , we construct @xmath776 , where @xmath773 is the inverse of @xmath774 and @xmath775 . @xmath777 solves equation  ( [ f310quinque ] ) with respect to a brownian motion depending on  @xmath243 .    by construction , the family @xmath778 converges a.s . to @xmath779 . using lebesgue dominated convergence theorems and the bounded convergence of @xmath780 and @xmath781 , we can take the limit when @xmath709 in expression ( [ f311 ] ) and obtain the desired result . [ r311a ] in particular , the corresponding laws of random variables @xmath782 are tight . again , we will adopt the same conventions as in section  [ s4 ] . we set @xmath783 . @xmath351 is the classical operator defined at ( [ e2.12 ] ) . let us consider @xmath784 as a formal operator . [ c311 ] let @xmath785 \times{\mathbb r})$ ] , @xmath786 . there is a generalized solution @xmath12 to @xmath787 , @xmath788 . this solution is unique and is given by     we now return to the original pde operator @xmath789 with distributional drift . we again denote by @xmath296 the same application defined in section  [ s5 ] and discuss existence and uniqueness of @xmath19-generalized solutions of related parabolic cauchy problems . a useful consequence of proposition  [ p39 ] is the following . [ t313 ] for @xmath790 \times{\mathbb r})$ ] or @xmath77 , we again set @xmath791 according to the conventions of section  . again , we consider @xmath792 as a formal operator . let @xmath793 \times{\mathbb r})$ ] , @xmath794    there is a unique solution @xmath753 \times{\mathbb r})$ ] to @xmath795 \\[-8pt ] \nonumber u(t,\cdot ) & = & u^0.\end{aligned}\ ] ] moreover , @xmath796 solves @xmath797 \\[-8pt ] \nonumber \tilde{u } ( t,\cdot ) & = & \tilde{u}^0.\end{aligned}\ ] ]    in accordance with section  [ s4 ] , let @xmath798 be an approximating sequence which is related to @xmath799 . let us consider the pde operators @xmath800 defined at ( [ e2.2 ] ) . let @xmath801 be a sequence in @xmath802 such that @xmath803 , @xmath804 in a bounded way and for which there are classical solutions @xmath744 of @xmath805 we recall that those sequences always exist because of remark  [ r21](c ) . we set @xmath806 by lemma @xmath807 , we have @xmath808 where @xmath809 by proposition  [ p39 ] , and corollary [ c311 ] , @xmath810 in a bounded way , where @xmath811 this concludes the proof of the proposition . we now discuss how @xmath19-generalized solutions are transformed under the action of the function @xmath375 introduced at ( [ efk ] ) . a similar result to lemma [ t216 ] for the elliptic case is the following . [ p313 ] for @xmath790 \times{\mathbb r})$ ] or @xmath77 , we set @xmath812 . we set @xmath813 and consider the formal operator @xmath814 informally , we can write @xmath815    let @xmath793 \times{\mathbb r})$ ] , @xmath794    let @xmath12 be the unique @xmath19-generalized solution in @xmath816 \times{\mathbb r})$ ] to @xmath817 \\[-8pt ] \nonumber u(t,\cdot ) & = & u^0.\end{aligned}\ ] ] then @xmath818 solves @xmath819    let @xmath11 be the unique solution to @xmath820 which exists because of theorem [ t313 ] , taking @xmath821 . we define @xmath822 such that @xmath823 again , ( [ nonexplo ] ) implies that @xmath252 is bijective on @xmath71 . this case corresponds to example  ( a ) in remark  [ r2.6 ] with @xmath824 . we set @xmath825 . again , by theorem [ t313 ] , we have @xmath826 where @xmath827 and @xmath828 since @xmath829 this yields @xmath830 on the other hand , @xmath831 since @xmath832 we can therefore conclude that @xmath833 . since problem ( [ f316a ] ) has a unique solution , @xmath834 , where @xmath12 solves ( [ f316 ] ) and @xmath835 finally , @xmath836    [ probrep ] the unique @xmath19-generalized solution to admits a probabilistic representation in the sense that @xmath837 where @xmath545 is the solution to the martingale problem related to @xmath44 at time @xmath541 and point @xmath275 . the result follows from theorem [ t313 ] , corollary [ c311 ] and proposition  [ p3.3 ] , which collectively imply the following . if @xmath201 is a solution to the martingale problem related to @xmath44 at point @xmath275 at time @xmath541 , then @xmath592 solves the stochastic differential equation ( [ e3.3 ] ) with initial condition @xmath838 at time @xmath541 . we now discuss the existence of a density law for the solutions @xmath545 of the martingale problem related to @xmath44 . first , we suppose that @xmath44 is an operator in divergence form with @xmath839 and that there are positive constants such that @xmath840 . we will say , in this case , that @xmath44 _ has the aronson form_. this terminology refers to the fundamental paper @xcite concerning exponential estimates of fundamental solutions of nondegenerate parabolic equations . we begin with some properties ( partly classical ) stated in @xcite . we observe that point ( ix ) is slightly modified with respect to @xcite , but this new configuration can be immediately deduced from the proof in @xcite . this preparatory work will be applied to the operator @xmath621 introduced in ( [ e2.21 ] ) .    [ l5.1 ] we suppose that @xmath841 . let @xmath842 , @xmath843 , be smooth functions such that @xmath844 and @xmath845 in @xmath282 , as at the beginning of section  [ s4 ] . we set @xmath846 . there exists a family of probability measures @xmath847 [ resp . , @xmath848 enjoying the following properties :    @xmath849 , @xmath850 ;    ( aronson estimates ) there exists @xmath851 , depending only on constants  @xmath852 , @xmath853 , with @xmath854    we have @xmath855 and @xmath856 where @xmath857 ( resp . , @xmath858 ) is called the related to the previous parabolic linear equation ;    we have @xmath859    the map @xmath860 is continuous from @xmath8610,\infty [ \times \mathbb{r}^2 $ ] to @xmath862 ;    the @xmath863 are smooth on @xmath8610,\infty [ \times\mathbb{r}^2 $ ] ;    we have @xmath864 uniformly on each compact subset of @xmath8610,\infty[\times\mathbb{r}^2 $ ] ;    @xmath865 holds for every @xmath866 and every @xmath867 ;    @xmath868    the previous lemma allows us to establish the following . [ tdistr ] let @xmath869 be the solution to the martingale problem related to @xmath44 at time @xmath541 and point @xmath275 . suppose that @xmath44 to be of divergence type , having the aronson form . then there is fundamental solution @xmath870 of @xmath871 with the following properties :    letting @xmath785 \times{\mathbb r})$ ] , @xmath872 , the @xmath19-generalized solution @xmath12 to @xmath873 , @xmath788 , is given by @xmath874    the law of @xmath875 has @xmath876 as density with respect to lebesgue measure . let @xmath877 be the fundamental solution corresponding to the parabolic equation associated with @xmath878 , as introduced in section  [ s4 ] . we observe that @xmath879 converges in a bounded way to @xmath880 . we define @xmath881 points ( vi ) and ( ii ) of lemma [ l5.1 ] imply that functions @xmath744 belong to@xmath882 , so they are classical solutions to @xmath883 according to points ( ii ) and ( vii ) of the same lemma , one can prove that @xmath744 converges in a bounded way to @xmath12 defined by ( [ f312terd ] ) . in fact , the coefficients @xmath884 are lower and upper bounded with a common constant , related to @xmath852 and @xmath853 . therefore , this @xmath12 is the @xmath19-generalized solution of the cauchy problem being considered , which is known to exist . by uniqueness , point  ( i ) is established . setting @xmath885 , point ( i ) implies that @xmath886 is the @xmath19-generalized solution to @xmath887 with @xmath888 . by proposition  [ probrep ] , in particular , using the probabilistic representation , we get@xmath889 . [ rdistr ] if @xmath44 is in the divergence form , as before , then @xmath890 such that there exists @xmath891 with @xmath892 . this is a consequence of lemma [ l28bis ] and the fact that @xmath893 . hereafter , we will consider a general pde operator @xmath44 with distributional drift , as in section  [ s4 ] , for which the assumption  ( [ aronson ] ) below holds . we observe that the pde operator in divergence form of the type @xmath895 , where @xmath896 , has the aronson form , so the previous theorem can be applied . [ tdistr1 ] let @xmath545 be the solution to the martingale problem related to @xmath44 at time @xmath541 and point @xmath275 . suppose that @xmath44 fulfills assumption  . then there exists a kernel @xmath897 such that :    the law of @xmath898 has @xmath899 as density with respect to lebesgue measure for each @xmath900s , t]$ ] ;    letting @xmath785 \times{\mathbb r})$ ] , @xmath901 , the @xmath902-generalized solution @xmath12 to @xmath873 , @xmath788 , is given by @xmath903    \(i ) proposition  [ pmardiv ] says that @xmath904 solves the martingale problem with respect to @xmath621 . let @xmath905 be the fundamental solution associated with the parabolic pde @xmath906 . the first point then follows from the next observation . [ rcv ] by means of a change of variable , it is easy to see that the density law of @xmath898 equals @xmath907    \(ii ) this is a consequence of point ( i ) , fubini s theorem and proposition  [ probrep ] .    at this point , we need a lemma which extends to the kernel @xmath908 the integrability property of the kernel @xmath909 stated in ( [ rdistr ] ) concerning the divergence case . [ ddistr1 ] let @xmath908 be the kernel introduced in theorem  . then :    it is continuous in all variables @xmath9100,t[\ , \times\ , { \mathbb r}^2 $ ] ;    it fulfills aronson estimates ;    @xmath911 . we recall , by remark  [ rcv ] , that @xmath912 where @xmath913 is the fundamental solution associated with the operator @xmath914 , @xmath915 . this , and point ( v ) of lemma [ l5.1 ] , directly imply the validity of the first point . taking into account assumption  , aronson estimates for @xmath916 and the fact that @xmath917 result ( ii ) follows easily .    with the same conventions as before , we have @xmath918    so , for @xmath764 , @xmath919    \(iii ) follows after integration with respect to @xmath24 and because of lemma  [ l5.1](ix ) . [ pdistr1 ] let @xmath785 \times{\mathbb r } ) \cap l^1([0,t ] \times { \mathbb r})$ ] , @xmath920 . let @xmath921 \times{\mathbb r}\rightarrow{\mathbb r}$ ] be the @xmath19-generalized solution to @xmath763,@xmath922 . then :    @xmath923 ;    @xmath924 is absolutely continuous , @xmath925 and in particular , for a.e . @xmath926 $ ] , @xmath927 is square integrable . [ rdistr1 ] previous assumptions imply that @xmath125 and @xmath650 are also square integrable . proof of proposition  [ pdistr1 ] we recall the expression given in theorem [ tdistr1 ] , @xmath928 using lemma [ ddistr1 ] and classical integration theorems , we have @xmath929 \\[-8pt ] \nonumber & & { } + \int_t^t dr \int_{\mathbb r } \,ds\ , g(s , x_1)\ , \partial_x p_{t - s } ( x , x_1)\,dx_1 .\end{aligned}\ ] ] using jensen s inequality , we have @xmath930 therefore , @xmath931 using aronson estimates , this quantity is bounded by @xmath932 where @xmath79 is the gaussian @xmath933 density . this is clearly equal to @xmath934 this establishes point . concerning point ( b ) , in order to simplify the framework we will suppose that @xmath885 . expression ( [ f312tera ] ) implies that @xmath935 jensen s inequality implies that @xmath936 integrating with respect to @xmath24 gives @xmath937 this quantity is finite due to lemma . as in the previous section , we will adopt assumption  . at this point , we wish to investigate the link between @xmath19-generalized solutions and the notion of spde s weak solutions for a corresponding cauchy problem . we will adopt the same conventions as in section  [ s4 ] . in this section , we will suppose that coefficients @xmath938 are realizations of stochastic processes indexed by  @xmath71 . let us consider the formal operator @xmath383 , where @xmath44 acts on the second variable . we consider the equation @xmath939 \\[-8pt ] \nonumber u(t,\cdot ) & = & u^0.\end{aligned}\ ] ]    the aim of this section is to show that a @xmath19-generalized solution to ( [ e3.9quater ] ) provides , when @xmath20 , a solution to the ( stochastic ) pde of the type ( [ ei-0 ] ) , as defined in the definition [ dei-1rig ] , that is , with the help of a symmetric integral via regularization , as defined in section  [ s3 ] . we denote by @xmath940 the linear space of @xmath941 real functions with compact support . the link between the spde ( [ ei-0 ] ) and ( [ ei-0dual ] ) is given in the following . [ pspdes ] let @xmath942 , @xmath943 , @xmath926 $ ] , @xmath764 be two continuous random fields a.s . in @xmath9440,t [ \times{\mathbb r } ) $ ] such that @xmath945 . @xmath11 is a solution to the spde if and only if @xmath11 is a solution to the spde . we observe that @xmath946 . the proof is elementary . the only point to check is the following : @xmath947 this follows by the definition of symmetric integral and the following , obvious , identity : @xmath948 for every @xmath949 . we continue with a lemma , still supposing @xmath39 to be general . [ lspdes ] let @xmath49 ( resp . , @xmath735 ) be a random field with parameter @xmath950 \times{\mathbb r}$ ] ( resp . , @xmath764 ) whose paths are bounded and continuous . let @xmath938 be continuous stochastic processes such that @xmath291 is defined a.s . and assumption   is satisfied . let @xmath12 be the random field which is a.s . the @xmath19-generalized solution to  ( [ e3.9quater ] ) . the following then holds : @xmath951    for every @xmath952 . we fix a realization @xmath13 . theorem [ tdistr1 ] says that the unique solution to equation ( [ e3.9quater ] ) is given by @xmath953 where @xmath954 is the density law of the solution to the martingale problem related to @xmath44 at point @xmath275 at time @xmath541 . proposition  [ pdistr1](b ) implies that @xmath955 exists and is integrable on @xmath8610,t [ \times{\mathbb r}$ ] . according to proposition  [ p313 ] , we know that @xmath956 is a @xmath19-generalized solution to @xmath957 \\[-8pt ] \nonumber \bar{u } ( t,\cdot ) & = & \bar u^0,\end{aligned}\ ] ] where @xmath958 on the other hand , @xmath959 can be represented via ( [ f312terd ] ) in theorem [ tdistr ] through fundamental solutions @xmath960 of @xmath961 since the previous equation holds in the schwarz distribution sense , by inspection , it is not difficult to show that @xmath959 is a solution ( in the sense of distributions ) to ( [ c0div ] ) , which means that we have the following : @xmath962 \\[-8pt ] \nonumber & & \qquad = \int_t^t ds \int_{\mathbb r}\alpha(z ) \bar\lambda(s , z)\end{aligned}\ ] ] for every test function @xmath952 , @xmath963 $ ] . we recall , in particular , that @xmath964 is in @xmath9650,t [ \times{\mathbb r})$ ] . we set @xmath966 expression ( [ c0div1 ] ) shows that @xmath967 in the sense of distributions . so for each @xmath926 $ ] , @xmath968 is of class @xmath45 .    for @xmath926 $ ] and @xmath764 , we set @xmath969 , @xmath970 . we recall that @xmath971 therefore , @xmath972 so that @xmath973 therefore , @xmath974 and so @xmath975 is of class @xmath45 . since @xmath976 , ( [ ediv1 ] ) gives @xmath977    consequently , @xmath978 we integrate the previous expression against a test function @xmath979 to obtain @xmath980    remark  [ r1.0 ] and integration by parts for the symmetric integral provided by remark  [ r1.1](c ) allow us to conclude the proof of the lemma .    finally , we are able to state the theorem concerning the existence of weak solutions for the spde . [ tspdes ] let @xmath49 ( resp . , @xmath735 ) be a random field with parameter in @xmath950 \times{\mathbb r}$ ] ( resp . , @xmath764 ) whose paths are bounded and continuous . we suppose that @xmath981 and that @xmath7 is a ( two - sided ) zero strong cubic variation process such that there are two finite and strictly positive random variables @xmath982 with @xmath983 a.s . let @xmath12 be the random field which is @xmath13 a.s . a @xmath19-generalized solution to   for @xmath984 . we set @xmath945 . then @xmath11 is a ( weak ) solution of the spde  . proposition  [ pspdes ] says that it will be enough to verify that @xmath985 for every test function @xmath25 and every @xmath926 $ ] .    after making the identification @xmath21 , the previous lemma [ lspdes ] says that @xmath986    since @xmath7 is a zero strong cubic variation process , proposition  [ pstacub ] implies that @xmath987 is also a zero strong cubic variation process . then the it chain rule from proposition  [ crule ] , applied with @xmath988 and remark  [ r1.0 ] say that the right - hand side of previous expression gives @xmath989 this concludes the proof . we would like to thank an anonymous referee and the editor for their careful reading and stimulating remarks . the first named author is grateful to dr . juliet ryan for her precious help in correcting several language mistakes . flandoli , f. , russo , f. and wolf , j. ( 2004 ) . some sdes with distributional drift . ii . lyons  zheng structure , it formula and semimartingale characterization . _ random oper . stochastic equations _ * 12 * 145184 . gradinaru , m. , russo , f. and vallois , p. ( 2003 ) . generalized covariations , local time and stratonovich it s formula for fractional brownian motion with hurst index @xmath990 . probab . _ * 31 * 17721820 .
a new class of random partial differential equations of parabolic type is considered , where the stochastic term consists of an irregular noisy drift , not necessarily gaussian , for which a suitable interpretation is provided . after freezing a realization of the drift ( stochastic process ) , 
 we study existence and uniqueness ( in some appropriate sense ) of the associated parabolic equation and a probabilistic interpretation is investigated .    and    .